The Diffiety Institute Preprint Series

Preprint DIPS–6/2004 June 22, 2004

Available via INTERNET: http://diffiety.ac.ru; http://diffiety.org

The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

Methods of geometry of differential equations in analysis of the integrable field theory models

by

Arthemy V. Kiselev

arXiv.org/nlin.SI/0406036

METHODS OF GEOMETRY OF DIFFERENTIAL EQUATIONS IN ANALYSIS OF THE INTEGRABLE

FIELD THEORY MODELS

ARTHEMY V. KISELEV

Abstract. In this paper, we investigate the algebraic and geometric properties of the hyperbolic Toda equations u_{xy} = exp(Ku) associated with nondegenerate symmetrizable matrices K. A hierarchy of analogs to the potential modified Korteweg–de Vries equation u_{t} = u_{xxx} + u_{x}^{3} is constructed, and its relation with the hierarchy for the Korteweg–de Vries equation T_{t} = T_{xxx}+T T_{x} is established. Group–theoretic structures for the dispersionless (2+1)- dimensional Toda equation u_{xy} = exp(−u_{zz}) are obtained. Geometric properties of the multi–component nonlinear Schrödinger equation type systems Ψ_{t} = iΨ_{xx} + if(|Ψ|) Ψ (multi–soliton complexes) are described.

Contents

Introduction 4 Part I. The Korteweg–de Vries type equations associated with

the Toda systems 28

Chapter 1. Conservation laws and the Noether symmetries of

the Toda equations 28

1. The Toda equation 29 2. The Noether symmetries of the Toda equation 40 3. Recursion operators for the Toda equation 44 Chapter 2. The Korteweg–de Vries hierarchies and the Toda

equations 47

4. Analogs of the potential modified Korteweg–de Vries equation 49

5. The Hamiltonian formalism for the Euler equations 61 6. Hyperbolic Euler equations 63 7. Properties of the Korteweg–de Vries hierarchies 70

1991 Mathematics Subject Classification. 35Q53, 37K05. Key words and phrases. The Toda equation, Korteweg–de Vries equation, nonlin-

ear Schrödinger equation, symmetries, conservation laws, Hamiltonian structures, Bäcklund transformations, zero–curvature representations.

This paper is submitted for publication in the Issue “Geometry of PDE” in: Fundamental’naya i Prikladnaya Matematika / Jour. of Math. Sci. (2004).

Partially supported by the scholarship of the Government of the Russian Federation, the INTAS grant YS 2001/2-33, and the Lecce University grant n. 650 CP/D.

Part II. Group–theoretic properties of the mathematical

physics equations: methods and applications 81

Chapter 3. Symmetries, solutions, and conservation laws for

differential equations 81

8. Nonlinear Schrödinger equation 81 9. The dispersionless Toda equation 84 Chapter 4. Bäcklund transformations and zero–curvature

representations 97

10. Bäcklund transformations and their deformations 97 11. Integrating Bäcklund transformations in nonlocal variables 104 12. Zero–curvature representations 111 Appendix A. Geometric methods of solving boundary–value

problems 116

Final remarks 126 References 129

The Toda equation ([93]) and the Toda type equations associated with the semisimple Lie algebras ([66]) play an important role in the models of modern conformal field theory. We recall that the Toda equations appear in gravity theory ([2, 11]), in the Yang–Mills field theory ([67]), in differential geometry ([29, 41]), and in the classification problems for nonlinear partial differential equations ([100]). Also, the Toda equations are related with the integrable dynamical systems ([22]), the Frobenius manifolds, and the associative algebra structures ([16]). The detailed study of the following systems that appear in the above–mentioned theories is reduced to the study of the Toda equations: these systems are the antiselfdual vacuum Einstein equations, the Yang–Mills equations, the Gauss–Mainardi–Codazzi equations for complex curves in the Kähler manifolds, the dynamical equations for the Laplace invariants of differential equations, the Korteweg–de Vries equation, and the WDVV (Witten–Dijkraaf–H. Verlinde–E. Verlinde) equation, respectively.

The algebraic approach towards the description of properties of the two–dimensional hyperbolic Toda equations

u_{xy} = exp(Ku)

SYMMETRIES AND CONSERVATION LAWS IN INTEGRABLE MODELS 3

was developed in the papers by A. N. Leznov and M. V. Saveliev ([66]), V. G. Drinfel’d and V. V. Sokolov ([22]), B. A. Dubrovin ([16]) et al. In these papers, the Toda equations are interpreted as the flat connection equations on semisimple complex Lie algebras (or the Kac– Moody algebras) such that K is the corresponding Cartan matrix. Then the Toda equations are said to be associated with the Lie algebras (respectively, the Kac–Moody algebras). These equations are exactly integrable ([66]). In the fundamental paper [22], the integrable Drinfel’d–Sokolov hierarchies were assigned to the Toda equations associated with the Kac–Moody algebras. The Drinfel’d–Sokolov equations are the bi–Hamiltonian Korteweg–de Vries type equations. Still, the Lie–algebraic approach does not provide the exhaustive description of geometric properties of the Toda equations themselves. Indeed, the structure of generators of the Noether symmetries Lie subalgebra for the Toda equation, the existence of recursion operators for its symmetry algebra, or the relation between conservation laws for the Toda equation and the Hamiltonian structures for the Korteweg–de Vries equation was not revealed. In particular, until recently it remained unnoticed that all these properties of the Toda equation are preserved for the general case of the Toda equations u_{xy} = exp(Ku) associated with nondegenerate symmetrizable matrices K, which are not assumed to be the Cartan matrices.

The homological methods developed by I. S.Krasil’shchik, V. V. Lychagin, A. M. Vinogradov ([10, 62, 63, 94]), and their scientific school are a powerful tool in the analysis of algebraic and geometric structures for differential equations. Owing to essential achievements of this theory, it was quite natural to apply the homological methods to the study of the Toda equation and related systems.

In this article, we carry out the detailed investigation of the geometric properties of the hyperbolic Toda equations. Also, we construct new Hamiltonian evolution systems associated with the Toda equations and establish their nontrivial links with other important classes of the mathematical physics equations such as the Korteweg–de Vries equation.

Introduction

The discrete one–dimensional Toda lattice ([93])

q̈_{n} = exp(q_{n−1} − q_{n}) − exp(q_{n} − q_{n+1}),  n ∈ ℤ, q_{n} = q_{n}(τ) (1)

is a model nonlinear integrable equation in mathematical physics. The equation

ε^{2} · q_{ττ} = exp(q(z − ε) − q(z)) − exp(q(z) − q(z + ε))  (2)

is a continuous analog of Eq. (1). Here ε ≥ 0 and q = q(τ, z). Equation (2) is then obtained by using the relation q_{n}(τ) = q(τ, nε). The equation

u_{ττ} = ±D_{z}^{2} ∘ exp(u), u = u(τ, z),

is the limit for the family of equations (2) as ε → +0. Here D_{z} is the total derivative with respect to z and u_{n}(τ) ≡ q_{n−1} − q_{n} = u(τ, nε). In the paper [16] Dubrovin proved that Eq. (2) is reduced to the nonlinear Schrödinger equation if ε = i. In this paper, we consider the multi– component analogs (multi–soliton complexes, see [1])

Ψ_{t} = iΨ_{xx} + if(|Ψ|) Ψ,  (3)

of the nonlinear Schrödinger equation. Here Ψ is the m-component vector, i = \sqrt{−1}, and f ∈ C^{∞}(ℝ).

The passage from one–dimensional equation (1) to the two–dimensional Toda equations

u_{xy} = exp(Ku)  (4)

was described in the paper [66]. By [66], the acceleration ∂^{2}/∂τ^{2} with respect to the time t ∈ ℝ is replaced by the d’Alambertian ∂^{2}/∂x∂y; we also assume (x, y) ∈ ℂ^{2}. The Toda fields u^{j}(x, y) are counted by the superscript j ∈ [1, r]. Suppose ⟨α⃗_{i}⟩ is the rood system of a semisimple complex Lie algebra 𝔤 of rank r and

K = ∥k_{ij} = \frac{2(α⃗_{i}, α⃗_{j})}{|α⃗_{j}|^{2}} ∥

is its Cartan matrix. In [66], equations (4) were supposed to be associated with the Cartan matrices by default. In the sequel, we extend the existing picture and analyse the properties of the Toda equations (4) such that K is a nondegenerate symmetrizable matrix (see precise definitions below).

Let the matrix K in Eq. (4) be the Cartan matrix of the Lie algebra of type A_{r}. Then we set

u^{j}(x, y) = u(x, y, z)|_{z=jε}

for each 1 ≤ j ≤ r. The continuous limit of Eq. (4) as r → ∞ and ε → +0 is the dispersionless Toda equation

u_{xy} = exp(−u_{zz}).  (5)

This equation appeares in many problems of mathematical physics, for example, in the gravity theory ([11], see also the paper [87] and references therein).

In this paper, we investigate algebraic and geometric properties of the Toda equations (4–5) and the nonlinear Schrödinger equation (3). Also, we describe the relation between the Toda equation (4) and the hierarchies of the Korteweg–de Vries equations (9) and (16) (see below). We apply modern cohomological methods and algorights in our study of the geometric properties of all these equations. By using the algebraic approach, we reject the idea of a decisive break–through and the inevitably immense calculations in coordinates. We operate with the notions of homological algebra in the category of infinite prolongations of differential equations instead.

The results exposed in this article are also found in the sequence of publications [44]–[51] and [46]–[60].

The paper is organized as follows. In the introduction, we fix notation, formulate necessary definitions, and give a brief list of the final results.

In Chapter 1, we describe important properties of the symmetry algebra as well as conservation laws for the Toda equations. In Sec. 1, we review some known properties of the hyperbolic Toda equations associated with the semisimple Lie algebras. Also, we define the Toda equations associated with nondegenerate symmetrizable matrices. In what follows, we study these equations that require less restrictions upon the matrix K. In Sec. 2 within Chapter 1, we obtain the Noether symmetries of the Lagrangian for the Toda equations. In Sec. 3, we construct a continuum of recursion operators for the Toda equation.

In Chapter 2, we construct the commutative hierarchy 𝔄 of analogs for the potential modified Kortweg–de Vries equation. This hierarchy is identified with a commutative subalgebra of the Noether symmetries algebra for the Toda equations. Also, some aspects of the Hamiltonian formalism for the hyperbolic Toda equations themselves are discussed. Then we relate the symmetry hierarchy 𝔄 with the higher Korteweg–de Vries equations.

In Chapter 3, we illustrate the methods of geometry of partial differential equations. Namely, we investigate the properties of the dispersionless Toda equation and the multi–component analog of the nonlinear Schrödinger equation, which is related with the former equation.

In Chapter 4, we consider Bäcklund transformations for the Toda equation associated with the algebra 𝔰𝔩_{2}(ℂ) (the Liouville equation) and related systems. Also, we obtain one–parametric families of these transformations. Examples of integrating Bäcklund transformations in nonlocal variables are given in Chapter 4, zero–curvature representations are pointed out, and the relations between all these structures are analysed.

In Appendix A, several geometrical methods of solving the boundary problems for equations of the mathematical physics are listed.

Basic definitions and notation. First let us introduce some notions from the geometry of partial differential equations. We follow [4, 5, 10, 62, 63, 94] and the articles [45, 47].

Differential equations and their symmetries. Let E = {F^{α}(x, u, p) = 0, 1 ≤ α ≤ r} be a partial differential equation of order k imposed by the smooth functions F^{α} on n independent variables x⃗ = ^{t}(x^{1}, …, x^{n}), r dependent variables u = ^{t}(u^{1}, …, u^{r}), and the derivatives

p = {p_{σ}^{j} | p_{σ}^{j} = ∂^{|σ|}u^{j}/∂(x^{1})^{i_{1}} … ∂(x^{n})^{i_{n}}, σ = {i_{1}, …, i_{n}}, |σ| = i_{1} + ⋯ + i_{n} ≤ k}.

Consider the trivial r-dimensional smooth fibre bundle

π: ℝ^{r} × ℝ^{n} → ℝ^{n}.

The space J^{k}(π) of kth jets of the bundle π is the union ⋃_{x} J_{x}^{k}, where J_{x}^{k} is the set of all equivalence classes [s]_{x}^{k} of sections s in the bundle π such that the sections s are tangent with order k at the point x ∈ ℝ^{n}. Define the sequence of smooth fibre bundles

π_{k+1,k}: J^{k+1}(π) → J^{k}(π)

by the formula

π_{k+1,k}([s]_{x}^{k+1}) = [s]_{x}^{k}.

We denote by π_{k} the smooth fibre bundle π_{k}: J^{k}(π) → ℝ^{n} defined by π_{k}([s]_{x}^{k}) = x. Also, by F_{k}(π) we denote the ring of smooth (C^{∞}) functions on J^{k}(π). Finally, we set F_{−∞}(π) = C^{∞}(ℝ^{n}).

We treat the variables x, u, and p as coordinates in the jet space J^{k}(π). By the Hadamars lemma ([38]), two sections s, s′ ∈ Γ(π) are equivalent in J_{x}^{k}(π) iff their partial derivatives at the point x coincide up to order k.

A submanifold

E = {F^{α} = 0} ⊂ J^{k}(π)

is a differential equation of order not greater than k imposed on r functions of n independent variables. An equation E ⊂ J^{k}(π) is regular if the mapping

π_{k}|_{E}: E → M = ℝ^{n}

is a surjection.

Definition 1. Consider the sections s of the fiber bundle π such that [s]_{x}^{k} = θ for some point θ ∈ J^{k}(π). Consider the tangent planes to the graphs Γ_{s}^{k} of the kth jets of s. The linear span of all these tangent planes is the Cartan plane C_{θ} = C_{θ}^{k} at the point θ. The union C of the mappings θ ↦ C_{θ} by all θ ∈ J^{k}(π) is the Cartan distribution.

Definition 2. The subset

E^{(l)} = {θ_{k+l} = [s]_{x}^{k+l}, s ∈ Γ(π) |

| j_{k}(s)(x) is tangent to E at the point θ_{k} = [s]_{x}^{k} with order ≥ l}

in J^{k+l}(π) is the lth prolongation of an equation E ⊂ J^{k}(π). The inverse limit

E^{∞} = proj lim_{l→∞} E^{(l)}

with respect to the projections

π_{l+1,l}: J^{l+1}(π) → J^{l}(π)

is called the infinite prolongation of the equation E. In what follows, we omit the superscript ∞ if it is clear from the context that the infinite prolongation E^{∞} but not the differential equation E is considered.

The infinite prolongation of the empty equation {0 = 0} ≃ J^{0}(π) is called the infinite jets space J^{∞}(π). The projections

π_{∞,k}: J^{∞}(π) → J^{k}(π)

and

π_{∞}: J^{∞}(π) → ℝ^{n}

are defined by the formulas

π_{∞,k}(θ_{∞}) = θ_{k}

and

π_{∞}(θ_{∞}) = x,

respectively, where

θ_{∞} = {x, θ_{k} ∈ J^{k}(π) | k ∈ ℕ} ∈ J^{∞}(π).

The algebra F(π) of smooth functions on J^{∞}(π) is defined by using the following procedure. We set

F(π) = ⋃_{k} F_{k}(π), k ∈ {−∞} ∪ ℕ.

The module Λ^{i}(π) of differential i-forms on J^{∞}(π) is defined by the relation

Λ^{i}(π) = ⨁_{k≥0} Λ^{i}(J^{k}(π)).

The restriction C_{E} of the Cartan distribution C onto the equation E^{∞} is an n-dimensional Frobenius distribution. This distribution defines the decomposition of the tangent space to E^{∞} into the horizontal and vertial subspaces. The horizontal component is generated by the restrictions of the total derivatives

D_{i} = ˆ \frac{∂}{∂x^{i}}

onto E^{∞}. These restrictions are denoted by D̄_{i}. The dual description of the Cartan distribution C_{E} in the language of differential forms is also useful. The de Rham differential on E^{∞} is representable as the restriction onto E^{∞} of the sum composed by the horizontal differential

d_{h} = ∑_{i=1}^{n} dx^{i} ⊗ D_{i}

(this is the lifting of the differential on the base of the fibre bundle π) and the Cartan differential

d_{C} = d − d_{h}.

Therefore the space Λ^{l}(E) of differential l-forms on the equation E^{∞} is the direct sum

Λ^{l}(E) = ⨁_{i+j=l} Λ̄^{i}(E) ⊗ C^{j}Λ(E)

of the horizontal i-forms Λ̄^{i}(E) and the Cartan j-forms C^{j}Λ(E). The Cartan forms ω_{σ}^{j} ≡ d_{C}(u_{σ}^{j}) constitute a basis in C^{1}Λ(E). Here u_{σ}^{j} are coordinates on E^{∞}.

The horizontal differential d̄_{h} generates the horizontal de Rham complex

0 → F(π) ⟶^{d_{h}} Λ̄^{1}(π) ⟶^{d_{h}} … ⟶^{d_{h}} Λ̄^{n}(π) → 0

of the space J^{∞}(π). The cohomologies of this complex are called the horizontal cohomologies and are denoted by H̄^{i}(π). We denote by H̄^{i}(E) the cohomologies of the restriction of this complex onto the equation E^{∞}. From the definition it follows that the equivalence classes [η] ∈ H̄^{n−1}(E) are conservation laws for the equation E.

Definition 3. (1) The evolutionary derivation is the operator

�_{φ} = ∑_{j,σ} D_{σ}(φ^{j}) ∂/∂p_{σ}^{j},

where φ^{j} ∈ C^{∞}(J^{k}(π)) for some k and D_{σ} is the composition of total derivatives D_{i} that corresponds to the multiindex σ. We denote by κ the F(π)-module Γ(π) ⊗_{F_{−∞}} F(π). Also, we denote

κ̂ = Hom_{F(π)}(κ, Λ̄^{n}(π)).

(2) Suppose Δ_{ψ} is the nonlinear differential operator determined by a function ψ ∈ C^{∞}(J^{k}(π)). The operator ℓ_{ψ} that acts by the rule

ℓ_{ψ}(φ) = �_{φ}(ψ)

is called the universal linearization operator of Δ_{ψ}. In coordinates, we have

ℓ_{ψ} = ∥∑_{σ} \frac{∂ψ^{i}}{∂u_{σ}^{j}} · D_{σ} · 1_{ij}∥ .

Any Lie field X that preserves the Cartan distribution C can be decomposed to the sum X = �_{φ} + Y, where Y ∈ C and �_{φ} is the evolutionary vector field. Any infinitesimal transformation of the space J^{0}(π) can be extended up to the Lie field. In coordinates, the lifting rules are the following. The field

X̂ = ∑_{i} a_{i} D_{i} + ∑_{i,j} �_{b_{j}−a_{i}p_{i}^{j}}

is assigned to the field

X^{0} = ∑_{i} a_{i} \frac{∂}{∂x^{i}} + ∑_{j} b_{j} \frac{∂}{∂u^{j}} .

An example is found in Eq. (35) on page 26.

Definition 4. A symmetry of an equation E^{∞} is a π-vertical vector field X such that X preserves the Cartan distribution C_{E} = ⋃_{θ∈E^{∞}} C_{θ}:

[X, C_{E}] ⊂ C_{E}.

Theorem 1 ([10]). Suppose E ⊂ J^{k}(π) is an equation

{F^{1} = 0, …, F^{r} = 0},

such that

π_{∞,0}(E^{∞}) = J^{0}(π).

Then the symmetry Lie algebra sym E^{∞} is isomorphic to the Lie algebra of solutions to the system of the determining equations

�_{φ}(F) = 0 on E

or, equivalently, of the equations

ℓ_{F} (φ) = 0 on E

owing to the definition of the linearization operator ℓ. Here φ ∈ κ|_{E^{∞}}. The bracket

{φ, ψ}_{E^{∞}} = (�_{φ}(ψ) − �_{ψ}(φ)) |_{E^{∞}}.

endowes the algebra sym E^{∞} of solutions φ with the Lie algebra structure.

In what follows, we identify the generating sections

φ = ^{t}(φ^{1}, …, φ^{r})

of symmetries

�_{φ} = ∑_{i,σ} D̄_{σ}(φ^{i}) · \frac{∂}{∂u_{σ}^{i}}

(we denote u_{σ}^{i} ≡ D_{σ}(u^{i})) of differential equations with the symmetries �_{φ} themselves (see [22, 49]). Roughly speaking, the component φ^{i} of a generating section φ measure the velocity u_{t}^{i} of the dependent variable u^{i} along the ”integral trajectories” of the field �_{φ}.

Definition 5. Suppose φ(u, …, u_{σ}) is a symmetry of a differential equation E, where σ is a multiindex and

u_{σ} = ∂^{|σ|}u/∂(x^{1})^{σ_{1}} … ∂(x^{n})^{σ_{n}}

is the derivative of the dependent variable u. Assume that the symmetry φ has the flow

A_{τ}: u(x, 0) ↦ u(x, τ).

This flow is defined on solutions of the equation u_{τ} = φ and maps the solutions s(x) = u(x, τ)|_{τ=0} of equation E to solutions of the same equation at points τ > 0. A solution s(x) of the equation E is called φ-invariant if s(x) is a stationary solution of the evolution equation

u_{τ} = φ(u, …, u_{σ}).

Therefore, the search of the φ-invariant solutions for a given equation E = {F = 0} is reduced to solving the system {F = 0, φ = 0}.

No we specify an important class of Hamiltonian equations within the set of evolution equations u_{t} = f(t, x , u, p). Here we formulate the definitions of the Poisson bracket and the Hamiltonian operators class (see [10, 26, 42] for further detailes).

Definition 6. Let A ∈ CDiff(κ̂(π), κ(π)) be an (r×r)-matrix operator in total derivatives:

A = ∥A^{ij}∥, A^{ij} = A_{σ}^{ij} · D_{σ},

and let L_{1}, L_{2} ∈ H̄^{n}(π) be two Lagrangians. By definition, set the Poisson bracket (the variational bracket) on H̄^{n}(π) by the formula

\{L_{1}, L_{2}\}_{A} = ⟨E(L_{1}), A(E(L_{2}))⟩ = [∑_{i,j} \frac{δL_{1}}{δu^{i}} · A^{ij} (\frac{δL_{2}}{δu^{j}} ) dx ], (6)

where ⟨·, ·⟩ is the natural coupling κ(π)×κ̂(π) → H̄^{n}(π) and [·] denotes the equivalence class of differential forms.

Definition 7. Suppose the operator A is subject to the assumptions of the previous definition. Then the operator A is called Hamiltonian if the Poisson bracket defined in Eq. (6) endows H̄^{n}(π) with the Lie algebra structure over ℝ such that the followin relations hold

{L_{1}, L_{2}}_{A} + {L_{2}, L_{1}}_{A} = 0,  (7a)
{{L_{1}, L_{2}}_{A}, L_{3}}_{A} + {{L_{2}, L_{3}}_{A}, L_{1}}_{A} + {{L_{3}, L_{1}}_{A}, L_{2}}_{A} = 0. (7b)

The bracket {·, ·}_{A} is the Hamiltonian structure.

Two Hamiltionian operators A_{1}, A_{2} are compatible if λA_{1} + μA_{2} is a Hamiltonian operator again for any λ, μ ∈ ℝ.

Condition (7a) is satisfied iff A+A^{∗} = 0. Refer [10, 63, 62, 42] for the suitable criteria^{1} which check whether Jacobi’s identity (7b) holds for a given operator A. For example, any skew–symmetric C-differential operator A with constant coefficients is Hamiltonian.

Definition 8. The evolution equation

u_{t} = A(E_{u}(H))  (8)

is a Hamiltonian evolution equation assigned to the Hamiltonian H ∈ H̄^{n}(π) by the Hamiltonian operator A.

Example 1 ([70]). The Korteweg–de Vries equation

T_{t} = −β T_{xxx} + 3T T_{x},  β = const,  (9)

is hamiltonian with respect to the pair

B̂_{1} = D_{x},  B̂_{2} = −β D_{x}^{3} + 2T · D_{x} + T_{x}  (10)

of the compatible Hamiltonian operators. Indeed, we have

T_{t} = B̂_{1} ∘ E_{T} ((\frac{1}{2}βT_{x}^{2} + \frac{1}{2}T^{3}) dx) = B̂_{2} ∘ E_{T} (\frac{1}{2}T^{2} dx).

Conservation laws. In this subsection, we formulate important definitions and statements concerning conservation laws for differential equations. Also, we recall the relation between symmetries and conservation laws for the Euler equations.

Definition 9. A conservation law

[η] ∈ H̄^{n−1}(E) ≡ {ω ∈ Λ̄^{n−1}(E) | d̄_{h}(ω) = 0}/

/ {ω ∈ Λ̄^{n−1}(E) | ω = d̄_{h}γ, γ ∈ Λ̄^{n−2}(E)}

for an equation E is an equivalence class of horizontal (n − 1)-forms η ∈ Λ̄^{n−1}(E) closed on E,

d̄_{h}η = ∇(F) dx .

Here

d̄_{h} = ∑_{i=1}^{n} dx^{i} ⊗ D̄_{i}

is the restriction of the horizontal differential d_{h} on E, D_{i} is the total derivative with respect to x^{i}, ∇ is an operator in total derivatives, and the equivalence is the factorization by exact forms d̄_{h}γ, γ ∈ Λ̄^{n−2}(E). Representatives η of the equivalence classes [η] ∈ H̄^{n−1}(E) are called conserved currents for the equation E.

1The of notion of the Poisson bi-vectors is a useful intrument in the Hamiltonian operators theory (see [26, 42]). For example, the first Hamiltonian structure for the Korteweg–de Vries equation (9) is 1 ∧ D_{x} and the second structure is 1 ∧ B̂_{2}, see Eq. (10). A bi-vector A is Poisson if and only if A satisfies the equation [[A, A]] = 0, where [[·, ·]] is the Schouten bracket. A pait of the Poisson bi-vectors A_{1} and A_{2} is compatible if [[A_{1}, A_{2}]] = 0. In the papers [52, 53], the equation [[A, A]] = 0 was considered in a more general situation such that A is not necessarily a bi-vector.

Example 2 ([50]). The horizontal 2-form

η = u_{xz} exp(−u_{zz}) dx ∧ dy + (\frac{1}{2}u_{xz}^{2} − u_{xx}) dx ∧ dz

is a conserved current for the dispersionless Toda equation

u_{xy} = exp(−u_{zz}).

Indeed, the relation

D̄_{y}(\frac{1}{2}u_{xz}^{2} − u_{xx}) = D̄_{z}(u_{xz} exp(−u_{zz}))  (11)

holds on that equation.

Definition 10. A regular equation E = {F = 0} is called ℓ-normal if the condition

∇ ∘ ℓ̄_{F} = 0

implies ∇ = 0.

Proposition 2 ([10]). Let n be the number of independent variables x^{1}, …, x^{n} and let E be a regular equation. Consider a coordinate neighbouhood Ω(θ^{∞}) ⊂ E^{∞} of a point θ^{∞} ∈ E^{∞} and suppose that there is a set {v}of the internal coordinates v on Ω such that the total derivatives D_{i}(v) can be expressed in terms of these coordinates {v} for any i, 1 ≤ i < n. Then the equation E is ℓ-normal.

If E is an ℓ-normal equation, then the compatibility complex ([10, 63]) for the equation E has length 2, and the equation satisfies the assumptions of the ’2-line’ theorem.

Remark 1. The Maxwell, the Yang–Mills, and the Einstein equations are not ℓ-normal since there is a nontrivial dependence between the equations that is provided by the gauge symmetry pseudogroup.

Now we need a test that checks whether a given equation is ℓ-normal. We have

Example 3 ([10, 63]). The evolution equations are ℓ-normal.

Conservation laws [η] for the ℓ-normal equations are described by their generating sections ψ⃗ ≡ ∇^{∗}(1) ∈ κ̂. We see that

d_{h}η = ⟨∇(F), 1⟩ = ⟨F, ∇^{∗}(1)⟩ + d_{h}γ.

Then the coupling

⟨∇^{∗}(1), F ⟩ = d_{h}(η − γ)  (12)

is an exact horizontal n-form. Suppose η is trivial and therefore ∇ = 0. Then, obviously, then ψ⃗ = 0.

Lemma 3 ([94]). Let E = {F = 0} be an ℓ-normal equation and assume that H^{n}(E) ⊂ H̄^{n}(E) (e.g., H^{n}(E) = 0). Suppose that the generating section ψ⃗ of a current η is zero. Then the conserved current η is trivial.

In what follows, we shall use the following remarkable theorem many times.

Theorem 4 ([94]). Let E = {F = 0} be an ℓ-normal equation in the fibre bundle π: ℝ^{m} × ℝ^{n} → ℝ^{n}. Then the generating sections ψ⃗ of conservation laws [η] satisfy the equation

ℓ̄_{F}^{∗} (ψ⃗_{η}) = 0,  (13)

where ℓ_{F}^{∗} is the operator formally adjoint to ℓ_{F} and ℓ̄_{F}^{∗} is the restriction of ℓ_{F}^{∗} onto the equation E.

Proof. Apply the Euler operator E to both sides of Eq. (12). Thence we obtain

0 = E(⟨ψ, F ⟩) = ℓ_{⟨ψ,F ⟩}^{∗}(1) = ℓ_{F}^{∗} (ψ) + ℓ_{ψ}^{∗}(F) = 0  (14)

by the Leibnitz rule. Now restrict Eq. (14) onto the prolongation of F = 0 and obtain the determining equation (13) imposed on the generating sections and satisfied on the initial equation E. □

The generating sections ψ_{η} are often named gradients of the conservation laws [η] (see [22]) by the following reason. The generating sections for evolution equations lie in the image of the Euler operator (that is, the ”gradient” δ/δu) applied to the corresponding conserved density.

Lemma 5 ([63, 96]). Let E = {u_{t} = f(t, x, u, u_{1}, …)} be an evolution equation. Assume that

η = η_{0} dx + ∑_{i=1}^{n}(−1)^{i−1}η_{i} dt ∧ dx^{1} ∧ ⋯ ∧ d̂x^{i} ∧ ⋯ ∧ dx^{n}

is a conserved current for E: d̄_{h}(η) = 0, that is,

D̄_{t}(η_{0}) + ∑_{i} D̄_{x^{i}}(η_{i}) = 0.

Then its generating function is

ψ_{η} = E(η_{0}) ≡ ℓ_{η_{0}}^{∗}(1).

The proof of Lemma 5 is straightforward.

The Euler–Lagrange equation E_{E−L} assigned to a Lagrangian L ∈ H̄^{n}(π) is

E_{E−L} = {G ≡ E_{u}(L) = 0}.  (15)

We recall that the Helmholz condition ℓ_{G} = ℓ_{G}^{∗} is valid for the image G of the Euler operator E.

The correlation between conservation laws [η], their generating sections ψ_{η}, and the Noether symmetries φ_{L} ∈ sym L of an Euler–Lagrange equation E = {E(L = 0} is guided by the following version of the Noether theorem.

Theorem 6 ([6]). Let E = {E(L) = 0} be the Euler-Lagrange equation for a Lagrangian L. Then the evolutionary derivation �_{φ} is a Noether symmetry of the Lagrangian L:

�_{φ}(L) = 0

if and only if φ is the generating section of a conservation law [η]: d_{h}η = 0 on E.

Proof. Suppose φ is the generating section of a conservation laws [η] for the equation E:

d_{h}(η) = ⟨1, □(F)⟩ = ⟨□^{∗}(1), F ⟩ + d_{h}(γ),

where the coupling ⟨·, ·⟩ takes values in the space Λ̄^{n}(π) of horizontal n-forms. Thence, ⟨φ, F ⟩ = d_{h}(η −γ) is an exact horizontal n-form. We have F = E(L) = ℓ_{L}^{∗}(1) by the initial assumption, whence we easily obtain

⟨φ, F ⟩ = ⟨φ, ℓ_{L}^{∗}(1)⟩ = ⟨ℓ_{L}(φ), 1⟩ + d_{h}β = ⟨�_{φ}(L), 1⟩ + d_{h}β.

Therefore, φ is a Noether symmetry. The second implication in the theorem’s statement, the sufficiency, is proved by inverting the reasonings. □

Coverings. Let E be a differential equation in the fibre bundle ℝ^{n} × ℝ^{m} → ℝ^{n} and E^{∞} be the infinite prolongation of E. The n-dimensional Cartan plane C_{θ} ⊂ T_{θ}(E^{∞}) is defined at each point θ ∈ E^{∞}. The Cartan distribution C_{E} on E^{∞} is Frobenius:

[C_{E}, C_{E}] ⊂ C_{E}.

In local coordinates, C_{E} is defined by n vector fields D̄_{1}, …, D̄_{n}, where D̄_{i} is the restriction of the total derivative with respect to the ith independent variable x^{i} onto E^{∞}.

Definition 11 ([10]). The equation Ẽ^{∞} with the n-dimensional Cartan distribution C̃ and the regular map τ are called a covering over the equation E^{∞} if at any point θ ∈ Ẽ^{∞} the tangent map τ_{∗,θ} is an isomorphim of the plane C̃_{θ} onto the Cartan plane C_{τ(θ)} on the equation E^{∞} at the point τ(θ). The equation Ẽ is called the covering equation. The dimension of the fibre in the bundle τ is the dimension of the covering.

In coordinates, the structure of a covering is realized in the following way. Locally, we can realize the manifold Ẽ and the mapping τ: Ẽ → E^{∞} as the direct product E^{∞} × W (W ⊆ ℝ^{N} is an open subset, 0 < N ≤ ∞) and the natural projection E^{∞} ×W → E^{∞}, respectively. Then we describe the distribution C̃ on Ẽ by the vector fields

D̃_{i} = D̄_{i} + ∑_{j=1}^{N} X_{ij} \frac{∂}{∂s^{j}} , i = 1, …, n,

where X_{ij} ∈ C^{∞}(Ẽ) are the coefficients of the τ-vertical fields on Ẽ and s_{1}, …, and s_{N} are the Cartesian coordinates in ℝ^{N}. Then, the Frobenius condition [C̃, C̃] ⊂ C̃ of integrability for the distribution C̃ is equivalent to the system of relations

[D̃_{i}, D̃_{j}] = 0, i, j = 1, …, n,

which are equivalent to the equations

D̃_{i}(X_{jk}) = D̃_{j}(X_{ik})

that hold for all i, j = 1, …, n, 0 ≤ k ≤ N.

The coordinates s_{i} are called nonlocal variables. In coordinates x^{i}, u_{σ}^{j}, and s_{j} the rules D̃_{i}(s_{j}) = X_{ij} to differentiate the nonlocal variables s_{j} as well as the initial equation E^{∞} define the covering equation Ẽ.

Example 4. Again, consider the Korteweg–de Vries equation (9). Extend the set of variables t, x, T_{j} ≡ D_{x}^{j}(T) with the ”nonlocality” s = ∫ T dx:

s_{x} = T, s_{t} = −β s_{xxx} + \frac{3}{2}s_{x}^{2}. (16)

We see that the equation that covers (9) is the potential Korteweg–de Vries equation.

A symmetry of the covering equation Ẽ is a nonlocal symmetry of the equation E^{∞}. Suppose that the field X̂ is a symmetry of an equation E^{∞} and τ: Ẽ → E^{∞} is a covering. Two radically different cases are possible:

(1) the symmetry X̂ of the equation E^{∞} can be extended up to a symmetry X̃ of the covering equation Ẽ, and

(2) the converse: any lifting of the symmetry X̂ is not a symmetry

of the covering equation. In the second case, the field X̂ generates the one-parametric family of equations Ẽ_{t} that cover the equation E^{∞} for any t.

Recursion operators. In this subsection, we describe an explicit method ([62]) of constructing recursion operators for differential equations. The Cartan generating forms approach developed by I. S. Krasil’shchik is the key-point of our reasonings.

Let E = {F = 0} be a determined system of differential equations, that is, the number m of dependent variables in this equation is equal to the number r of equations. Suppose φ ∈ sym E^{∞} is a symmetry of this equation and consider an r-component column

ω = ^{t}(ω^{1}, …, ω^{r})

such that its components

ω^{i} ∈ C^{∞}(E^{∞}) ⊗ C^{1}Λ(E)

are the Cartan 1-forms that vanish on the Cartan distribution C_{E} on E^{∞} and the coefficients at these forms are functions on E^{∞}. Obviously, the component-wise action �_{φ} ω ≡ φ′ is again an element of the module κ of evolutionary derivations. The condition for φ′ to be a symmetry of the initial equation E is

ℓ̄_{F}^{[1]}(ω) = 0,  (17)

where ℓ̄_{F}^{[1]} is the restriction of the linearization operator onto C^{1}Λ(E).

Now we formulate the rule that assigns differential operators R in total derivatives to the columns (the generating forms) ω. Suppose the components ω^{i} of a generating form ω = ^{t}(ω^{1}, …, ω^{r}) are

ω^{i} = ∑_{j,σ} a_{σ}^{ij} ω_{σ}^{j}.

Then we obtain

�_{φ} ω = ^{t}(…, ∑_{j,σ} a_{σ}^{ij} D_{σ}(φ^{j}), …).

We see that the component-wise action of the Cartan forms ω^{i} is equivalent to the action of the operators

∑_{j,σ} a_{σ}^{ij} ℓ_{u_{σ}^{j}}

on the components of the initial generating section φ = ^{t}(φ^{1}, …, φ^{r}). Hence we deduce that the formal differential recursion operator R is the (r × r)-matrix,

R = ∥∑_{σ} a_{σ}^{ij} D_{σ}∥ .

Still, we observe that “usually” equation (17) admits only the trivial solution

ω_{∅} = ^{t}(ω_{∅}^{1}, …, ω_{∅}^{r})

that corresponds to the identity recursion operator

id: φ ↦ φ′ = φ.

The explanation is that the recursion operators for known equations of mathematical physics usually contain summands that involve the negative powers D_{x^{α}}^{−1} of the total derivatives D_{σ}.

Example 5. The recursion operator R_{KdV} = B̂_{2}∘B̂_{1}^{−1} for the Korteweg– de Vries equation (9) is equal to

R_{KdV} = −β D_{x}^{2} + 2T + T_{x} · D_{x}^{−1}.  (18a)

The recursion operator R_{pKdV} for the potential Korteweg–de Vries equation (16) is

R_{pKdV} = −β D_{x}^{2} + 2s_{x} − D_{x}^{−1} ∘ s_{xx}.  (18b)

From the geometric viewpoint, the situation is like follows. We obtain nontrivial solutions of Eq. (17) by extending the set of local coordinates ⟨x^{1}, …, x^{n}, u_{σ}^{j}⟩ by the “nonlocal” variables s^{i}. These nonlocalities are defined by the compatible rules of their derivation with respect to the independent variables. Also, to each s^{i} we assign the Cartan form

d_{C}(s^{i}) = ds^{i} − ∑_{j=1}^{n} s_{x^{j}}^{i} dx^{j},

which now belongs to some larger distribution C^{1}Λ(Ẽ).

Consider the restriction ℓ̃_{F}^{[1]} of the linearization operator ℓ_{F} onto the extended set of the Cartan forms. Substitute the new linearization operator in equation (17) and assume that a nontrivial solution ω appeared. Now we see that the generating form ω is coupled with some prolongation

˜�_{φ,A(φ)} = �_{φ} + ∑_{i} A_{i}(φ) · \frac{∂}{∂s^{i}} + ⋯

of the initial evolutionary field �_{φ}. Possibly, this prolongation involves a larger (ingeneral, an infinite) set of nonlocal variables, see the paper [43]. The result

φ′ = ˜�_{φ,A(φ)} ω

of the action of ω satisfies the equation

ℓ̃_{F} (φ) = 0.

These sections φ′(x, u_{σ}^{j}, s^{i}, …) are called shadows of nonlocal symmetries. In general, there are shadows that cannot be extended up to a true nonlocal symmetry by using a prescribed set of the nonlocal variables s^{i}. An example is found in Proposition 58 on page 98.

Example 6. Consider the following one–dimensional covering over the potential Korteweg–de Vries equation (16). Introduce the nonlocal variable ζ such that

ζ_{x} = −\frac{1}{2}s_{1}^{2},  ζ_{t} = βs_{1}s_{3} − \frac{1}{2}βs_{2}^{2} − s_{1}^{3},  (19)

where s_{j} ≡ D_{x}^{j}(s) and we similarly denote ζ_{j} ≡ D̃_{x}^{j}(ζ). One easily checks that the covering equation is the Krichever–Novikov type equation

ζ_{t} = −β ζ_{3} + \frac{3}{4}βζ_{2}^{2}ζ_{1}^{−1} − 2\sqrt{2}(−ζ_{1})^{3/2}.  (20)

Then the equation

ℓ̃_{pKdV}^{[1]}(ω) = 0

admits a nontrivial solution in the extended set of variables t, x, s_{j}, and ζ. Indeed, the Cartan generating form of the recursion operator for Eq. (16) is

ω_{pKdV} = −β d_{C}(s_{2}) + s_{1} d_{C}(s) − d_{C}(ζ).  (21)

Now we describe the relation between recursion operators R in total derivatives, which are convenient in applications, and solutions ω solutions of the determining equation

ℓ̃_{F}^{[1]}(ω) = 0.

It is sufficient to solve this problem for the Cartan forms d_{C}(s^{i}). Now it is clear that the problem is reduced to calculation of the linearization ℓ_{s^{i}} with respect to the nonlocal variable s^{i}. To do this, we use the following lemma.

Lemma 7. Let s ∈ F(π) be a function; fix a superscript i ∈ [1, n] of the independent variable x^{i}. Then the relation

ℓ_{D_{i}(s)} = D_{i} ∘ ℓ_{s}  (22)

holds.

Proof. Suppose φ ∈ κ, then we obtain

ℓ_{D_{i}(s)}(φ) = �_{φ} ∘ D_{i}(s) = D_{i} ∘ �_{φ}(s) = D_{i} ∘ ℓ_{s}(φ),

whence follows relation (22). □

Lemma 7 defines the rule

ℓ̃_{s^{j}} = D̃_{i}^{−1} ∘ ℓ̃_{D̃_{i}(s^{j})}  (23)

of calculating the linearization of a nonlocality s^{j}. Here i ∈ [1, n] is arbitrary. Therefore, to each generating form ω = ^{t}(ω^{1}, …, ω^{r}), where

ω^{i} = ∑_{j,σ} a_{σ}^{ij} ω_{σ}^{j} + ∑_{j} a^{ij} d_{C}(s^{j}),

we assign the matrix operator

R = ∥∑_{j,σ} a_{σ}^{ij} · D_{σ} · ℓ_{u^{j}} + ∑_{j} a^{ij} · D_{x^{k}}^{−1} ∘ ℓ_{D_{x^{k}} (s^{j})}∥,

where the index i enumerates the rows and 1 ≤ k ≤ n is arbitrary.

Example 7. Recursion operator (18b) corresponds to the generating Cartan form (21). Operator (18a) corresponds to the Cartan form

ω_{KdV} = −β d_{C}(T_{2}) + 2T d_{C}(T) + T_{1} d_{C}(s).

Bäcklund transformations. The notion of a covering is very useful in description of Bäcklund transformations between differential equations.

Definition 12 ([10]). Let E_{i} ⊂ J^{k_{i}}(π_{i}), i = 1, 2, be two differential equations and τ_{i}: Ẽ → E_{i}^{∞} be coverings with the same total space Ẽ. Then the diagram

B(Ẽ, τ_{i}, E_{i}) = {E_{1} ⟵^{τ_{1}} Ẽ ⟶^{τ_{2}} E_{2}}  (24)

is called a Bäcklund transformation B(Ẽ, τ_{i}, E_{i}) between the equations E_{i}. Diagram (24) is called a Bäcklund autotransformation if E_{1}^{∞} = E_{2}^{∞} = E^{∞}.

In what follows, any Bäcklund transformation between equations E and E′ is a system of differential relations imposed on the unknown functions u and u′ such that the following property holds. Suppose the function u is a solution to the equation E and both functions u and u′ satisfy these relations, then the function u′ is a solution to the equation E′, and vice versa.

Example 8. Bäcklund autotransformation for the two–dimensional Laplace equation Δ_{2}v = 0 is defined by the relations

v′_{y} = v′′_{x}, v′_{x} = −v′′_{y}.

Remark 2. Let τ_{j}: Ẽ_{j} → E_{j}^{∞}, j = 1, 2, be two coverings and μ: Ẽ_{1} → Ẽ_{2} be a diffeomorphism that maps the Cartan distribution C_{τ_{1}}D(Ẽ_{1}) into C_{τ_{2}}D(Ẽ_{2}). Then the diagram B(Ẽ_{1}, τ_{1}, τ_{2} ∘ μ, E_{j}) is also a Bäcklund transformation between the equations E_{j} and these coverings τ_{1} and τ_{2} ∘ μ are called equivalent.

Remark 3. Let τ: Ẽ → E^{∞} be a covering and μ: Ẽ → Ẽ be a nontrivial diffeomorphism of manifolds that preserves the Cartan distribution, e.g., a discrete symmetry that cannot be restricted onto E^{∞}. Then the diagram

E ⟵^{τ} Ẽ ⟶^{μ} Ẽ ⟶^{τ} E  (25)

is also Bäcklund autotransformation for E. In Chapter 4, we apply this contruction to the study of Bäcklund autotransformation for the Liouville equation u_{xy} = exp(2u).

Now we give a brief exposition of the mail results obtained in the present article.

Main results. In Chapter 1, we consider a problem standard in the geometry of differential equations. Namely, we describe relations between symmetries, conservation laws, the Noether symmetries, and recursion operators for the Toda equations. The structures obtained in this chapter are essentially used in what follows.

Let K = ∥k_{ij}, 1 ≤ i, j ≤ r∥ be a nondegenerate (r × r)-matrix and K^{−1} = ∥k^{ij}∥ be its inverse. Suppose there is the set a⃗ of real numbers {a_{i} ≠ 0, 1 ≤ i ≤ r} such that the symmetry condition κ_{ij} = κ_{ji} holds fot the matrix κ = ∥κ_{ij}∥ whose elements are

κ_{ij} = a_{i} · k_{ij}.

Then we say that the matrix K is symmetrizable.

The hyperbolic Toda equations associated with a nondegenerate symmetrizable (r × r)-matrix K are

E_{Toda} = {F_{i} ≡ u_{xy}^{i} − exp(∑_{j=1}^{r} k_{ij}u^{j}) = 0, 1 ≤ i ≤ r}.  (26)

In particular, suppose 𝔤 is a semisimple Lie algebra of rank r, let {α_{i}, 1 ≤ i ≤ r} be the system of its simple roots, and K = ∥k_{ij} = 2(α_{i}, α_{j}) · |α_{j}|^{−2}, 1 ≤ i, j ≤ r∥ be the Cartan matrix of the algebra 𝔤. Tne we set a_{i} = |α_{i}|^{−2}. The Toda equation (26) associated with this matrix K is said to be ([66]) associated with the Lie algebra 𝔤.

The Toda equations (26) are the Euler–Lagrange equations with respect to the action functional

L_{Toda} = ∫ L_{Toda} dx ∧ dy

with the density

L_{Toda} = −\frac{1}{2} ∑_{i,j=1}^{r} κ_{ij}u_{x}^{i}u_{y}^{j} − ∑_{i=1}^{r} a_{i} · exp(∑_{j=1}^{r} k_{ij}u^{j}).

The canonical Hamiltonian structure is known for the Toda equations (see [77]).

Suppose K is a nondegenerate symmetrizable matrix. Then the Toda equation (26) associated with K admits at least one integral ([100]), that is, an expession dependent of u_{σ}^{j} whose total derivative D̄_{y} vanishes on this equation. Namely, we have

T = \frac{1}{2} ∑_{i,j=1}^{r} κ_{ij}u_{x}^{i}u_{x}^{j} − ∑_{i=1}^{r} a_{i} · u_{xx}^{i} ∈ ker D̄_{y}.  (27)

By definition, put

T_{j} ≡ D̄_{x}^{j}(T).

The differential consequences T_{j} to the functional T generate the subspace T ⊂ ker D̄_{y} within the kernel of the total derivative D̄_{y}. Indeed, any smooth function Q supplies the functional

Q(x, T) ≡ Q(x, T, T_{1}, …, T_{μ}) ∈ ker D̄_{y}.

We say that the nondegenerate symmetrizable matrix K is generic if the integral T in Eq. (27) is a unique solution to the equation D̄_{y}(T) = 0 on the corresponding Toda equation (26).

Still, by a special choice of the matrix K one can achieve the situation such that the functional T will not be a unique integral for the Toda equation (26). By [90], a necessary and sufficient condition for r nontrivial independent solutions Ω^{i} of the equation

D̄_{y}(Ω^{i}) = 0

to exist is that K be the Cartan matrix of a semisimple Lie algebra 𝔤. The Toda equations associated with 𝔤 are exactly integrable ([65, 77]). From now on, by Ω ⊆ ker D̄_{y} we denote the subspace in ker D̄_{y}, which is differentially generated by all solutions Ω^{i} to the equation D̄_{y}(Ω^{i}) = 0. The total number of these solutions is denoted by q: 1 ≤ i ≤ q ≤ r. We always assume Ω^{1} ≡ T.

By Δ⃗ = |Δ^{i}|, where Δ^{i} = ∑_{j=1}^{r} k^{ij}, we denote the vector of the con-

formal dimensions of the Toda fields exp(u) ≡ ^{t}( exp(u^{1}), …, exp(u^{r})). The transformation

\begin{matrix}x & ↦ X (x), \\ y & ↦ Y(y), \\ u^{i}(x, y) & ↦ ũ^{i} = u^{i}(X , Y) + Δ^{i} log X′(x)Y′(y) \end{matrix}  (28)

is a finite conformal symmetry of the Toda equation E_{Toda}. The generating section φ of the infinitesimal form of conformal transformation (28) is

φ = □(ϕ(x)),

where the vector–valued, first–order differential operator □ is

□ = u_{x} + Δ · D̄_{x}  (29)

and ϕ is an arbitrary smooth function. The structure of the Lie algebra sym E_{Toda} generators is described in the paper [75]. We have

(1) Suppose K is a generic nondegenerate (r × r)-matrix and the operator □ is assigned to this matrix by using formula (29). Then any symmetry of the Toda equation (26) is

φ = □(ϕ(x, Ω)),  (30)

where ϕ is an arbitrary smooth function that depends on a set of the integrals Ω ⊂ ker D̄_{y}.

(2) Suppose the matrix K is subject to additional constraints such that the Toda equation (26) admits q independent integrals Ω^{i} ∈ ker D̄_{y}, where 1 ≤ i ≤ q ≤ r and

Ω = {Ω_{j}^{i} ≡ D̄_{x}^{j}(Ω^{i})} ⊃ T.

We also assume that the first–order operator □ is the (r × q)-matrix and satisfies some additional restrictions (see [75]). Then symmetries φ of the Toda equation are of the form (30) with respect to the new notation.

The Jacobi bracket on the symmetries φ ∈ sym E_{Toda} induces a bracket on the arguments of the operator □. Suppose

φ_{1} = □ (ϕ_{1}(x, T))

and

φ_{2} = □ (ϕ_{2}(x, T)).

Then we have

{φ_{1}, φ_{2}} = □ (ϕ_{{1,2}}),

where

ϕ_{{1,2}} = �_{φ_{1}}(ϕ_{2}) − �_{φ_{2}}(ϕ_{1}) + D̄_{x}(ϕ_{1}) ϕ_{2} − ϕ_{1} D̄_{x}(ϕ_{2}).

Here ϕ_{{1,2}} = ϕ_{{1,2}}(x, T), since the evolution Ṫ_{ϕ} of integral (27) along any symmetry φ = □ (ϕ) is equal to

Ṫ_{ϕ} = −β D̄_{x}^{3}(ϕ) + T D̄_{x}(ϕ) + D̄_{x}(T · ϕ).  (31)

Here and below we use the notation

β ≡ ∑_{i=1}^{r} a_{i} · Δ^{i}.

Consider the operator applied to the function ϕ in the right-hand side of Eq. (31). We see that this operator is the second Hamiltonian structure B̂_{2} for the Korteweg–de Vries equation (9).

In this paper, we prove that the generating sections ψ_{η} of conservation laws [η] and the Noether symmetries φ_{L} for the Toda equation (26) are related by the equation ψ_{η} = κ φ_{L}. Hence we obtain

Theorem (theorem 18 on page 41). (1) Suppose Ω^{i} is an integral for the Toda equation (55) such that D_{y}(Ω^{i}) = ˜∇_{i}(F). Then there is the operator ∇_{i} such that

˜∇_{i} = ∇_{i} ∘ □^{∗} ∘ κ.

In particular, consider the integral Ω^{1} = T defined in Eq. (58). Then

∇_{1} = 1.

(2) The Noether symmetries of the Toda equation are

φ_{L} = □ ∘ ∑_{i} ∇_{i}^{∗} ∘ E_{Ω^{i}}(Q(x, Ω̄)),

where Ω^{i} ∈ ker D̄_{y} are the integrals for the equation E_{Toda},

E_{Ω^{i}} = ∑_{j≥0}(−1)^{j}D_{x}^{j} · \frac{∂}{∂Ω_{j}^{i}}

is the Euler operator with respect to Ω^{i}, Ω̄ is an arbitrary set of differential consequences to Ω^{i}, and Q is a smooth function.

Example 9 ([57]). The Noether symmetries φ_{L} of the Toda equation (55) associated with a generic nondegenerate symmetrizable matrix K are

φ_{L} = □ ∘ E_{T} (Q(x, T))

up to the transformation x ↔ y.

In Sec. 3, we construct a continuum of recursion operators for the symmetry algebra of the Toda equations. These operators are parameterized by arbitrary smooth functions. Although the structure of the symmetry algebra itself is known, see Eq. (30), presence of the recursion operators gives us additional information about the Toda equation and permits to establish the relation between E_{Toda} and other mathematical physics equations.

Theorem (theorem 24 on page 45). (1) Equation (55) admits continuum of local recursion operators

R: sym E_{Toda} → sym E_{Toda},

which are

R = □ ∘ ∑_{i,j} f_{ij}(x, Ω) · D̄_{x}^{j} ∘ ℓ_{Ω^{i}}.

Here f_{ij} are arbitrary smooth functions and the linearizations ℓ_{Ω^{i}} with respect to the integrals Ω^{i} for the Toda equation are

ℓ_{Ω^{i}} = (…, \underbrace{∑_{σ} ∂Ω^{i}/∂u_{σ}^{k} · D̄_{σ}}_{kth component}, …).  (32)

(2) There is a continuum of nonlocal recursion operators for Eq. (26), which are constructed in the following way. Assign the nonlocal variables s^{i} to the integrals Ω^{i} by the compatible differentiation rules

s_{x}^{i} = Ω^{i}, s_{y}^{i} = 0.

The linearizations ℓ_{s^{i}} are defined by the formulas

ℓ_{s^{i}} = D̄_{x}^{−1} ∘ ℓ_{Ω^{i}},

where ℓ_{Ω^{i}} are calculated by Eq. (32). Then the required recursion operators are

R = □ ∘ ∑_{i} f_{i}(x, s, Ω) · D̄_{x}^{−1} ∘ ℓ_{Ω^{i}},

where the functions f_{i} are arbitrary. In general, these operators do not preserve the locality of elements (30) of the symmetry algebra sym E_{Toda}.

In Chapter 2, we construct the commutative Hamiltonian hierarchy 𝔄 of r-component analogs for the potential modified Korteweg–de Vries equation. This hierarchy is related with the Korteweg–de Vries hierarchy for equation (9) that describes the dynamics of the integral (27).

First, we consider an example ([48]) with r = 1. Namely, we consider the scalar Liouville equation

u_{xy} = exp(2u)

and construct the symmetry sequence for this equation. This sequence is identified with the hierarchy 𝔄 of the potential modified Korteweg–de Vries equation

u_{t} = −\frac{1}{2}u_{xxx} + u_{x}^{3}

Also, we assign the hierarchy 𝔅 of the Korteweg–de Vries equation

T_{t} = −\frac{1}{2}T_{xxx} + 3T T_{x}

to the initial Liouville equation.

Now suppose r ≥ 1. The hierarchy 𝔄 is constructed in Sec. 4 by using the following procedure. Introduce the nonlocal variable s such that s_{x} = T and s_{y} = 1. Choose the initial function ϕ_{−1} = 1. Then construct two sequences,

ϕ_{i} = D_{x}^{−1}(Ṫ_{ϕ_{i−1}}),

see Eq. (31), and

φ_{i} = □(ϕ_{i−1}).

We denote the first sequence ϕ_{i} by 𝔅. This is the commutative bi– Hamiltonian hierarchy of local higher symmetries for the potential Korteweg–de Vries equation

s_{t} = −β s_{xxx} + \frac{3}{2}s_{x}^{2}, β = ∑_{i} a_{i}Δ^{i}.

The second sequence φ_{i} is denoted by 𝔄. This is the required hierarchy of the higher analogs for the potential modified Korteweg–de Vries type equation

u_{t} = φ_{1}.

The following theorem is valid.

Theorem (theorem 41 on page 59). The elements φ_{i} of the sequence 𝔄 ⊂ sym E_{Toda} compose a commutative Lie algebra:

[𝔄, 𝔄] = 0.

Then we describe the elements φ_{k} ∈ 𝔄 such that k < 0. We prove that the symmetry φ_{−1} is defined by the Toda equation itself represented in the Hamiltonian form

u_{y} = A_{1} ∘ E_{u}((a⃗ · exp(Ku)) dx),

where

A_{1} = κ^{−1} · D̄_{x}^{−1}

is the first Hamiltonian structure for the equation within the hierarchy 𝔄 and a⃗ = ^{t}(a_{1}, …, a_{r}), κ = ∥a_{i}k_{ij}∥.

Theorem. 1 (theorem 47 on page 73). The generators φ_{k} of the commutative Lie algebra 𝔄 are the Noether symmetries of the Toda equation: φ_{k} ∈ sym L_{Toda}.

2 (proposition 49 on page 75). Suppose k ≥ 0 is arbitrary. Then the kth term ϕ_{k} = E_{T} (h_{k} dx) of the hierarchy 𝔅 is a conserved density for the kth higher potential modified Korteweg–de Vries equation.

Now we formulate the most remarkable relation between the hierarchy 𝔄 for the potential modified Korteweg–de Vries equation (93) and the hierarchy 𝔅 for scalar equation (91).

Theorem. 1 (proposition 51 on page 78). Each Noether symmetry

φ_{L} = □ ∘ E_{T} (Q(x, T)) ∈ sym L_{Toda}

of the Toda equation associated with a nondegenerate symmetrizable matrix K is Hamiltonian with respect to the Hamiltonian structure A_{1} = κ^{−1} · D_{x}^{−1} and the Hamiltonian H = [Q(x, T)]:

φ_{L} = A_{1} ∘ E_{u}(H).

2 (theorem 52 on page 78). The Hamiltonian [h_{k} dx] for the kth higher Korteweg–de Vries equation is the Hamiltonian for the Noether symmetry φ_{k} ∈ 𝔄 for any integer k ≥ 0.

In Chapter 3, we apply the methods of geometry of partial differential equations to the study of the properties of equations (5) and (3).

In Sec. 8 of Chapter 3 we obtain the geometric structures for the m-component analog

Ψ_{t} = iΨ_{xx} + if(|Ψ|) Ψ

of the nonlinear Schrödinger equation (3). By [98], this equation admits a commutative bi–Hamiltonian hierarchy of higher symmetries and an infinite set of the conserved densities in involution if f = id. Suppose f is arbitrary, then this is not true. We compute the symmetry algebra for this equation if f is the homogeneous function of weight Δ; this case is actual for applications in physics. Also, we point out m^{2} conserved currents

η_{ij} = Ψ^{i}Ψ̄^{j} dx + i (Ψ_{x}^{i} Ψ̄^{j} − Ψ^{i} Ψ̄_{x}^{j}) dt

for this equation. Here f is arbitrary. These currents generalize the known conservation laws of energy for an ith component Ψ^{i} in the multi–component nonliear Schrödinger equation.

In Sec. 9 of the same chapter we consider the dispersionless Toda equation (5),

u_{xy} = exp(−u_{zz}),

and realize the following scheme for it. First, we find the Lie algebra of its classical symmetries and assign classes of exact solutions to these symmetries; then we construct conserved currents for this equation. The results of our calculations are rather cumbersome. They are found on pages 86–81 and also in the paper [50].

In Chapter 4, recent geometric concepts in the theory of Bäcklund transformations and zero–curvature representations are illustrated for the scalar Liouville equation

E_{Liou} = {u_{xy} = exp(2u)},  (33)

which is the Toda equation associated with the Lie algebra 𝔤 = 𝔰𝔩_{2}(ℂ).

In Sec. 10 we construct one–parametric families of Bäcklund (auto)- transformations Ẽ_{t} for equation (33). These transformations are

Ẽ_{t} = \{ \begin{matrix}(ũ − u)_{x} & = e^{−t} exp(ũ + u), \\ (ũ + u)_{y} & = 2e^{t} sinh(ũ − u) \end{matrix}\} .  (34)

By definition, put

u_{k} ≡ \frac{∂^{k}u}{∂x^{k}}, u\overline{k} ≡ \frac{∂^{k}u}{∂y^{k}}, k ∈ ℕ.

Consider the scaling symmetry

X̂ = −x \frac{∂}{∂x} + y \frac{∂}{∂y} + ∑_{k≥1} ku_{k} \frac{∂}{∂u_{k}} − ∑_{k≥1} ku\overline{k} \frac{∂}{∂u\overline{k}}  (35)

of the Liouville equation (33). Our reasonings are based on the following property of X̂. This symmetry cannot be extended up to a symmetry of the covering equation Ẽ_{t}. Recently I. S. Krasil’shchik described ([37]) the scheme of generating one–parametric families of coverings over differential equations. We have

Theorem ([37]). Let τ: Ẽ → E be a covering and A_{t}: Ẽ → Ẽ be a smooth family of diffeomorphisms such that A_{0} = id and

τ_{t} = τ ∘ A_{t}: Ẽ → E

is the covering for any t ∈ ℝ. Then the evolution of the Cartan connection form U_{τ_{t}} is

\frac{dU_{τ_{t}}}{dt} = [[X̂_{t}, U_{τ_{t}}]]^{FN},  (36)

where X̂_{t} is a τ_{t}-shadow for any t ∈ ℝ and [[·, ·]]^{FN} is the Frölicher– Nijenhuis bracket, see Eq. (165) on page 99.

We prove that scaling symmetry (35) is the τ_{t}-shadow such that the evolution of the connection form U_{t} in the covering τ_{t}: Ẽ_{t} → E_{Liou}, see Eq. (34), is given by Eq. (36). The proof is straighforward. It is based on a useful identity in total derivatives.

Theorem (lemma 62 on page 101). Let u(x) and f(u) be smooth functions and D_{x} be the total derivative with respect to x. Denote u_{k} ≡ D_{x}^{k}(u(x)), k ≥ 0, u_{0} ≡ u. Then the relation

n · D_{x}^{n}(f(u)) = ∑_{m=1}^{n} m u_{m} \frac{∂}{∂u_{m}} D_{x}^{n}(f(u))

holds for any integer n ≥ 1.

In Sec. 11, we consider the problem of constructing the pairs of solutions to the hyperbolic Liouville equation (33) and the wave equation

s_{xy} = 0

such that these solutions are related by the Bäcklund transformation. Namely, we obtain the set of the nonlocal variables such that Bäcklund transformations are successfully integrated and each of the solutions related by Bäcklund transformation is obtained explicitly in terms of these variables.

In Sec. 12, we analyse the correspondence between Bäcklund transformations and zero–curvature representations. We use two distinct representations of the Lie algebra 𝔤 = 𝔰𝔩_{2}(ℂ) generated by the elements ⟨e, h, f⟩. The first representation is the representation of 𝔤 in the traceless matrices:

ϱ(e) = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) ,  ϱ(h) = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) ,  ϱ(f) = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) .

The second representation involves vector firlds on the straight line:

ρ(e) = 1 · \frac{∂}{∂Ξ},  ρ(h) = −2Ξ · \frac{∂}{∂Ξ},  ρ(f) = −Ξ^{2} · \frac{∂}{∂Ξ}.

We point out Bäcklund transformations assigned to the known zero– curvature representations, and vice versa.

In Appendix A, we list several geometrical methods of solving the boundary problems for equations of the mathematical physics. Constructing solutions invariant w.r.t. symmetries of the problem is discussed. A method based on the respesentation of the equation at hand in the evolutionary form is pointed out. The methods based on the deformation of the boundary problem along discrete or continuous parameters are described. Among these methods, we note the direct iterations, the boundary conditions homotopy, the relaxation method, and the deformation of the initial equation. Then, comparative analysis of the results of computer experiments in applying these methods is carried out.

Part I. The Korteweg–de Vries type equations associated with the Toda systems

In Part I, we analyse the properties of the symmetry algebra sym E_{Toda} for the hyperbolic Toda equations E_{Toda} associated with nondegenerate symmetrizable matrices K. By using the canonical nonlocal recursion operator for the algebra sym E_{Toda}, we construct its commutative Lie subalgebra 𝔄 generated by the local higher Noether symmetries of the latter equation. We identify this subalgebra with the Hamiltonian hierarchy of the higher analogs for the r-component potential modified Korteweg–de Vries equation associated with K. Then we relate the hierarchy 𝔄 with the commutative bi-Hamiltonian hierarchy 𝔅 of the scalar Korteweg–de Vries equation. The hierarchy 𝔅 itself is a commutative Lie subalgebra of the Noether symmetries of the scalar wave equation. By using all these relations between the hyperbolic equations and the evolution hierarchies of their symmetries, we prove that the hierarchies 𝔄 and 𝔅 share the same sequence of the Hamiltonians and demonstrate that the Toda equation itself is the first nonlocal term in the symmetry algebra 𝔄.

Chapter 1. Conservation laws and the Noether symmetries of the Toda equations

In this chapter, we study the geometric properties of the two–dimensional Toda equations u_{xy} = exp(Ku) (in particular, of the Toda equations associated with the complex semisimple Lie algebras, see [66, 83]). Namely, we analyse the relations between their conservation laws ([68, 90]), Noether’s symmetries of the Lagrangian functional ([86]), and the recursion operators for the symmetry algebra of the latter equations.

In Sec. 1, we pass from the scalar Liouville equation to the hyperbolic Toda equations associated with generic nondegenerate symmetrizable matrices K. Further, we consider properties of the Lagrangian for these equations. Then we point out the minimal integral T ∈ ker D̄_{y} and its differential span T ⊂ D̄_{y}. Finally, we assign generators of the Lie algebra sym E_{Toda} to elements Ω of the kernel ker D̄_{y}.

In Sec. 2, we construct a one-to-one correspondence between the generating sections of conservation laws and the Noether symmetries of the Lagrangian L_{Toda} for the Toda equation. The description of the Lie algebra sym L_{Toda} ⊂ sym E_{Toda} of the Noether symmetries for the Toda equation associated with a generic matrix K is based on this correspondence. Also, we establish some properties which are common to all integrals Ω^{i} ∈ ker D̄_{y} of the Toda equation associated with the Cartan matrix K of a semisimple Lie algebra.

Finally, in Sec. 3 we construct a continuum of the recursion operators for the symmetry algebra of the Toda equation. There are two types of the recursion operators: they are either local or nonlocal with respect to the total dervatives.

The exposition follows the papers [54, 46, 57].

1. The Toda equation

The Liouville equation and its generalizations. The Liouville equation

E_{Liou} = {u_{ξξ} + u_{ηη} = exp(2u)}  (37)

is a model exactly integrable nonlinear differential equation that appeares in many parts of mathematics and mathematical physics. The study of this equation was initiated by Liouville ([69]) and H. Poincaré ([80]). One of the problems that they posed, the uniformization problem for algebraic curves (compact Riemannian surfaces), was investigated later by Kazdan and Warner ([41]). Suppose the Lagrangian

L_{Liou} = −\frac{1}{2}[(u_{ξ}^{2} + u_{η}^{2} + exp(2u)) dξ ∧ dη]

is regularized as described in ([101]) and then calculated on a classical solution of the Liouville equation. In this case, L_{Liou} is known ([101]) to be the generating function for the accessor parameters that characterize the uniformization of a Riemannian surface of genus 0. Also, the Lagrangian L_{Liou} for Eq. (37) is the potential for the Weil–Peterson metric on the Teichmüller space of marked Riemannian surfaces (ibid).

Equation (37) plays an important role in modern field theory. In particular, the quantum Liouville field appeares as a quantum anomaly in the string theory ([82]). Further, consider the free self–dual Yang– Mills equation

F_{μν} = ∗F_{μν},

where F_{μν} is the stress tensor. Then, finding the N-instanton solutions, which minimize the action for the latter equation, also leads ([97]) to Eq. (37).

In Riemannian geometry, the Liouville equation (37) is the Gauss equation for a conformal metric on the Lobachevsky plane (see [23] and also [41]).

Example 10. Consider two pointwise equivalent Riemannian metrics

ds_{j}^{2} = f_{j}(x, y)(dx^{2} + dy^{2}),

where f_{j} > 0 and j is either 1 or 2, on open twofolds of constant Gaussian curvature

K_{j} = −(2f_{j})^{−1}Δ log f_{j} = const_{j} .

Here Δ is the Laplace operator. Namely, suppose f_{2} = f_{1} · exp(2u), then u satisfies the equation

Δu = −K_{2}f_{1} exp(2u) − K_{1}f_{1}.  (38)

Obviously, the two–dimensional elliptic Liouville equation corresponds to the case of the flat metric such that f_{1} ≡ 1 (i.e., K_{1} ≡ 0) and the Lobachevsky plane metric (K_{2} ≡ −1). This interpretation of Eq. (37) is treated as a limit case (K_{1} < 0) in the superconductivity theory in description of the Abrikosov curls in the 2D model ([41]). The Liouville equation appeares in the study of the Kadomtsev–Pogutse equations. The latter equation is a reduction of the general magneto–hydrodynamic system, where some detailes that are inessential in the stability problem for high–temperature plasma in TOKAMAK are omitted ([33]).

For any solution of the elliptic Liouville equation (37) there is a Lobachevsky plane model that is conformally equivalent to the Euclidean plane with the diagonal metric g_{ij} = δ_{ij}. Consider the standard Poincaré model in the upper half-plane y > 0 such that f_{2} = 1/y^{2}. Then we get a particular solution u = − log y of Eq. (37).

Suppose f_{1} ≡ 1, K_{1} ≡ 0, and K_{2} ≡ +1. Then formula (38) describes the conformal equivalence between the Eucludean metric δ_{ij} on the plane 𝔼^{2} and the metric g_{ij} = exp(2U) δ_{ij} on the two-dimensional sphere S^{2}. The function U(x, y) satisfies the equation

U_{xx} + U_{yy} + exp(2U) = 0  (39)

that differs from Eq. (37) by the sign at the exponent (or, equivalently, by the multiplication of the independent variables by i = \sqrt{−1}). We shall say that Eq. (39) is the Liouville scal^{+}-equation.

Remark 4. Consider the orthogonal metric

ds^{2} = α^{2} dx^{2} + β^{2} dy^{2}

on the sphere of raduis ρ. Then, Eq. (39) is a particular (conformal) case of the equation ([18])

(α^{−1}β_{x})_{x} + (β^{−1}α_{y})_{y} + ρ^{−2}αβ = 0  (40)

that describes an orthogonal coordinate net on this sphere. Indeed, α = β = ρ exp(U) is a solution of Eq. (40) if U(x, y) is an arbitrary solution of Eq. (39). Obviously, this solution is not unique. The pair

α = sin V, β = ρV_{y},

where V satisfies the sine-Gordon equation

V_{xy} = sin V,

or the pair

α = ρ, β = ρ sin x

are also solutions to Eq. (40).

Suppose E is a differential equation. Throughout this article we denote the number of dependent variables u^{j} (the unknown functions) by r and suppose that these r functions depend on n base coordinates (usually they are denoted by x^{1}, …, x^{n}).

In this paper, we consider the generalizations of the two–dimensional scalar Liouville equation (37) to the cases n ≥ 2 and r ≥ 1. We investigate the properties of these generalizations by using modern methods and tools of the PDE geometry.

We obtain a generalization of Eq. (37) for n ≥ 2 by the following procedure. Recall that the Liouville equation is the condition for the Euclidean metric on 𝔼^{n} to be conformally equivalent to the conformal metric on an n-dimensional manifold of constant scalar curvature

scal ≡ R = const .  (41)

We fix the value R = −2 (such that the Gaussian curvature is K = −1 at n = 2) and preserve the correlation with the two–dimensional case of Eq. (37). We also put

ds^{2} = exp(2u)∑_{k}dx^{k}dx^{k}.  (42)

Relation (41) is a nonlinear PDE for the function u(x^{1}, …, x^{n}).

Theorem 8 ([46]). Condition (41) is

(n − 1)Δu + \frac{1}{2}(n − 1)(n − 2)(grad u)^{2} = exp(2u),  (43)

where Δ is the Laplace operator in the Euclidean space 𝔼^{n} and the scalar product (·, ·) is also induced by the Euclidean metric.

Proof. The scalar curvature R of metric (42) is given by the formula

R = ∑_{i,q} exp(−2u)R_{qqi}^{i},

see [23]. Then we have

R_{qqi}^{i} = ∂_{i} Γ_{qq}^{i} − ∂_{q} Γ_{qi}^{i} + Γ_{pi}^{i}Γ_{qq}^{p} − Γ_{pq}^{i}Γ_{qi}^{p},

where

Γ_{ij}^{k} = ∂_{i}u δ_{j}^{k} + ∂_{j}u δ_{i}^{k} − ∂_{l}u δ_{ij}δ^{kl}

are the Christoffel symbols. After a straightforward calculation of the sums in q, i, and p = 1 … n, we get (43). □

The Toda equations. M. Toda ([93]) considered the integrable nonlinear dynamical system described by Eq. (1), i.e., the one–dimensional lattice with the exponential interaction. Nowadays, vastest literature is devoted to the study of the properties of system (1) and its various generalizations (see [22, 100, 66, 77] and references therein). In order to obtain the two–dimensional Toda system out of the nonperiodic Toda lattice

q_{tt}^{i} = \frac{∂H}{∂q^{i}} , i ∈ ℤ,

where the Hamiltonian H is

H = −∑_{i} exp(q^{i} − q^{i+1}),

we replace the “acceleration” d^{2}/dt^{2} with the operator ∂^{2}/∂x∂y. Different stages of this replacement process and the discovery of the Toda equations associated with the Lie algebras (and the Kac–Moody algebras as well, see [39]) are discussed in the papers [67, 65, 68, 66, 7, 8, 27, 28, 29], see also [22, 77].

In what follows, we derive the generalizations of Eq. (37) from a nontrivial viewpoint. The resulting number r of the dependent variables u^{1}, …, u^{r} is r ≥ 1. Namely, we use the notion of the Laplace invariants ([100]). To start with, we simplify the calculations by making the change of coordinates such that the Liouville equation acquires the hyperbolic form u_{xy} = exp(u). From now on, we study the hyperbolic equations and their symmetries. The difference between the elliptic and the hyperbolic cases vanishes after the complexification. Still, the study of the symmetry properties of PDE does not require the presence of a complex structure. Therefore we assume all equations to be real in agreement with the definition on page 6.

Following [100], we calculate the principal Laplace invariants H_{0} and H_{1} for a hyperbolic equation

u_{xy} = f(x, y, u, u_{x}, u_{y});

we have f = exp(u). Then we get

H_{0} =^{def} − D̄_{y} (\frac{∂f}{∂u_{y}} ) + \frac{∂f}{∂u_{x}} \frac{∂f}{∂u_{y}} + \frac{∂f}{∂u} = exp(u),
H_{1} =^{def} − D̄_{x} (\frac{∂f}{∂u_{x}} ) + \frac{∂f}{∂u_{x}} \frac{∂f}{∂u_{y}} + \frac{∂f}{∂u} = exp(u).

The definition of other Laplace’s invariants follows from the equations

D̄_{xy}(log H_{i}) = −H_{i−1} + 2H_{i} − H_{i+1}, i ∈ ℤ.  (44)

We recall that the quasilinear equation u_{xy} = f is of the Liouvillean type if its sequence of the Laplace invariants is finite, i.e., there are some p ≥ 1 and q ≥ 0 such that H_{p} = H_{−q} ≡ 0. One easily checks that the sequence H_{i} for the Liouville equation vanishes at once: H_{i} ≡ 0 if i is neither 0 nor 1. This observation motivates the above definition^{2}. Now suppose −q < i < p and make the substitution H_{i} = exp(U^{i}). Now, restrict equation (44) onto the graphs of jets of the sections U in the jet bundle ℝ^{r} × ℝ^{2} → ℝ^{2}. Then the total derivatives D̄_{i} are mapped to the derivations ∂/∂x^{i}. We finally obtain the system

U_{xy}^{0} = 2 exp(U^{0}) − exp(U^{1}), U_{xy}^{1} = − exp(U^{0}) + 2 exp(U^{1}).

2We also note that the matrix Laplace invariants are defined for system (46) of hyperbolic differential equations, see [100].

Generally, we obtain the system of equations

U_{xy}^{i} = ∑_{j=−q+1}^{p−1}k_{ij} exp(U^{j})

such that the structure of the nondegenerate (p + q − 1) × (p + q − 1)- matrix K = ∥k_{ij}∥ is

k_{ii} = 2,  k_{i,i+1} = k_{i.i−1} = −1,  k_{ij} = 0 if |i − j| > 1. (45)

We see that K is the Cartan matrix of the Lie algebra of type A_{r−1}. Now we shift the index i that enumerates the variables U^{i} and make the change ([66]) of the form U = K · u. Thus, we get the system

u_{xy}^{i} = exp(−u^{i−1} + 2u^{i} − u^{i+1}), 1 ≤ i ≤ r, u^{0} = u^{r+1} ≡ 0.

This is the two–dimensional Toda system associated with the type A_{r−1} Lie algebra, which is denoted by 𝔤. Generally, we assign the Toda equations

u_{xy} = exp(Ku)  (46)

to a semisimple Lie algebra with the Cartan matrix K by following the geometric scheme ([66]), which is discussed below. In Chapter 4, we use the contructions supplied by this scheme and analyse the relationship between the canonical zero–curvature representations and Bäcklund transformations for a certain class of differential equations.

First, we fix some notation. Let 𝔤 be a semisimple complex Lie algebra of rank r. By {α_{i}, 1 ≤ i ≤ r} denote its system of simple roots and let K be the Cartan matrix of 𝔤. We thus have

K = ∥k_{ij} = 2(α_{i}, α_{j}) · |α_{j}|^{−2}, 1 ≤ i, j ≤ r∥.

By K^{−1} = ∥k^{ij}∥ denote the inverse matrix for K and denote its elements by k^{ij}. Suppose A, B ∈ 𝔤. Then we assume that

θ = A dz + B dz̄  (47)

is the flat connection form in the principal fibre bundle G → M, where G is the Lie group of the Lie algebra 𝔤. We thus have

dθ + \frac{1}{2}[θ, θ] = 0.  (48)

In terms of the extended total derivatives we have, then,

[∂ + A, ¯∂ + B] = 0 ⇔ −¯∂A + ∂B + [A, B] = 0.  (49)

Suppose further that H_{j} are the Cartan generators and E_{j}, F_{j} are the Chevalley generators of 𝔤, respectively, for all j such that 1 ≤ j ≤ r = rank 𝔤. The commutation relations

\begin{aligned}[H_{i}, H_{j}] & = 0, & [H_{i}, E_{j}] & = k_{ji}E_{j}, \\ [H_{i}, F_{j}] & = −k_{ji}F_{j}, & [E_{i}, F_{j}] & = δ_{i,j}H_{i} \end{aligned}  (50)

hold. We choose the following ansatz for the connection coefficients A and B:

\begin{aligned}A & = ∑_{j=1}^{r} (a_{h}^{j}(z, z̄) · H_{j} + a_{e}^{j}(z, z̄) · E_{j}) , \\ B & = ∑_{j=1}^{r} (b_{h}^{j}(z, z̄) · H_{j} + b_{f}^{j}(z, z̄) · F_{j}) . \end{aligned}  (51)

Then from Eq. (49) we get the relations

∂b_{h}^{j} − ¯∂a_{h}^{j} + a_{e}^{j}b_{f}^{j} = 0,  (52a)
¯∂log a_{e}^{j} = −∑_{i=1}^{r}k_{ji} b_{h}^{i},  ∂ log b_{f}^{j} = ∑_{i=1}^{r}k_{ji} a_{h}^{i}  (52b)

for the coefficients of H_{j}, E_{j}, and F_{j}, respectively, for any j such that 1 ≤ j ≤ r. From Eq. (52b) we obtain

a_{h}^{i} = ∑_{j=1}^{r}k^{ij} ∂ log b_{f}^{j}, b_{h}^{i} = −∑_{j=1}^{r}k^{ij} ¯∂log a_{e}^{j},

and from Eq. (52a) we get the relation

∑_{j=1}^{r}k^{ij} ∂ ¯∂log (a_{e}^{j} · b_{f}^{j}) = a_{e}^{i} · b_{f}^{i}  (53)

for any i, 1 ≤ i ≤ r. By definition, put

u_{i} = ∑_{j=1}^{r}k^{ij} log (a_{e}^{j} · b_{f}^{j}) .  (54)

Now substitute Eq. (54) in Eq. (53). We finally get the Toda equation (see Eq. (46)) associated with the Lie algebra 𝔤 ([66]). In the sequel, we use the coordinates x and y as synonims for the complex coordinates z and z̄, respectively. Also, we suggest the following convention: all symmetries, conservation laws, and similar structures for the Toda equations are treated up to the discrete symmetry x ↔ y.

Now suppose K = ∥k_{ij}, 1 ≤ i, j ≤ r∥ is a nondegenerate (r × r)- matrix and K^{−1} = ∥k^{ij}∥ is its inverse. Assume that there is an rcomponent vector a⃗ such that a_{i} ≠ 0 for any 1 ≤ i ≤ r and the symmetry condition κ_{ij} ≡ a_{i} · k_{ij} = κ_{ji} holds for the matrix κ = ∥κ_{ij}∥. Then we say that this matrix K is symmetrizable, see also [89]. By κ̂ we denote the operator of left multiplication by the nondegenerate matrix κ.

The hyperbolic Toda equations associated with a nondegenerate symmetrizable (r × r)-matrix K are

E_{Toda} = {F^{i} ≡ u_{xy}^{i} − exp(∑_{j=1}^{r} k_{ij}u^{j}) = 0, 1 ≤ i ≤ r}.  (55)

In particular, suppose 𝔤 is a semisimple Lie algebra of rank r, let {α_{i}, 1 ≤ i ≤ r} be the system of its simple roots, and denote the Cartan matrix of the algebra 𝔤 by

K = ∥k_{ij} = \frac{2(α_{i}, α_{j})}{|α_{j}|^{2}}, 1 ≤ i, j ≤ r∥ .

Then we set a_{i} = |α_{i}|^{−2}; we thus have

κ_{ij} = \frac{2(α_{i}, α_{j})}{|α_{i}|^{2} · |α_{j}|^{2}} = κ_{ji}.

Lagrangian formalism for the Toda equation. The Toda equation E_{Toda} is the Euler-Lagrange equation in the following sence. Consider the action functional

L_{Toda} = ∫ L_{Toda} dx ∧ dy

with the density

L_{Toda} = −\frac{1}{8} ∑_{i,j} ∑_{μ,ν} g^{μν}κ_{ij}u_{;μ}^{i}u_{;ν}^{j} + a_{i}^{2} · exp(∑_{j=1}^{r} k_{ij}u^{j}).

Here u_{;μ}^{j} ≡ D_{μ}(u^{i}) and g^{μν} = ( \begin{matrix}0 & 2 \\ 2 & 0 \end{matrix} ) is the inverse of the metric tensor

g_{μν} = ( \begin{matrix}0 & \frac{1}{2} \\ \frac{1}{2} & 0 \end{matrix} ) that defines the flat metric ds^{2} = dx dy on the base of the jet bundle π. In local coordinates, the density of the Lagrangian is

L_{Toda} = −\frac{1}{2} ∑_{i,j=1}^{r} κ_{ij}u_{x}^{i}u_{y}^{j} − ∑_{i=1}^{r} a_{i} · exp(∑_{j=1}^{r} k_{ij}u^{j}).  (56)

Assign the Euler-Lagrange equations

E_{u}(L_{Toda}) = |∑_{j} κ_{ij}F^{j}| = κ · F = 0  (57)

to the Lagrangian L_{Toda}. Then these equations are equivalent to equation (55) since the matrices K and κ are nondegenerate simultaneously due to the assumption a_{i} ≠ 0.

Suppose E is a differential equation, φ ∈ ker ℓ̄_{E} is its symmetry and ψ ∈ ker ℓ̄_{E}^{∗} is the generating section of its conservation law. Now we obtain the transformation rules for φ and ψ with respect to the reparameterizations that preserve the equation’s manifold E and the ideal E^{∞} of its differential consequences. The following lemma is valid.

Lemma 9 ([57]). Suppose E = {G^{i} = 0, 1 ≤ i ≤ r} be a nonoverdetermined differential equation and let

G^{i} = A_{j}^{i}F^{j}

be a nondegenerate transformation of the relations that define the equation E. Then the following two conditions hold:

(1) The identities

ℓ_{G} = A · ℓ_{F} , ℓ_{G}^{∗} = ℓ_{F}^{∗} ·^{t}A,

are fulfilled. Here A is the reparameterization matrix for the relations that describe E.

(2) Suppose φ_{G} ∈ ker ℓ̄_{G} is a symmetry of the equation E and ψ_{G} ∈ ker ℓ̄_{G}^{∗} is an arbitrary solution to Eq. (13) for E = {G = 0}. Now, consider the reparameterization G = AF of the relations that define the equation E ≃ {F = 0}. Then φ_{F} = φ_{G} is a symmetry of the equation E again: φ_{F} ∈ ker ℓ̄_{F}, while a solution ψ_{G} of Eq. (13) is transformed by the rule

ϕ_{G} ↦ ψ_{F} = ^{t}A · ψ_{G} ∈ ker ℓ̄_{F}^{∗} .

Proof. By using the definition of the linearization operator ℓ_{G} for G = AF, we get

ℓ_{G} = ℓ_{AF} = A · ℓ_{F} .

Therefore,

ℓ_{G}^{∗} = (A · ℓ_{F})^{∗} = ℓ_{F}^{∗} ∘ A^{∗} = ℓ_{F}^{∗} ∘^{t}A.

If the matrix A is nondegenerate, then the condition ℓ̄_{G}(φ_{G}) = 0 is equivalent to A · ℓ̄_{F} (φ_{G}) = 0, whence we get φ_{F} = φ_{G}. From the assumption ℓ̄_{F}^{∗} (^{t}A · ψ_{G}) = 0 we deduce the formula ψ_{F} = ^{t}A · ψ_{G} for the rules of solution transformation for the equation ℓ̄_{F}^{∗} (ψ_{F}) = 0. □

Now we apply the Noether theorem (see Theorem 6 on page 14):

Corollary 10. Let the assumptions of Theorem 6 and Lemma 9 hold. Let ψ ∈ ker ℓ̄_{F}^{∗} be the generating section of a conservation law for the Euler-Lagrange equation E = {F = 0}. Then there is the Noether symmetry φ ∈ ker ℓ̄_{F} of the equation E such that the relation ψ = ^{t}A·φ holds.

The minimal integral for the Toda equation. Consider the Toda equation associated with a nondegenerate symmetrizable matrix K. One easily checks that Eq. (55) admits at least one integral ([100]), that is, an expession dependent of u_{σ}^{j} whose total derivative D̄_{y} vanishes on this equation. Namely, we have

T = \frac{1}{2} ∑_{i,j=1}^{r}κ_{ij}u_{x}^{i}u_{x}^{j} − ∑_{i=1}^{r}a_{i} · u_{xx}^{i} ∈ ker D̄_{y}.  (58)

Both components T and T̄ of the traceless energy–momentum tensor Θ for the Euler equations (57) are well known to be of form (58) up to the complex conjugation:

Θ = T dx + T̄ dy,

see, e.g., [77]. In Sec. 5 of Chapter 2 we shall discuss some aspects of the Hamiltonian formalism for the Toda equations and derive (58) from density (56) of the Lagrangian. Meanwhile, we construct the differential span T of the minimal integral T ∈ ker D̄_{y}. By definition, put

T_{j} ≡ D̄_{x}^{j}(T).

The differential consequences T_{j} to the functional T generate the subspace T ⊂ ker D̄_{y} within the kernel of the total derivative D̄_{y}. Indeed, any smooth function Q supplies the functional

Q(x, T) ≡ Q(x, T, T_{1}, …, T_{μ}) ∈ ker D̄_{y}.

We say that the nondegenerate symmetrizable matrix K is generic if the integral T in Eq. (58) is a unique solution to the equation D̄_{y}(T) = 0 on the corresponding Toda equation (55).

We emphasize that by a special choice of the matrix K one can achieve the situation such that the functional T will not be a unique integral for the Toda equation (55). The criteria for the equality dim ker D̄_{y} = 2 to hold at r = 2 are found in the paper [68]. From now on, by Ω ⊆ ker D̄_{y} we denote the subspace in ker D̄_{y}, which is differentially generated by all solutions Ω^{i} to the equation D̄_{y}(Ω^{i}) = 0. The total number of these solutions is denoted by q: 1 ≤ i ≤ q ≤ r. We always assume Ω^{1} ≡ T.

Example 11 ([68]). Consider the Cartan matrix K = ( \begin{matrix}2 & −1 \\ −1 & 2 \end{matrix} ) of the Lie algebra 𝔰𝔩_{3}(ℂ). Then the Toda equation (55) associated with K admits two integrals. The first integral Ω^{1} = T, which is defined in Eq. (58), acquires the form

Ω^{1} = (u_{x}^{1})^{2} − u_{x}^{1}u_{x}^{2} + (u_{x}^{2})^{2} − u_{xx}^{1} − u_{xx}^{2},  (59a)

while the second integral is

Ω^{2} = u_{xxx}^{1} + u_{x}^{1} · (u_{xx}^{2} − 2u_{xx}^{1}) + (u_{x}^{1})^{2} · u_{x}^{2} − u_{x}^{1} · (u_{x}^{2})^{2}.  (59b)

The integral Ω^{2} is a solution to the equation D̄_{y}(Ω) = 0 but does not follow from Ω^{1}.

By [90], a necessary and sufficient condition for r nontrivial independent solutions Ω^{i} of the equation D̄_{y}(Ω^{i}) = 0 to exist is that K be the Cartan matrix of a semisimple Lie algebra 𝔤. The Toda equations associated with 𝔤 are exactly integrable ([65]).

The symmetry algebra of the Toda equations. In this subsection, we assign classes of infinitesimal symmetries of the Toda equation to the functional span Ω of differential consequences to the integrals Ω^{i}, which were introduced in the previous subsection. Then, in Sec. 2 we find out which out these symmetries are the Noether symmetries of the Lagrangian L_{Toda}.

By Δ⃗ = |Δ^{i}| denote the vector of the conformal dimensions ([82])

Δ^{i} = ∑_{j=1}^{r} k^{ij}

of the Toda fields exp(u) ≡ ^{t}( exp(u^{1}), …, exp(u^{r})). This notation is well–defined due to

Proposition 11 ([8]). (1) The transformation

\begin{matrix}x & ↦ X (x), \\ y & ↦ Y(y), \\ u^{i}(x, y) & ↦ ũ^{i} = u^{i}(X , Y) + Δ^{i} log X′(x)Y′(y) \end{matrix}  (60)

is a finite conformal symmetry of the Toda equation E_{Toda}.

(2) The Lagrangian L_{Toda} = ∫ L_{Toda} dx∧dy is invariant with respect to this change.

(3) By definition, put β ≡ ∑_{i=1}^{r}a_{i}·Δ^{i}; under diffeomorphism (60), the

component T of the energy–momentum tensor Θ is transformed by the rule

T [u] ↦ (X′(x))^{2} · T [ũ(X , Y)] − β · (\frac{X′′′(x)}{X′(x)} − \frac{3}{2}(\frac{X′′(x)}{X′(x)})^{2}). (61)

Remark 5. The symmetry properties, Eq. (60) and (61), of the Toda equation associated with the Lie algebras were considered in the paper [8]. The present formulation of Proposition 11 is an extension of the cited result to the Toda equation (55), which is now associated with an arbitrary nondegenerate symmetrizable r × r-matrix K. The coefficients Δ⃗ and β are already adapted to this general case.

The infinitesimal variant of Proposition 11 is

Proposition 12. 1 ([75, 77]). The infinitesimal components of conformal symmetries (60) of the Toda equation are

φ_{0}^{f} = □(f(x))

up to the transformation x ↔ y. Here f is an arbitrary smooth function and

□ = u⃗_{x} + Δ⃗ · D̄_{x}  (62)

is a vector–valued, first–order differential operator. 2 ([45]). Each point symmetry φ_{0}^{f} is a Noether symmetry of the Lagrangian L_{Toda}:

�_{φ_{0}^{f}} (L_{Toda} dx) ∈ im d_{h}.

3 ([77, 7]). The functional T, which was defined in Eq. (58), is a density of the Hamiltonian for the infinitesimal conformal symmetry φ_{0}^{f}:

φ_{0}^{f} = A_{1} · E_{u}(T · f(x) dx), ��� A_{1} = κ̂^{−1} · D_{x}^{−1}.  (63)

The third statement in Proposition 12 was formulated in local coordinates in the paper [77]. In Sec. 5 of Chapter 2 we establish the relation between the canonical Hamiltonian formalism for the Toda equation and the Hamiltonian operator A_{1} above. In Theorem 52 on page 52 we find out that equality (63) is the root part of the infinite sequence of relations between the hierarchy of the higher analogs of the Korteweg–de Vries equation, see Eq. (9) on page 11, and the commutative hierarchy within the Noether symmetries algebra for the Toda equation (this hierarchy is constructed in Sec. 4).

Lemma 13. Suppose K is a symmetrizable r × r-matrix and the operator □ is defined by Eq. (62). Then the relations

\begin{aligned}ℓ̄_{F} ∘ □ & = D̄_{x} ∘ □ ∘ D̄_{y}, \\ ℓ̄_{F}^{∗} ∘ κ̂ ∘ □ & = D̄_{x} ∘ κ̂ ∘ □ ∘ D̄_{y} \end{aligned}  (64)

hold.

Proof. Here we express the first relation in local coordinates:

ℓ̄_{F} ∘ □ = ∥δ_{ij} D̄_{xy} − k_{ij} exp(∑_{l} k_{il}u^{l})∥ · |u_{x}^{j} + Δ^{j} · D̄_{x}| =

= u_{x} D̄_{xy} + u_{xx} D̄_{y} + u_{xy} D̄_{x} + u_{xxy} + Δ⃗ D̄_{xxy} −

− |∑_{j} k_{ij}u_{x}^{i} exp(∑_{l} k_{il}u^{l})| − |∑_{j,p} k_{ij}k^{jp} exp(∑_{l} k_{il}u^{l})| · D̄_{x} =

= D̄_{x} ∘ |u_{x} + Δ⃗ D̄_{x}| ∘ D̄_{y},

Q. E. D.

The second relation is deduced from the first one by using Lemma 9 and the Helmholtz condition

ℓ̄_{E(L_{Toda})} = ℓ̄_{E(L_{Toda})}^{∗}

that holds since the matrix κ = ^{t}κ is symmetric. □

Corollary 14. The vector–functions

φ = □(ϕ(x, Ω))  (65)

are symmetries of the Toda equation: φ ∈ sym E_{Toda} for any function ϕ that depends on an arbitrary subset Ω of the integrals Ω_{j}^{i} ≡ D̄_{x}^{j}(Ω^{i}) ∈ ker D̄_{y}.

Formula (65) gives the description of the whole symmetry algebra sym E_{Toda}:

Proposition 15 ([75]). (1) Let the assumptions of Lemma 13 hold;

then any symmetry of Eq. (55) is (65).

(2) Suppose further K is such that there exist q independent integrals Ω^{i} ∈ ker D̄_{y}, where 1 < q ≤ r. Put

f^{i} = exp(∑_{j=1}^{r} k_{ij}u^{j}) and f_{j}^{i} = k_{ij} · f^{i}.

Assume further that there are r constant (r ×q)-matrices M_{α} = ∥(M_{α})_{j}^{i}∥, 1 ≤ i ≤ r, 1 ≤ j ≤ q, and a constant (r × q)-matrix Δ = ∥Δ_{j}^{i}∥, 1 ≤ i ≤ r, 1 ≤ j ≤ q, such that rank Δ = q and the equations

Δ_{i}^{β} f_{β}^{α} = (M_{β})_{i}^{α} f^{β}, f_{β}^{α} (M_{γ})_{i}^{β} = (M_{β})_{i}^{α} f_{γ}^{β}

hold. Now, construct the first–order, (r × q)-matrix differential operator

□ = ∑_{α=1}^{r}M_{α} · u_{x}^{α} + Δ · D̄_{x}  (66)

and consider a vector ϕ = |ϕ^{i}(x, Ω)|, 1 ≤ i ≤ q. Under these assumptions, the sections defined in Eq. (65) exhaust all symmetries of the Toda equation E_{Toda}.

Therefore, in both cases we have

sym E_{Toda} ≃ {φ^{i} = □_{j}^{i}ϕ^{j}(x, Ω) mod (x ↔ y)},

where the number of columns in the operator □ is equal to the number q of the independent integrals Ω^{l} and ϕ is an arbitrary vector.

Corollary 16. Any solution ψ to the equation ℓ̄_{F}^{∗} (ψ) = 0 for system (55) is of the form

ψ = κ̂(□(ϕ(x, Ω))).  (67)

We stress that the problem of finding the integrals Ω is primary for the Toda equations, and the search for the symmetries φ as well as the selection of the Noether symmetries φ_{L} follow this problem. We also note that each conformal symmetry (60) of the Toda equation is Noether, i.e., preserves the Lagrangian L_{Toda}. Still, not each section φ of type (65) is a Noether symmetry of Eq. (55).

2. The Noether symmetries of the Toda equation

First, we recall Example 3 on page 12 and obtain an important property of the Toda equation. This property permits the application of the generating sections machinery in description of the conservation laws and the Noether symmetries of the Toda equation. Namely, we have

Lemma 17. The Toda equation E_{Toda} is ℓ-normal.

Proof. By Example 3 on page 12, it is sufficient to represent the equation E_{Toda} in the evolutionary form. Let

ξ = x + y, η = x − y

be the new independent variables. The way to choose them is such that the equations

u_{ξξ}^{i} − u_{ηη}^{i} = exp (∑_{j=1}^{r} k_{ij}u^{j})

hold. Now, put v^{i} ≡ u_{η}^{i}; then the equations E_{ev} ⊂ J^{2}(ℝ^{2}, ℝ^{2r}) of the form

{u_{η}^{i} = v^{i}, v_{η}^{i} = u_{ξξ}^{i} − exp(∑_{j=1}^{r} k_{ij}u^{j})}

are the required evolutionary representation of Eq. (55). □

From relation (57) and Corollary 10 we deduce the relation ψ = κ φ_{L} between the Noether symmetries and the generating sections of conservation laws for the Toda equations. This observation allows to specify a property common for all the integrals Ω^{i} for Eq. (55). Namely, suppose

d_{h}(Ω^{i} dx) = − ˜∇_{i}(F) dx ∧ dy

for any admissible i and consider the conservation law [η] = [Q(x, Ω) dx]. Then we have

d_{h}Q(x, Ω) dx = −∑_{i,j} \frac{∂Q}{∂Ω_{j}^{i}} D_{x}^{j} ∘ ˜∇_{i}(F) dx ∧ dy.

By definition, the generating section ψ_{η} of the conservation law [η] is

ψ_{η} = − ∑_{i,j}(−1)^{j}\binom{}{˜∇_{i}}^{∗} ∘ D_{x}^{j} (\frac{∂Q}{∂Ω_{j}^{i}} ) = − ∑_{i} \binom{}{˜∇_{i}}^{∗} ∘ E_{Ω^{i}}(Q). (68)

Now we compare Eq. (67) with Eq. (68) and, by using Theorem 6 and Lemma 9, we obtain

Theorem 18. (1) Suppose Ω^{i} is an integral for the Toda equation (55) such that D_{y}(Ω^{i}) = ˜∇_{i}(F). Then there is the operator ∇_{i} such that

˜∇_{i} = ∇_{i} ∘ □^{∗} ∘ κ̂.

In particular, consider the integral Ω^{1} = T defined in Eq. (58). Then

∇_{1} = 1.

(2) The Noether symmetries of the Toda equation are

φ_{L} = □ ∘ ∑_{i}∇_{i}^{∗} ∘ E_{Ω^{i}}(Q(x, Ω)),

where Ω^{i} ∈ ker D̄_{y} are the integrals for the equation E_{Toda},

E_{Ω^{i}} = ∑_{j≥0}(−1)^{j}D_{x}^{j} · \frac{∂}{∂Ω_{j}^{i}}

is the Euler operator with respect to Ω^{i}, Ω is an arbitrary set of differential consequences to Ω^{i}, and Q is a smooth function.

Example 12. Again, consider the Toda equation (55) associated with the root system A_{2}. Then we have r = 2, a_{i} = |α_{i}|^{−2} = 1 for i = 1, 2, and

K = (\begin{matrix}2 & −1 \\ −1 & 2 \end{matrix}) , K^{−1} = \frac{1}{3} · (\begin{matrix}2 & 1 \\ 1 & 2 \end{matrix}) , Δ⃗ = \binom{1}{1}.

Put □ = u⃗_{x} + Δ⃗ · D̄_{x}, see (62). The integrals Ω^{1} and Ω^{2} are defined in Eq. (59). One easily checks that D_{y}(Ω^{1}) = □^{∗} ∘ κ̂(F). Therefore, ∇_{1} = 1 and

D_{y}(Ω^{2}) = −D_{x} ∘ □^{∗} ∘ κ̂(F),

thence, ∇_{2} = −D_{x}. We emphasize that the integral Ω^{2} is not equivalent to −D_{x}(Ω^{1}).

Remark 6. In Theorem 18 we established the relation ∇_{1} = 1 to hold for the minimal integral T, which is defined in Eq. (58). Now we restrict all previous reasonings onto the subspace {Q(x, T)} ⊂ ker D̄_{y} of the kernel of the total derivative. Thus, we consider the subspace that is generated by the integral T and its differential consequences. Then, by Theorem 6, the conservation laws [Q dx] for the Toda equation E_{Toda} are in one-to-one correspondence with the Noether symmetries

φ_{L} = □ ∘ E_{T} (Q(x, T)).  (69)

In other words, the Noether symmetries φ_{L} of the Toda equation (55) associated with a generic nondegenerate symmetrizable matrix K are (69) up to the transformation x ↔ y. This statement is an extension of the relation ([86]) between the Noether symmetries and conservation laws for the scalar Liouville equation onto r ≥ 1. We note that our scheme of reasonings is essentially more simple than the straightforward proof for the Liouville equation (ibid).

In the rest part of this section, we generalize the method of the paper [86]. We solve the equation

�_{φ}(L_{Toda}) = 0

directly with respect to φ and thus find out which of the symmetries φ = □(ϕ(x, T)) ∈ sym E_{Toda} preserve the Lagrangian L_{Toda} for the Toda equations. We follow the paper [54].

Detailed analysis of some algebraic aspects for Eq. (55) shows (technically, we calculate the term E_{2}^{0,2}(E_{Toda}) of the Vinogradov’s C-spectral sequence, see [94]) that the following two conditions are equivalent:

�_{φ}(L_{Toda}) = 0 ⟺ E_{u}(�_{φ}(L_{Toda})) = 0,

i.e., a density is a total direvgence iff its variation is zero. We note that this question was not discussed in [86]. Further on, we get

Lemma 19. The relation E_{u}(�_{φ}(L_{Toda})) = 0 is equivalent to the condition

−E_{u} (D_{y}(T) · ϕ(x, T, …, T_{m})) = 0,

SYMMETRIES AND CONSERVATION LAWS IN INTEGRABLE MODELS 43

where φ = □(ϕ) and the total derivative D_{y} is not restricted onto E_{Toda}. The proof of Lemma 19 is based on multiple use of the relation

∑_{j=1}^{r} κ_{ij}Δ^{j} = a_{i}.

Lemma 20. The highest order m of the derivative T_{m} in the set of ϕ’s arguments is even: m = 2μ, and ϕ = ϕ(x, T, …, T_{2μ}).

Proof. Put G_{i} = \frac{δ}{δu^{i}} (�_{φ}(L_{Toda})) and by u_{(k,l)}^{j} denote the derivative D_{x}^{k} ∘ D_{y}^{l}(u^{j}). Then we have

\frac{∂G_{i}}{∂u_{(m+4,1)}^{j}} = − [(−1)^{3} + (−1)^{m+2}] · a_{i} a_{j} · \frac{∂ϕ}{∂T_{m}} = 0,

whence (−1)^{3} + (−1)^{m+2} = 0 and thus m is even. □ Lemma 21. The derivatives

\frac{∂^{2}ϕ}{∂T_{m}∂T_{l}}

vanish for all l such that μ < l ≤ m. Proof. It suffices to check that

\frac{∂G_{i}}{∂u_{2m+4}^{j}} , \frac{∂G_{i}}{∂u_{2m+2}^{j}} , …, \frac{∂G_{i}}{∂u_{m+6}^{j}}

are equal to 0. This can be done straighforwardly. □

Still, the derivative

\frac{∂^{2}ϕ}{∂T_{μ}∂T_{2μ}}

is, in general, nontrivial owing to Lemma 22. The identity

E_{u}(−D_{y}(T) · E_{T} (P (x, T, …, T_{μ}))) ≡ 0

holds for any function P. Proof. Indeed, we have

D_{y}(T)·E_{T} (P) dx∧dy = ⟨D_{y}(T), ℓ_{P}^{(T)}^{∗}(1)⟩ = ⟨ℓ_{P}^{(T)}(D_{y}(T)), 1⟩+d_{h}γ =

⟨�_{D_{y}(T)}^{(T)}P (x, T, …, T_{μ}), 1⟩ + d_{h}γ = ⟨D_{y}(P), 1⟩ + d_{h}γ ∈ ker E_{u}, where the coupling ⟨, ⟩ takes values in the horizontal 2-forms ω = f · dx∧dy, d_{h} = ∑_{i} dx^{i}⊗D_{i} is the horizontal differential, d_{h}γ ∈ ker E_{u} is an exact form, and both the evolutionary derivation and the linearizations are evaluated with respect to T. □

Proposition 23. A symmetry φ = □(ϕ(x, T, …, T_{m})) ∈ ker ℓ̄_{F} is a Noether symmetry of Toda’s equation (55) iff the following two conditions hold:

(1) m = 2μ, μ ≥ 0, and

(2) the function ϕ = E_{T} (Q(x, T, …, T_{μ})) ∈ im E_{T} lies in the image of the Euler operator with respect to T, where Q is an arbitrary smooth function.

Proof. We show that ϕ ∈ im E_{T} by induction. Choose P = P (m; x, T, …, T_{μ}) such that

\frac{∂^{2}P (m)}{∂T_{μ}^{2}} = (−1)^{μ} · \frac{∂ϕ}{∂T_{m}}  (70)

and put ϕ̃ =^{def} ϕ − E_{T} (P (m)), then we get

\frac{∂ϕ̃}{∂T_{m}} ≡ 0.

By Lemma 20, the equation

E_{u}(D_{y}(T) · ϕ̃(x, T, …, T_{m−2})) = 0

holds. By using Lemma 21, we choose P (m) in accordance with (70) and apply Lemma 22. Therefore, we decrease the order m = 2μ to 0 with the step 2 inductively. Finally, we obtain

ϕ = E_{T} (∑_{i=0}^{μ} P (2i; x, T, …, T_{i})).

The proof is complete. □

Thus, we have obtained another description of the Noether symmetries’ class (69) for the Toda equation assigned to a generic matrix K.

3. Recursion operators for the Toda equation

In this section, we construct a continuum of the recustion operators, which are either local or nonlocal with respect to D_{x}, for the symmetry algebra of the Toda equation. Although the structure of the symmetry algebra itself is known, see Eq. (65), presence of the recursion operators gives us additional information about the Toda equation and permits to establish the relation between E_{Toda} and other mathematical physics equations.

The explicit method by J. Krasil’shchik that allows construction of the recursion operators for symmetry algebras of differential equations was briefly described in the Introduction on pages 15–18. The theorem below contains the result of application of this method to the Toda equation (55) associated with a nondegenerate symmetrizable (r × r)- matrix.

Theorem 24 ([57]). (1) Equation (55) admits a continuum of local recursion operators R: sym E_{Toda} → sym E_{Toda}, which are

R = □ ∘ ∑_{i,j}f_{ij}(x, Ω) · D̄_{x}^{j} ∘ ℓ_{Ω^{i}}.

Here f_{ij} are arbitrary smooth functions and the linearizations ℓ_{Ω^{i}} with respect to the integrals Ω^{i} for the Toda equation are

ℓ_{Ω^{i}} = (…, \underbrace{∑_{σ} \frac{∂Ω^{i}}{∂u_{σ}^{k}} · D̄_{σ}}_{kth component}, …).  (71)

(2) There is a continuum of nonlocal recursion operators for Eq. (55), which are constructed in the following way. Assign the nonlocal variables s^{i} to the integrals Ω^{i} by the compatible differentiation rules s_{x}^{i} = Ω^{i} and s_{y}^{i} = 0. The linearizations ℓ_{s^{i}} are defined by the formulas

ℓ_{s^{i}} = D̄_{x}^{−1} ∘ ℓ_{Ω^{i}},

where ℓ_{Ω^{i}} are calculated by Eq. (71). Then the required recursion operators are

R = □ ∘ ∑_{i}f_{i}(x, s, Ω) · D̄_{x}^{−1} ∘ ℓ_{Ω^{i}},

where the functions f_{i} are arbitrary. In general, these operators do not preserve the locality of elements (65) of the symmetry algebra sym E_{Toda}.

Proof. First, enlarge the set of the dependent variables u_{σ}^{j} by introducing the nonlocals s^{i} such that their derivatives are

s_{x}^{i} = Ω^{i},  s_{y} = 0.  (72)

(In fact, any definition of s_{y}^{i} is allowed if it is compatible with the condition s_{xy}^{i} = s_{yx}^{i} = 0). Then, extend the total derivatives:

D̃_{x} = D̄_{x} + ∑_{i} Ω^{i} \frac{∂}{∂s^{i}} ,  D̃_{y} = D̄_{y},

such that [D̃_{x}, D̃_{y}] = 0. The Cartan’s flat connection is now defined on the equation

Ẽ^{∞} = {D̃_{x}^{k+1}(s^{i}) = D̄_{x}^{k}(Ω^{i}), D̃_{x}^{k} ∘ D̃_{y}(s^{i}) = 0, k ≥ 0;

D̄_{σ}(F) = 0, |σ| ≥ 0}.

The Cartan’s generating 1-forms

ω_{Toda} ∈ C^{∞}(Ẽ^{∞}) ⊗ CΛ^{1}(Ẽ^{∞})

for the recursion operators R_{Toda} satisfy the determining equation

ℓ̃_{Toda}^{[1]}(ω_{Toda}) = 0,

where ℓ̃_{Toda}^{[1]} is the restriction of the linearization of Eq. (55) onto H_{C}^{1,0}(Ẽ). From the factorization

ℓ̄_{Toda} ∘ □ = D̄_{x} ∘ □ ∘ D̄_{y}

in Eq. (64) it follows that any Cartan’s 1-form

ω_{Toda} = □(∑_{i≥0} f_{i}(x, s, …, D̃_{x}^{k_{i}}(s^{j})) · d_{C}(D̃_{x}^{i}(s^{l})))  (73)

is an element of the kernel ker ℓ̃_{Toda}^{[1]}, whence we get the statement of the theorem. In particular, suppose that each f_{i} does not depend on the nonlocal variables s. Then the resulting recursion operator is local. □

Example 13. Consider the integral T defined in Eq. (58). Its linearization is

ℓ_{T} = (…, \underbrace{∑_{j=1}^{r} κ_{ij}u_{x}^{j} · D̄_{x} − a_{i} · D̄_{x}^{2}}_{ith component}, …) = □^{∗} ∘ D_{x} ∘ κ̂. (74)

Now, introduce the nonlocal variable s: we set s_{x} = T and s_{y} = 1. The compatibility condition for the variable s is

s_{xy} = 0.  (75)

Finally, we construct the recursion operator

R_{Toda} = □ ∘ D̄_{x}^{−1} ∘ ℓ_{T} .

Apply the operator R_{Toda} to the translation u_{x} ∈ sym E_{Toda}. We obtain the symmetry sequence

φ_{k} = □(ϕ_{k−1})

that corresponds to the sequence of functions

ϕ_{−1} = 1, ϕ_{0} = s_{1}, ϕ_{1} = −βs_{3} + \frac{3}{2}s_{1}^{2},

ϕ_{2} = β^{2}s_{5} − \frac{5}{2}βs_{2}^{2} − 5βs_{1}s_{3} + \frac{5}{2}s_{1}^{3},

etc. In Chapter 2 we shall investigate the properties of this recursion operator R_{Toda} together with the properties of the symmetry sequence

𝔄 = {φ_{k} ≡ R_{Toda}^{k}(φ_{0}), φ_{0} = u⃗_{x}}

in detailes. Meanwhile, we claim that the symmetries φ_{k} of the Toda equation, which are obtained by multiple action of R_{Toda} to the translation φ_{0} = u⃗_{x}, are local, Hamiltonian, and commute with each other. Also, in the next section we shall establish the relation between the symmetry sequence 𝔄, the Korteweg–de Vries equations (9) and (16), and recursion operators (18) for the latter equations.

Chapter 2. The Korteweg–de Vries hierarchies and the Toda equations

In this chapter, we construct the commutative Hamiltonian hierarchy 𝔄 of multi–component analogs for the potential modified Korteweg–de Vries equation. This hierarchy is identified with the commutative Lie subalgebra in sym E_{Toda} composed by certain Noether’s symmetries. Also, we discuss some aspects of the Hamiltonian formalism for the Toda equations themselves and establish a link between the hierarchy 𝔄 and the higher Korteweg–de Vries equations for Eq. (9). The exposition follows the papers [51, 60].

We start our study of the relation between the Toda equation (55) and classical equations (9) and (16) of mathematical physics, the Korteweg–de Vries equations, with the following

Example 14. Consider the hyperbolic Liouville equation

E_{Liou} = {u_{xy} − exp(2u) = 0}.  (76)

The minimal integral, see Eq. (58), for the latter equation is ([99, 100])

T = u_{1}^{2} − u_{2}, D̄_{y}(T) = 0.  (77)

Introduce the nonlocal variable s such that

s_{x} = T,  s_{y} = 1,  (78)

and put

ϑ ≡ 2u_{1}.  (79)

Then, consider the symmetry

φ = (u_{1} + \frac{1}{2} D̄_{x})(T)

of the Liouville equation and calculate the evolution of the variables u, ϑ, T, and s along this symmetry. We get

u_{t} = −\frac{1}{2}u_{3} + u_{1}^{3}  (potential mKdV)  (80)
T_{t} = −\frac{1}{2}T_{3} + 3T T_{1}  (KdV)  (81)
s_{t} = −\frac{1}{2}s_{3} + \frac{3}{2}s_{1}^{2}  (potential KdV).  (82)

The Miura transformation (see [76, 84]) acquires the form

ϑ_{1} = ∓2T ∓ \frac{1}{2}ϑ^{2},  ϑ_{t} = ±T_{2} − (ϑT_{1} + ϑ_{1}T)  (83a)
T = ∓\frac{1}{2}ϑ_{1} − \frac{1}{4}ϑ^{2}.  (83b)

One can treat relations (83) as Bäcklund transformation between the Korteweg–de Vries equation, see Eq. (81), and the equation

ϑ_{t} = −\frac{1}{2}ϑ_{3} + \frac{3}{4}ϑ^{2}ϑ_{1}  (modified KdV).  (84)

The signs ‘±’ and ‘∓’ in Eq. (83) are induced by the symmetry ϑ ↦ −ϑ of equation (84). This discrete transformation provides Bäcklund autotransformation ([10, 95]) for Eq. (81).

The recursion operator

R_{Liou} = D_{x}^{2} − 2u_{1} + D_{x}^{−1}u_{1} D_{x}

is common for both equations (76) and (80), see [48, 40]. This operator generates the commutative Lie subalgebra

𝔄_{Liou} = span_{ℝ}⟨φ_{k} = R_{Liou}^{k}(φ_{0}), φ_{0} = u_{1}⟩

of local higher symmetries of the potential modified Korteweg–de Vries equation, see Eq. (80).

Now, to the Liouville equation we assign the variable 𝔳 such that

𝔳_{x} = exp(2u),  (85a)
E_{𝔳} = {𝔳_{y} = 𝔳^{2}}  (85b)

and therefore the compatibility condition 𝔳_{xy} = 𝔳_{yx} holds. Thence, the equation E_{Liou} is represented in the evolutionary form:

u_{t_{−1}} = φ_{−1} ≡ 𝔳,  (86)

where the variable t_{−1} ≡ y is the parameter and φ_{−1} is the shadow of the nonlocal symmetry

˜�_{φ_{−1},a_{−1}} ≡ ˜�_{φ_{−1}} + a_{−1} · \frac{∂}{∂𝔳}

such that a_{−1} = 𝔳^{2}. The solution to Eq. (85b) is 𝔳 = −(y + X (x))^{−1}. Apply transformation (60) of the form ỹ = Y(y) to the ”time” y. Then we obtain the potential

𝔳 = −\frac{Y′(y)}{(X (x) + Y(y))}

for the genreral solution u = \frac{1}{2} log 𝔳_{x} of the Liouville equation (see [69, 85]):

u = \frac{1}{2} log[\frac{X′(x) · Y′(y)}{(X (x) + Y(x))^{2}} ].  (87)

We recall that functional (77) is continuous on formal divergences u → ±∞ of solution (87), see the paper [46] and references therein. We estimate the scheme of constructing the general solution for the Liouville equation by using the potential 𝔳 to be really laconic and productive. A similar approach was used in [10] to derive the Cole–Hopf transformation for the Burgers equation, see Eq. (203) on page 121.

Solution (87) of Eq. (76) is the mapping

τ: {X_{y} = 0, Y_{x} = 0} → E_{Liou}.

Consider the evolutionary vector field �_{u_{t}} defined in Eq. (80). This field can be lifted onto the inverse image of the covering τ. Hence we obtain the equation Y_{t} = 0 and the Krichever–Novikov equation

X_{t} = −2X_{3} + 3X_{2}^{2}/X_{1} = −2X_{1} · {X , x},  (88)

where {X , x} is the Schwartz derivative. Also, the evolution equation

𝔳_{t} = a_{1} ≡ 𝔳_{3} − \frac{3}{2}𝔳_{2}^{2}𝔳_{1}^{−1}

holds.

The evolution equations that appear in Example 14 are systemized as shown in the diagram below:

The Krichever-Novikov Eq. (20)

The Krichever-Novikov Eq. (88), Y_{t} = 0

Eq.↓^{(19)}  ^{Eq.}↓^{(87)}

Potential KdV Eq. (82)

⟶_{(72)}^{Eq.}

Petential mKdV Eq. (80)

Eq.↓^{(78)}  ^{Eq.}↓^{(79)}
KdV Eq. (81)  ⟵_{(83)}^{Eq.}  Modified KdV Eq. (84) .

4. Analogs of the potential modified Korteweg–de Vries equation

4.1. Constructing the hierarchy 𝔄. First, we formulate an important property of the integrals Ω^{i} ∈ ker D_{y}|_{E} for a Liouvillean type equation E. Namely, we specify the evolution of these integrals along symmetries φ ∈ sym E of the equation E.

Lemma 25 ([100]). Consider a symmetry field �_{φ} of a Liouvillean type equation E. Then the evolution �_{φ}(Ω^{i}) of an arbitrary integral Ω^{i} ∈ ker D̄_{y} for E is an element of ker D̄_{y} again.

Proof. Indeed, we have

D̄_{y}(�_{φ}(Ω^{i})) = �_{φ}(D̄_{y}(Ω^{i})) = �_{φ}(0) = 0.
□

Example 15. Consider a symmetry φ = □(ϕ(x, T)) of the Toda equation, see Eq. (65) on page 39. Then the relation

Ṫ_{ϕ} ≡ �_{□(ϕ)}(T) = (−β D̄_{x}^{3} + T D̄_{x} + D_{x} ∘ T )(ϕ)  (89)

holds. Here

β ≡ ∑_{i=1}^{r} a_{i} · Δ^{i}, □ = u_{x} + Δ⃗ D̄_{x}, and Δ^{i} = ∑_{j} k^{ij}.

Also, consider the nonlocal variable s that was defined in Example 13 by the equations s_{x} = T and s_{y} = 1. Then the evolution s_{t} of the nonlocality is described by the formula

�φ⃗(s) = D_{x}^{−1} ∘ �φ⃗(T).

Suppose that the sequence φ⃗_{0}, φ⃗_{1}, φ⃗_{2} of symmetries of the Toda equation is assigned to the specially chosen set of the functions ϕ in formula (65). We set ϕ_{−1} = 1 and obtain the symmetry φ_{0} = □(ϕ_{−1}). Then, we calculate the evolution Ṫ_{ϕ_{−1}} that corresponds to this symmetry. Now we equal the next function ϕ_{0} to the evolution of the potential s. Then, we obtain the function ϕ_{1} and the symmetries φ_{1}, φ_{2} in a similar way. The result is shown in the next diagram:

φ⃗_{2} ←^{□} ϕ_{1} = −βs_{3} + \frac{3}{2}s_{1}^{2} ←^{D_{x}^{−1}} Ṫ_{ϕ_{0}} = −βT_{3} + 3T T_{1}
[figure: R_{Toda}; ℓ_{T}; R_{KdV}]
φ⃗_{1} = □(s_{1}) ← ^{□}  ϕ_{0} = s_{1} ← ^{D_{x}^{−1}} Ṫ_{ϕ_{−1}} = T_{1}
[figure: R_{Toda}; ℓ_{T}]
φ⃗_{0} = u_{x} ←  ^{□} ϕ_{−1} = 1

(90)

The following evolution equations are met in diagram (90): the Korteweg–de Vries equation is

E_{KdV} = {T_{t} = −βT_{3} + 3T · T_{1}}  (91)

(we recall that its recursion operator is (18a) on page 16); then, we also get the potential Korteweg–de Vries equation

E_{pKdV} = \{s_{t} = −βs_{3} + \frac{3}{2}s_{1}^{2}\} ,  (92)

and the equation

E_{pmKdV} = {u_{t} = □(T (u_{1}, u_{2})}.  (93)

Suppose K = ∥2∥ is the Cartan matrix of the Lie algebra 𝔤 = 𝔰𝔩_{2}(ℂ). Then the Toda equation (55) associated with this algebra is the hyperbolic Liouville equation (76), and equation (93) is nothing else than the scalar potential modified Korteweg–de Vries equation (82). Further on, suppose that the matrix K in Eq. (55) is an arbitrary nondegenerate symmetrizable (r × r)-matrix, not necessarily the Cartan matrix of a semisimple Lie algebra 𝔤 of rank r. Then we get the r-component system of the third-order evolution equations with the cubic nonlinearity. In local coordinates, this system is

u_{t}^{i} = \frac{1}{2} ∑_{p,q=1}^{r} a_{p} · \{k_{pq}u_{x}^{i}u_{x}^{p}u_{x}^{q} + 2(Δ^{i}k_{pq} − δ_{i,q})u_{xx}^{p}u_{x}^{q} − 2Δ^{i}u_{xxx}^{p}\} ,

where 1 ≤ i ≤ r; we recall that

Δ^{i} = ∑_{j} k^{ij} and a_{p}k_{pq} = a_{q}k_{qp}.

The analogs of the potential modified Korteweg–de Vries equation for r = 2. In this subsection, we consider evolution systems (93) in case r = 2 and the matrix K is symmetric.

Example 16. Let K be the symmetric matrix

K = (\begin{matrix}2 & λ \\ λ & 2 \end{matrix}) ,

thence,

Δ⃗ = \frac{1}{λ + 2}\binom{1}{1}.

Introduce new dependent variables

u = u^{1} + u^{2}, v = (λ + 2)(u^{1} − u^{2});

the inverse transformation is defined by the formulas

u^{1} = \frac{1}{2}u + \frac{1}{2(λ + 2)}v, u^{2} = \frac{1}{2}u − \frac{1}{2(λ + 2)}v.

Then, the normal form of Eq. (93) is

\begin{aligned}u_{t} & = −\frac{2}{λ + 2}u_{3} + \frac{2 − λ}{(λ + 2)^{3}} v_{1}v_{2} + \frac{λ + 2}{4}u_{1}^{3} + \frac{2 − λ}{4(λ + 2)^{2}} u_{1}v_{1}^{2}, \\ v_{t} & = −u_{2}v_{1} + \frac{λ + 2}{4}u_{1}^{2}v_{1} + \frac{2 − λ}{4(λ + 2)^{2}} v_{1}^{3}. \end{aligned} (94)

Further on, make the scaling transformation

t = (λ + 2)^{2} · t̃,  u = (λ + 2)^{−1} · ũ,  v = ṽ,  (95)

preserving the same notation x, t, u, and v.

Proposition 26. In coordinates (95), equations (94) become linear in λ and acquire the form

\begin{aligned}u_{t} & = −2u_{3} + 2v_{1}v_{2} + \frac{1}{2}u_{1}^{3} + \frac{1}{2}u_{1}v_{1}^{2} + λ(−v_{1}v_{2} + \frac{1}{4}u_{1}^{3} − \frac{1}{4}u_{1}v_{1}^{2}), \\ v_{t} & = −2u_{2}v_{1} + \frac{1}{2}u_{1}^{2}v_{1} + \frac{1}{2}v_{1}^{3} + λ(−u_{2}v_{1} + \frac{1}{4}u_{1}^{2}v_{1} − \frac{1}{4}v_{1}^{3}). \end{aligned} (96)

Remark 7. Consider the vector coefficient of λ in Eq. (96). Then this vector-valued function is not a symmetry of the flow that stands at λ^{0} (and vice versa). Namely, the following statements hold.

Proposition 27. (1) Consider the flow

u_{t} = −2u_{3} + 2v_{1}v_{2} + \frac{1}{2}u_{1}^{3} + \frac{1}{2}u_{1}v_{1}^{2},
v_{t} = −2u_{2}v_{1} + \frac{1}{2}u_{1}^{2}v_{1} + \frac{1}{2}v_{1}^{3}

at λ^{0} in Eq. (96). Then its symmetries

(\begin{matrix}φ^{1}, \\ φ^{2} \end{matrix}) (t, x, u, v, u_{1}, v_{1}, u_{2}, v_{2}, u_{3}, v_{3})

of order ≤ 3 are generated by

(\begin{matrix}tu_{t} + \frac{1}{3}xu_{x} \\ tv_{t} + \frac{1}{3}xv_{x} \end{matrix}) ,  (\begin{matrix}u_{t} \\ v_{t} \end{matrix}) ,  (\begin{matrix}u_{x} \\ v_{x} \end{matrix}) ,  (\begin{matrix}1 \\ 0 \end{matrix}) ,  (\begin{matrix}0 \\ 1 \end{matrix}) .

These generators are the scaling symmetry, two translations, and two shifts, respectively. The commutator with the scaling symmetry maps the translations along t and x to themselves.

(2) Consider the flow

u_{t} = −v_{1}v_{2} + \frac{1}{4}u_{1}^{3} − \frac{1}{4}u_{1}v_{1}^{2},
v_{t} = −u_{2}v_{1} + \frac{1}{4}u_{1}^{2}v_{1} − \frac{1}{4}v_{1}^{3}

at λ^{1} in Eq. (96). Then the symmetries

(\begin{matrix}φ^{1} \\ φ^{2} \end{matrix}) (t, x, u, v, u_{1}, v_{1}, u_{2}, v_{2}, u_{3}, v_{3})

of order ≤ 3 of this flow are generated by the sections

(\begin{matrix}tu_{t} + \frac{1}{3}xu_{x} \\ tv_{t} + \frac{1}{3}xv_{x} \end{matrix}) ,  (\begin{matrix}u_{t} \\ v_{t} \end{matrix}) ,  (\begin{matrix}u_{x} \\ v_{x} \end{matrix}) ,  (\begin{matrix}1 \\ 0 \end{matrix}) ,  (\begin{matrix}0 \\ 1 \end{matrix}) ,

which are the scaling, the translations, and the shifts, respectively.

The conservation laws for system (96) in the normal form are few.

Proposition 28. Suppose λ is generic (see Remark 8 below). Then there is a unique generating section

(\begin{matrix}ψ^{1} \\ ψ^{2} \end{matrix}) (t, x, u, v, u_{1}, v_{1}, u_{2}, v_{2}) = (\begin{matrix}u_{2} \\ \frac{λ−2}{λ+2} v_{2} \end{matrix})

of order ≤ 2 of a conservation law for the flow in Eq. (96); this section corresponds to the conserved density

H = u_{x}^{2} + \frac{λ − 2}{λ + 2} v_{x}^{2}.

In Sec. 7 we shall demonstrate that equation (94) is Hamiltonian with respect to the operator A_{1} = K^{−1} · D̄_{x}^{−1} (recall that the matrix K is assumed symmetric and therefore a_{i} = 1). Now we get the Hamiltonian representation of system (96) by using the transformation rule for Hamiltonian operators with respect to transformations of the dependent variables.

Lemma 29. Consider a Hamiltonian equation

u_{t} = A(E_{u}(H[u])).

Let ũ = Qu be a nondegenerate transformation of the dependent variables. Then the equation

ũ = Ã(Eũ(H[ũ]))

holds for the Hamiltonian operator

Ã = Q · A ·^{t}Q.

Example 17. Consider Eq. (96). Then we get the diagonal operator

(\begin{matrix}u \\ v \end{matrix})_{t} = 2 (\begin{matrix}λ + 2 & 0 \\ 0 & \frac{(λ+2)^{2}}{2−λ} \end{matrix} ) · D_{x}^{−1}(E_{(u,v)}(T^{2} dx)),

where the density h_{1} of the Hamiltonian T^{2} dx for the Korteweg–de Vries equation acquires the following form in the coordinates u, v:

h_{1} = \frac{1}{16}((λ − 2)v_{x}^{2} − (λ + 2)u_{x}^{2} + 4(λ + 2)u_{xx})^{2}.

Proposition 30. Again, suppose λ is generic (see Remark 8). The symmetries

(\begin{matrix}φ^{1} \\ φ^{2} \end{matrix}) (t, x, u, v, u_{1}, v_{1}, u_{2}, v_{2})

of order ≤ 2 of equation (96) are generated by the translation and the shifts, which are

(\begin{matrix}u_{x} \\ v_{x} \end{matrix}) ,  (\begin{matrix}1 \\ 0 \end{matrix}) ,  (\begin{matrix}0 \\ 1 \end{matrix}) ,  (97)

respectively.

Remark 8. We emphasize that the symmetry algebra (and even the classical symmetry algebra) of system (96) depends on the initial matrix K essentially. The cases λ = −1 (the matrix K corresponds to the Lie algebra A_{2}) and λ = ±2 (the matrix K is degenerate) are special. In what follows, we analyse them separately.

First, let λ = −1, that is, K is the Cartan matrix for the Lie algebra A_{2}. Then substitution (95) is

u = u^{1} + u^{2}, v = u^{1} − u^{2}.

After routine transformations, we obtain

\begin{aligned}u_{t} & = −u_{3} + \frac{3}{2}v_{1}v_{2} + \frac{1}{8}u_{1}^{3} + \frac{3}{8}u_{1}v_{1}^{2}, \\ v_{t} & =  − \frac{1}{2}u_{2}v_{1} + \frac{1}{8}u_{1}^{2}v_{1} + \frac{3}{8}v_{1}^{3}. \end{aligned}  (98)

Proposition 31. (1) The symmetries

(\begin{matrix}φ^{1} \\ φ^{2} \end{matrix}) (t, x, u, v, u_{1}, v_{1}, u_{2}, v_{2})

of order ≤ 2 of flow (98) are generated by the sections

(\begin{matrix}v \\ −\frac{1}{3}u + \frac{2}{3} log v_{x} \end{matrix}) ,  (\begin{matrix}u_{x} \\ v_{x} \end{matrix}) ,  (\begin{matrix}1 \\ 0 \end{matrix}) ,  (\begin{matrix}0 \\ 1 \end{matrix}) .

The first symmetry is nonpolynomial. The other two are the translation and the shift, respectively. The nonpolynomial symmetry flow can be also represented in the form

u_{xt}^{2} = exp(u + 3u_{tt}).

(2) The generating sections

(\begin{matrix}ψ^{1} \\ ψ^{2} \end{matrix}) (t, x, u, v, u_{1}, v_{1}, u_{2}, v_{2})

of order ≤ 2 of conservation laws for Eq. (98) are

(\begin{matrix}u_{2} \\ 3v_{2} \end{matrix}) , (\begin{matrix}Ψ(u, v, v_{x}) \\ 2∂Ψ/∂v + 2v_{2}v_{1}^{−1} ∂Ψ/∂v_{1} − u_{1} ∂Ψ/∂v_{1} \end{matrix}) ,

where the function Ψ satisfies the equation

\frac{∂Ψ}{∂u} + \frac{1}{2}v_{1} \frac{∂Ψ}{∂v_{1}} − \frac{1}{2}Ψ = 0.

Now, for the first time within this paper, suppose that the matrix K is degenerate. Quite untrivially, this degeneracy can occur in two distinct ways.

First, assume that λ = 2 and K = ( \begin{matrix}2 & 2 \\ 2 & 2 \end{matrix} ). Then system (94) is in triangle form^{3}:

\{ \begin{matrix}u_{t} & = −\frac{1}{2}u_{3} + u_{1}^{3}, \\ v_{t} & = −u_{2}v_{1} + u_{1}^{2}v_{1}. \end{matrix}  (99)

It consists of the potential modified Korteweg–de Vries equation and the auxiliary dispersionless component. We note that the variable v admits an arbitrary shift. Indeed, the section ^{t}(0, f(v)) is a symmetry of Eq. (99) if f is arbitrary. In addition, there are two more symmetries. They are the translation and the shift of u, see Eq. (97).

Proposition 32. If λ = 2, then the generating sections

(\begin{matrix}ψ^{1} \\ ψ^{2} \end{matrix}) (t, x, u, v, u_{1}, v_{1}, u_{2}, v_{2})

of order ≤ 2 of conservation laws for the flow in Eq. (99) are

(\begin{matrix}u_{2} \\ 0 \end{matrix}) ,  (\begin{matrix}exp(u) \\ 0 \end{matrix}) ,  (\begin{matrix}exp(−u) \\ 0 \end{matrix}) .

The corresponding conserved densities are

ρ_{1} = −\frac{1}{2}u_{x}^{2}, ρ_{2} = exp(u), ρ_{3} = − exp(−u),

respectively. The last two densities belong to the nonpolynomial conserved currents

η_{2} = exp(u) dx + (2u_{2} − u_{1}^{2}) exp(u) dt,
η_{3} = − exp(−u) dx + (2u_{2} + u_{1}^{2}) exp(−u) dt

φ⃗ = αu⃗_{x} + β(x)1,

3The classical symmetry algebra for the Toda equations associated with this matrix is composed by the sections where the constant α and the function β(x) are arbitrary.

for the potential modified Korteweg–de Vries equation

u_{t} = −2u_{3} + u_{1}^{3}

whose right–hand side is polynomial.

Now, suppose λ = −2, whence K = ( \begin{matrix}2 & −2 \\ −2 & 2 \end{matrix} ). Then equation (96) is

\{ \begin{matrix}u_{t} & = −2u_{3} + 4v_{1}v_{2} + u_{1}v_{1}^{2}, \\ v_{t} & = v_{x}^{3}. \end{matrix}  (100)

Proposition 33. The symmetries of system (96) for λ = −2 are described by Eq. (97). The generating sections

(\begin{matrix}ψ^{1} \\ ψ^{2} \end{matrix}) (t, x, u, v, u_{1}, v_{1}, u_{2}, v_{2})

of order ≤ 2 of conservation laws for the flow in right-hand side of Eq. (100) are, then,

(\begin{matrix}0 \\ Ψ(x, t, v, v_{x}, v_{xx}) \end{matrix}) .

Here the function Ψ is subject to the equation

\frac{∂Ψ}{∂t} − 2v_{1}^{3} \frac{∂Ψ}{∂v} − 3v_{1}^{2} \frac{∂Ψ}{∂x} + v_{1}v_{2}^{2} \frac{∂Ψ}{∂v_{2}} − 6v_{1}v_{2} Ψ = 0.

Factorizations of the recursion operators. Now we return to the problem of constructing the commutative hierarchy 𝔄, which is associated with the initial Toda equation E_{Toda}.

By construction, the symmetries φ_{i} are related by the recursion operator

R_{Toda} = □ ∘ D_{x}^{−1} ∘ ℓ_{T}  (101)

for the Toda equation (55). We met this operator in Example 13 on page 46. We fix^{4} this notation R_{Toda} onwards.

In this subsection, we demonstrate that each symmetry

φ⃗_{k} = □(ϕ_{k−1}) ∈ sym E_{Toda}

defines the next function ϕ_{k} explicitly by the relation ϕ_{k} = �φ⃗_{k}(s), and therefore, φ⃗_{k+1} = □(ϕ_{k}). We also prove that the sections ϕ_{k} compose the hierarchy of the potential Korteweg–de Vries equation (92).

First, we note that the functions ϕ_{−1}, ϕ_{0}, and ϕ_{1} are mapped successfully one to another by the recursion operator ([96])

R_{pKdV} = −β D_{x}^{2} + 2s_{1} − D_{x}^{−1} ∘ s_{2}  (102)

for equation (92). Recall that this operator was obtained in Introduction while illustrating the Cartan generating forms method ([62]) by I. S. Krasil’shchik.

4As we have already noted in Chapter 1, we treat the Toda equations, as well as the structures on them, up to the symmetry x ↔ y.

Lemma 34. The following two decompositions of the recursion operators hold:

R_{Toda} = □ ∘ ℓ_{s}, R_{pKdV} = ℓ_{s} ∘ □, (103)

where ℓ_{s} = D_{x}^{−1} ∘ ℓ_{T} by Eq. (23) and the linearization ℓ_{T} is defined in Eq. (74).

Proof. The first decomposition holds by construction. To establish another one, we use the relations

ℓ_{T} ∘ (u_{x} + Δ⃗D_{x}) =

= \frac{1}{2} ∑_{i,j} κ_{ij}u_{1}^{i} D_{x} ∘ u_{1}^{j} + \frac{1}{2} ∑_{i} a_{i}\{(∑_{j} k_{ij}u_{1}^{i})D_{x} ∘ u_{1}^{i} − 2D_{x}^{2} ∘ u_{1}^{i}\} +

+ \frac{1}{2} ∑_{i,j} κ_{ij}u_{1}^{i}Δ^{j}D_{x}^{2} + \frac{1}{2} ∑_{i} a_{i}\{(∑_{j} k_{ij}u_{1}^{j})Δ^{i}D_{x}^{2} − 2Δ^{i}D_{x}^{3}\} =

= − ∑_{i} a_{i}Δ^{i} D_{x}^{3} + \frac{1}{2} ∑_{i,j} a_{i}\{k_{ij}(u_{1}^{i}u_{2}^{j} + u_{2}^{i}u_{1}^{j}) − 2u_{3}^{i}\} +

+ \frac{1}{2} ∑_{i,j} a_{i}\{k_{ij}(u_{1}^{i}u_{1}^{j} + u_{1}^{i}u_{1}^{j}) − 4u_{3}^{i}\} D_{x} +

+ \frac{1}{2}\{∑_{i} a_{i} · (−2u_{1}^{i})D_{x}^{2} + ∑_{i,j} κ_{ij}Δ^{j}u_{1}^{i}D_{x}^{2} + ∑_{i,j} κ_{ij}Δ^{i}u_{1}^{j}D_{x}^{2}\} =

= −βD_{x}^{3} + T_{1} + 2T D_{x},

whence follows the second decomposition. □

We draw attention to the fact that the representation of the scalar operator R_{pKdV} in the form of the product of the vector–valued operator □ and the row ℓ_{s} of length r seems to have been unnoticed in the literature.

Now we identify infinitesimal symmetries of differential equations with autonomous evolution equations and, by using Lemma 34, we extend diagram (90) infinitely upwards. The result is displayed in the

diagram …  …

[figure: ℓ_{s}]
u_{t_{2}} = φ⃗_{2}^{↑^{R_{Toda}}}← ^{□} s_{t_{1}} = ϕ_{1} = −βs_{3} + \frac{3}{2}s_{1}^{2}^{↑^{R_{pKdV}}}
[figure: ℓ_{s}]
u_{t_{1}} = φ⃗_{1}^{↑^{R_{Toda}}}← ^{□} s_{t_{0}} = ϕ_{0} = s_{1}^{↑^{\begin{matrix}R_{pKdV}= \\ −βD_{x}^{2}+2s_{1}−D_{x}^{−1}∘s_{2} \end{matrix}}}
[figure: ℓ_{s}]
\underbrace{u_{t_{0}} = φ⃗_{0} = u_{x}}_{𝔄}^{↑^{R_{Toda}}} ←^{□} \underbrace{s_{t_{−1}} = ϕ_{−1} = 1.}_{𝔅}^{↑^{R_{pKdV}}}

(104)

The right-hand sides ϕ_{k} of the evolution equations s_{t_{k}} = ϕ_{k} are generated by the recursion operator R_{pKdV}, therefore they are higher symmetries of the potential Korteweg–de Vries equation E_{pKdV}.

The times t_{i} in the equations u_{t_{k}} = φ_{k} and s_{t_{k}} = ϕ_{k} within diagram (104) are correlated. Namely, we have

Theorem 35. The evolution �_{φ_{k}}(s) of the nonlocal variable s along the higher symmetry φ_{k} = □(ϕ_{k−1}) of the Toda equation (55) coincides with the evolution ϕ_{k} that is defined by the kth higher analog s_{t_{k}} = R_{pKdV}^{k+1}(1) of the potential Korteweg–de Vries equation (92):

�_{□(ϕ_{k−1})}(s) = ϕ_{k} = R_{pKdV}(ϕ_{k−1}).

Proof. From decompositions (103) it follows that

R_{pKdV}(ϕ_{k}) = D_{x}^{−1} (�_{□(ϕ_{k})}(T)) .
□

Corollary 36. 1. Each shadow φ⃗_{k} in the covering τ_{s}, see Eq. (72), over the Toda equation can be reconstructed up to the true nonlocal symmetry ˜�_{φ⃗_{k},ϕ_{k}} of the Toda equation.

2. All subdiagrams

φ⃗_{k+2} ⟵^{□} ϕ_{k+1}
R_{Toda}↑  ↑^{R_{pKdV}}
φ⃗_{k+1} ⟵^{□} ϕ_{k}

in diagram (104) are commutative: we have R_{Toda}(φ⃗_{k+1}) = □(ϕ_{k+1}), and the relations

R_{Toda} = □ ∘ R_{pKdV} ∘ □^{−1}, R_{pKdV} = □^{−1} ∘ R_{Toda} ∘ □

hold.

Remark 9. In the paper [40], the recursion operators R_{Toda} and R_{pKdV} were noted to be conjugate to each other in the scalar case (76-84).

Proof of the locality for the symmetry sequence 𝔄. All previous reasonings do not imply the locality of the elements ϕ_{k} of the sequence 𝔅 composed by higher symmetries of the potential Korteweg–de Vries equation (92). Indeed, these symmetries lie in the image of the operator D_{x}^{−1}. Therefore, the elements φ_{k+1} = □ (ϕ_{k}) of the sequence 𝔄 are not yet proved to be local, too.

Proposition 37 ([20]). Recursion operator (102) for the potential Korteweg–de Vries equation (92) generates the local in T sequence of the higher symmetries

ϕ_{k} = R_{pKdV}^{k+1}(ϕ_{−1}) = ϕ_{k}(T, …, T_{2k}),

where ϕ_{−1} = 1 is the shift of the dependent variable s by a constant.

There are several ways to prove Proposition 37. The method by I. S. Krasil’shchik ([61]) is based on the weak nonlocality of the recursion operator R_{pKdV} (see also Eq. (142) on page 84).

Corollary 38. The symmetries φ⃗_{k} = □(ϕ_{k−1}) ∈ 𝔄 of the Toda equation (55) are local and depend on the derivatives u_{σ}^{j}, |σ| ≥ 1 for all k ≥ 0.

We also note that in the paper [48] we analysed the case r = 1 and established the locality of the higher symmetries 𝔄 ⊂ sym E_{Liou} directly, leaving apart the discussion upon the properties of the potential equation (92).

4.2. Commutativity of the hierarchy 𝔄. A classical example of evolution equations that admit a commutative symmetry subalgebra is given in the following lemma.

Lemma 39 ([36]). Suppose E is the scalar evolution equation

u_{t} = u_{k} + f(u_{k−2}, …, u)  (105)

such that f is a polynomial. Consider the Lie subalgebra

⟨φ ∈ sym E | φ = φ(u_{σ})⟩ ⊆ sym E

of its symmetries that depend on the variable u and its derivatives only. Then this subalgebra is commutative.

We denote by 𝔅 the minimal Lie subalgebra generated by the symmetries ϕ_{k} of the potential Korteweg–de Vries equation; here k ≥ −1. From Lemma 39 it follows that the algebra 𝔅 is commutative:

{ϕ_{k}, ϕ_{l}} = �_{ϕ_{k}}(ϕ_{l}) − �_{ϕ_{l}}(ϕ_{k}) = 0,

therefore 𝔅 coincides with the linear span of its generators ϕ_{k}:

𝔅 = span_{ℝ}⟨ϕ_{k} | k ≥ −1⟩.

Now we analyse the commutation properties of the symmetries φ = □(ϕ(x, T)) of the Toda equations (55).

Lemma 40. Suppose φ′ = □ (ϕ′(x, T)) and φ′′ = □ (ϕ′′(x, T)). Then the Jacobi bracket {φ′, φ′′}, which was defined in Theorem 1, of the symmetries φ′ and φ′′ is

{φ′, φ′′} = □ (ϕ_{{1,2}}),

where the bracket

ϕ_{{1,2}} = �_{φ′}(ϕ′′) − �_{φ′′}(ϕ′) + D̄_{x}(ϕ′) ϕ′′ − ϕ′ D̄_{x}(ϕ′′)  (106)

on the arguments of the operator □ is induced by the Jacobi bracket {φ′, φ′′}. Moreover,

ϕ_{{1,2}} = ϕ_{{1,2}}(x, T)

by Eq. (89).

Further on, we denote by 𝔄 the minimal Lie algebra generated by φ⃗_{k} for all k ≥ 0.

Theorem 41. The Lie algebra 𝔄 ⊂ sym E_{Toda} is commutative: [𝔄, 𝔄] = 0, and thence

𝔄 = span_{ℝ}⟨φ⃗_{k} | k ≥ 0⟩.

Proof. Commute two symmetries �_{φ_{k_{1}}} and �_{φ_{k_{2}}}, apply the resulting evolutionary vector field to the variable s and take into account the relation between ϕ_{k} and φ_{k}. Hence we get

[�_{φ_{k_{1}}}, �_{φ_{k_{2}}}](s) = �_{φ_{k_{1}}}(ϕ_{k_{2}}) − �_{φ_{k_{2}}}(ϕ_{k_{1}}) =

�_{ϕ_{k_{1}}}^{(s)}(ϕ_{k_{2}}) − �_{ϕ_{k_{2}}}^{(s)}(ϕ_{k_{1}}) = {ϕ_{k_{1}}, ϕ_{k_{2}}} = 0.

Since T = s_{x}, thence

[�_{φ_{k_{1}}}, �_{φ_{k_{2}}}](T) = 0.  (107)

Now estimate the evolution of the integral T by using Lemma 40. First, consider the bracket {φ_{k_{1}}, φ_{k_{2}}} and then calculate Ṫ_{ϕ_{{k_{1},k_{2}}}}. Recall that φ_{k_{1}} = □(ϕ_{k_{1}−1}) and φ_{k_{2}} = □(ϕ_{k_{2}−1}). Therefore,

{φ_{k_{1}}, φ_{k_{2}}} = □(ϕ_{{k_{1},k_{2}}}),

where

ϕ_{{k_{1},k_{2}}} =

�_{ϕ_{k_{1}}}(ϕ_{k_{2}−1}) − �_{ϕ_{k_{2}}}(ϕ_{k_{1}−1}) + D̄_{x}(ϕ_{k_{1}−1}) ϕ_{k_{2}−1} − ϕ_{k_{1}−1} D̄_{x}(ϕ_{k_{2}−1})

owing to Eq. (106). From Example 15 it follows that

�_{{φ_{k_{1}},φ_{k_{2}}}}(T) = �_{□(ϕ_{{k_{1},k_{2}}})}(T) =

= (−β D̄_{x}^{3} + T D̄_{x} + D̄_{x} ∘ T )(ϕ_{{k_{1},k_{2}}}). (108)

Comparing Eq. (107) and Eq. (108), we get

(−β D̄_{x}^{3} + T D̄_{x} + D̄_{x} ∘ T ) ϕ_{{k_{1},k_{2}}} = 0.  (109)

In the left-hand side of Eq. (109) we obtain the operator B̂_{2} in total derivatives whose coefficients are elements of T. We apply this operator to ϕ_{{k_{1},k_{2}}}(T, …, T_{μ(k_{1},k_{2})}) and get 0 in the right-hand side Thence, ϕ_{{k_{1},k_{2}}} = 0 and therefore

{φ_{k_{1}}, φ_{k_{2}}} = □(0) = 0.

The indexes k_{1} and k_{2} are arbitrary, therefore 𝔄 is commutative. □

Proposition 42. Let E_{(0)} = {u_{t_{0}} = φ_{0}(u_{σ})} be an evolution equation. Suppose that the symmetry φ_{k} ∈ sym E_{(0)} is assigned to each k ≥ 0 and assume that φ_{k} is independent of the time t_{0} explicitly: φ_{k} = φ_{k}(u_{σ}). Then the following two statements are equivalent:

(1) The algebra 𝔄 = span_{ℝ}⟨φ_{k} | k ≥ 0⟩ is a commutative Lie algebra: {φ_{k}, φ_{l}} = 0.

(2) The evolutionary vector field �_{φ_{l}} is a symmetry of the autonomous evolution equation E_{(k)} = {u_{t_{k}} = φ_{k}} for each k, l ≥ 0.

Proof. First, identify the evolutionary vector field �_{φ_{k}} with the autonomous evolution equation u_{t_{k}} = φ_{k}. Then, consider the equality

�_{φ_{k}}(φ_{l}) = �_{φ_{l}}(φ_{k}).

In its left-hand side we have

�_{φ_{k}}(φ_{l}) = �_{u_{t_{k}}} (φ_{l}) = (D̄_{t_{k}} − \frac{∂}{∂t_{k}} )(φ_{l}) = D̄_{t_{k}}(φ_{l}),

since φ_{l} does not depend on any time t_{k}. In the right-hand side of the equality we get

�_{φ_{l}}(φ_{k}) = ℓ_{φ_{k}}(φ_{l}).

Therefore, the comuutativity condition {φ_{k}, φ_{l}} = 0 for the symmetries φ_{k} and φ_{l} is equivalent to the determining equation

(D̄_{t_{k}} − ℓ_{φ_{k}})(φ_{l}) = 0.

Thence φ_{l} is a symmetry of the equation E_{(k)} for any k, l ≥ 0. □

Corollary 43. For any k, l ≥ 0, the following propositions hold:

(1) The sections φ⃗_{k} ∈ 𝔄 are not only symmetries of the Toda equation, but also symmetries of all equations E_{(l)} = {u_{t_{l}} = φ⃗_{l}}:

φ⃗_{k} ∈ sym E_{(l)}^{∞}.

(2) The recursion operator R_{Toda} is common for the whole tower of the evolution equations E_{(l)}:

R_{Toda} ∈ Rec E_{(l)}.

In particular,

R_{Toda} = R_{pmKdV}.

Remark 10. In the paper [40], the scalar potential modified Korteweg– de Vries equation was considered in the gauge u_{t} = u_{3} + u_{1}^{3}. In this case, this equation shares the recursion operator with the sine-Gordon equation u_{xy} = sin u.

Remark 11. By using Corollary 38, we deduce that the section φ′_{−1} = const is a central extension of the commutative Lie subalgebra 𝔄 ⊂ sym E_{(l)}^{∞} of symmetries of the evolution equation E_{(l)} for any l ≥ 0, although const ∉ sym E_{Toda}.

Remark 12. By using formula (106), we establish a curious property of the Korteweg–de Vries equation (91). Recall that the proof of Theorem 41 is based on the equality ϕ_{{k_{1},k_{2}}} = 0 that holds for all elements ϕ_{k_{1}} and ϕ_{k_{2}} of the hierarchy 𝔅. Consider the first Hamiltonian structure B̂_{1} = D_{x} for Eq. (91). We know that the Hamiltonians of the higher Korteweg–de Vries equations are in involution with respect to this structure. Thence we have

⟨E_{T} (H_{i}), B̂_{1} ∘ E_{T} (H_{j})⟩ = 0 ⟺ ϕ_{i} · D_{x}(ϕ_{j}) ∈ im D_{x}.

Now we see that the last two summands in Eq. (106) are of this form. Therefore the sum

�_{ϕ_{k_{1}+1}}(ϕ_{k_{2}}) − �_{ϕ_{k_{2}+1}}(ϕ_{k_{1}}) ∈ im D_{x}

is always a total derivative. The feature we observe is that each of these summands above is a total derivative by itself, therefore generalizing the involutivity property. Namely, by straighforward calculation we obtain Ω_{ij} such that the relations

ϕ_{i} · \frac{∂}{∂T} (D_{x}(ϕ_{j})) = D_{x}(Ω_{ij})

hold for the elements of the hierarchy 𝔅. Several Ω_{ij} that correspond to i, j ≤ 2, and the initial elements

ϕ_{−1} = 1, ϕ_{0} = s_{1}, ϕ_{1} = −βs_{3} + \frac{3}{2}s_{1}^{2},

ϕ_{2} = β^{2}s_{5} − \frac{5}{2}βs_{2}^{2} − 5βs_{1}s_{3} + \frac{5}{2}s_{1}^{3}

of the sequence 𝔅 are found in the table below:

\begin{aligned}& j = 1 & j = 2 \\ i = −1 & 3T & −5βT_{2} + \frac{15}{2}T^{2} \\ i = 0 & \frac{3}{2}T^{2} & 5T^{3} − 5βT T_{2} + \frac{5}{2}βT_{1}^{2} \\ i = 1 & −\frac{3}{2}T_{1}^{2} + \frac{3}{2}T_{2}^{2} & \frac{5}{2}β^{2}T_{2}^{2} + \frac{35}{8}T^{4} − \frac{15}{2}βT^{2}T_{2} \\ i = 2 & \begin{matrix}− 3βT_{1}T_{3} + \frac{3}{2}T_{2}^{2} \\ + \frac{15}{8}T^{4} − \frac{15}{2}βT T_{1}^{2} \end{matrix} & \begin{matrix}− \frac{5}{2}β^{3}T_{3}^{2} + \frac{15}{2}T^{5} + 15β^{2}T T_{1}T_{3} \\ − \frac{25}{2}βT^{3}T_{2} − \frac{5}{2}β^{2}T_{1}^{2}T_{2} \\ + 5β^{2}T T_{2}^{2} − \frac{75}{4}βT^{2}T_{1}^{2}. \end{matrix} \end{aligned}

Obviously, Ω_{i,−1} = Ω_{i,0} ≡ 0 for all i.

5. The Hamiltonian formalism for the Euler equations

In this section, we consider the problem of constructing commutative Hamiltonian hierarchies of evolution equations associated with hyperbolic Euler systems (in particular, with the wave equation s_{xy} = 0 or the Toda equation (55), see [51, 59, 60]. Also, we interpret the canonical coordinate–momenta formalism, which is widely used in mathematical physics (see [9, 77, 16] and references therein), within nonlocal Hamiltonian operators language of the jet-bundle framework ([10, 42, 63]) preserving the distinction between the coordinates and momenta. Next, we consider hyperbolic Euler equations (i.e., we specify the ansatz for the Lagrangian density) and obtain the differential constraint between the dependent variables u^{i} and the momenta 𝔪_{j}. Then, we identify symmetries of these equations with autonomous potential evolution equations; also, we investigate the relation between Hamiltonian operators for potential and nonpotential evolution equations that describe the evolution of coordinates and momenta, respectively. The aim of our reasonings is to assign new jet bundle π′ and the jet space J^{∞}(π′) to the initial Euler equation E_{E−L} ⊂ J^{k}(π), see Eq. (15), such that the latter equation becomes an evolutionary vector field that belongs to κ(π′), while the evolutionary fields that correspond to the momenta 𝔪 become elements of the adjoint module κ̂(π′). Finally, we relate commutative Lie subalgebras of the Noether symmetries for hyperbolic Euler equations and pairs of Hamiltonian hierarchies composed by potential and nonpotential evolution equations.

5.1. Canonical formalism. Consider an abstract 2r-dimensional dynamical system with r dependent variables u^{i}, momenta 𝔪_{j}, the spatial coordinates x, and the time t, defined by the Poisson brackets ([9, 77])

\begin{aligned}\{u^{i}, u^{j}\}Ā & = 0, \\ \{𝔪_{i}, 𝔪_{j}\}Ā & = 0, \\ \{u^{i}(x, t), 𝔪_{j}(x′, t)\}Ā & = Ā_{j}^{i} δ(x − x′); \end{aligned}  (110)

where Ā is an (r × r)-matrix differential operator in total derivatives

D_{σ} = (D_{x^{1}})^{σ_{1}} ⋯ (D_{x^{n}})^{σ_{n}}

with respect to x (σ = (σ_{1}, …, σ_{n}) is a multiindex) and the skewsymmetric bracket acts as the derivation with respect to any of its arguments. Assume that H = [H dx] is a Hamiltonian with the density

H(x) = H(u(x), u_{σ}(x); 𝔪(x), D_{σ}𝔪(x)).

Then the dynamics

\begin{aligned}u̇ & = \{u(x), H(u(x′), 𝔪(x′))\}Ā, \\ 𝔪̇ & = \{𝔪(x), H(u(x′), 𝔪(x′))\}Ā \end{aligned}  (111)

of the variables u and 𝔪 is obtained in a standard way:

u̇ = \{u(x), H(x′)\}Ā = ∮_{C(x)} ∑_{σ}\{u(x), D_{σ}𝔪(x′)\}Ā · \frac{∂H}{∂ (D_{σ}𝔪(x′))} dx′.

Here C(x) is a small contour around the point x. We emphasize that this language of local coordinates and the δ-distribution is spread in many works on the Hamiltonian formalism application in the field theory; in the sequel, we pass to the invariant exposition soon. Meanwhile, integrating by parts we obtain

u̇ = Ā ∘ \frac{δH}{δ𝔪(x)}.  (112a)

Similarly we obtain the second relation

𝔪̇ = −Ā ∘ \frac{δH}{δu(x)}.  (112b)

These reasonings motivate the following definition of the variational bracket of two Hamiltonians with the densities H and H′, respectively: we set

\{H, H′\}Ā = \frac{δH}{δu} · Ā \frac{δH′}{δ𝔪} − \frac{δH}{δ𝔪} · Ā \frac{δH′}{δu} .

Remark 13. Usually, all dependent variables are treated uniformly. In our case, the momenta can be absorbed by the additional variables u^{r+j} = 𝔪_{j} for 1 ≤ j ≤ r. In other words, we enlarge the total number m = 2r of the coordinates u^{j} twice. The operators Ā and −Ā are also united in the (m × m)-matrix

A = (\begin{matrix}0 & Ā \\ Ā^{∗} & 0 \end{matrix}) ,

such that the dynamical equations take the form

u̇ = A ∘ \frac{δ(H(u))}{δu⃗},

where the variation δ/δu⃗ is calculated with respect to the new vector u⃗ ≡ ^{t}(u, 𝔪). Due to this reason, Remark 13 motivates definitions 6–8, see page 10 in the Introduction.

A traditional approach ([10, 42, 63]) to description of Hamitonian PDE dynamics does not appeal to any distinction between the dependent variables, coordinates u and momenta 𝔪. Nevertheless, by using the double set of m = 2r dependent varuables u and 𝔪 we describe several remarkable properties of models of mathematical physics, for example, of the Korteweg–de Vries equation (91), the modified Korteweg–de Vries equation (84), and similar equations. These aspects are discussed in Sec. 7 below.

6. Hyperbolic Euler equations

Consider a first-order Lagrangian

L = ∫ L(u, u_{x}, u_{y}; x, y) dx ∧ dy

with the density

L = −\frac{1}{2} ∑_{i,j} κ̄_{ij}u_{x}^{i}u_{y}^{j} + H(u; x, y),

where κ̄ is a nondegenerate, symmetric, constant (r × r)-matrix. We note that the notation κ̄ is correlated with the general exposition. Suppose r = 1 and H = 0, then we obtain the wave equation (75) (see Sec. 3). If κ̄ = κ = ∥a_{i}k_{ij}∥ and the function H is defined by formula (117), then we get the Toda equations.

Choose the independent variable y for the “time” coordinate (the nondegeneracy condition implies ∂L/∂u_{y} ≠ 0), leave x for the spatial coordinate on the base ℝ of the new jet bundle π′: ℝ × ℝ⟶^{u}ℝ with the old fiber coordinate u, and denote by 𝔪_{j} = ∂L/∂u_{y}^{j} the jth conjugate coordinate (momentum) for the jth dependent variable u^{j} for any 1 ≤ j ≤ r:

𝔪_{i} = −\frac{1}{2} ∑_{j=1}^{r} κ̄_{ij}u_{x}^{j}.  (113)

The differential constraint, Eq. (113), between coordinates and momenta for the initial equation (15) is our main tool that allows to construct the Hamiltonian structures.

Consider the Legendre transform

H dx ∧ dy = ⟨𝔪, \frac{∂L}{∂u_{y}} ⟩ − L

and assign the Hamiltonian

H(u, 𝔪) = [H dx]

to the Lagrangian L. We decompose its density H into the sum of two equal summands and use the relation

u = −2κ̄^{−1} D̄_{x}^{−1}(𝔪)

in one of the components, in agreement with (112):

H = \frac{1}{2}H[u] + \frac{1}{2}H[𝔪].

The hyperbolic Euler equation

E_{E−L} = {E_{u}(L) = 0}

is equivalent to the system

u_{y} = \frac{δH}{δ𝔪} ,  𝔪_{y} = −\frac{δH}{δu}  (114)

with respect to the canonical Hamiltonian structure Ā = 1. Owing to the relations

\frac{1}{2} \frac{δ}{δ𝔪} = \underbrace{κ̄^{−1} · D_{x}^{−1}}_{A_{1}} ∘\frac{δ}{δu}, \frac{δ}{δu} = \frac{1}{2} \underbrace{D_{x} ∘ κ̄}_{Â_{1}} ·\frac{δ}{δ𝔪},

the dynamical equations are separated:

\begin{aligned}u_{y} & = A_{1} ∘ E_{u}([H[u] dx]), \\ 𝔪_{y} & = −\frac{1}{2} Â_{1} ∘ E_{𝔪}([H[𝔪] dx]). \end{aligned}  (115)

Example 18. Suppose that r = 1, κ̄ = ∥1∥, and the Hamiltonian is trivial: H ≡ 0. Then the Hamiltonian operators

B_{1} = D_{x}^{−1},  B̂_{1} = D_{x}

assigned to wave equation (75) are mutually inverse.

Now we apply these reasonings to the Toda equations (55), which are assigned to Lagrangian (56). It turns out that their Hamiltonian representation (118) hints the existence of the minimal integral T, see Eq. (58). The conservation of T is induced by the conservation of Hamiltonian density (117) for Eq. (55).

We start with a general

Lemma 44. Assume that the density H of a Hamiltonian H = [H dx] for the Hamiltonian evolution equation u_{t} = A(E_{u}(H)) does not depend on the time t explicitly. Then the density H is conserved for this equation:

[D̄_{t}(H) dx] = 0.

Proof. By using the condition ∂H/∂t = 0, we calculate the derivative D̄_{t}(H):

D̄_{t}(H) = ⟨1, �_{A∘E_{u}(H)}(H)⟩ = ∑_{σ}⟨\frac{∂H}{∂u_{σ}} , D̄_{σ}(A ∘ E_{u}(H))⟩ =

integrating by parts, we obtain

= ∑_{σ}⟨(−1)^{σ} D̄_{σ}(\frac{∂H}{∂u_{σ}} ), A ∘ E_{u}(H)⟩ =

since the Hamiltonian operator A is skew–symmetric, A^{∗} = −A, we get

= −⟨A ∘ E_{u}(H), E_{u}(H)⟩.

Again, by using the definition of the Euler operator E_{u} and integrating by parts, we have

= −⟨∑_{σ} D̄_{σ}(A ∘ E_{u}(H)), \frac{∂H}{∂u_{σ}} ⟩ =

= −⟨�_{A∘E_{u}(H)}(H), 1⟩ = −D̄_{t}(H).

Moving the result to the left-hand side of the initial equality, we finally obtain the required condition 2D̄_{t}(H) ∈ im D̄_{x}. □

Example 19 ([59]). Choose the coordinate y for the “time”, define the momenta (see (113)) 𝔪 = ∂L/∂u_{y}:

𝔪_{i} = \frac{1}{2} ∑_{j=1}^{r} κ_{ij}u^{j},  (116)

and obtain the density H_{Toda} of the Hamiltonian H_{Toda}:

H_{Toda}(u, 𝔪) = −\frac{1}{2} ∑_{i=1}^{r} a_{i} exp (\frac{2}{a_{i}} D_{x}^{−1}(𝔪_{i})) − \frac{1}{2} ∑_{i=1}^{r} a_{i} exp(∑_{j=1}^{r} k_{ij}u^{j}).

We have already decomposed the density H into the sum of two components that depend on 𝔪_{i} and u^{j}, respectively. Then, the canonical Hamiltonian representation of the Toda equation E_{Toda} is

u̇^{i} = \frac{δH_{Toda}}{δ𝔪_{i}} = D_{x}^{−1}(exp(∑_{j=1}^{r} k_{ij}u^{j})),

𝔪̇_{i} = −\frac{δH_{Toda}}{δu^{i}} = −\frac{1}{2} ∑_{j=1}^{r} κ_{ij} exp(∑_{l=1}^{r} k_{jl}u^{l}).

Therefore, in terms of the dependent variables u, we have

H_{Toda}(u) = ∑_{i=1}^{r} a_{i} exp(∑_{j=1}^{r} k_{ij}u^{j})  (117)

and

u̇ = A_{1} ∘ E_{u}(H_{Toda}(u)),  (118)

where A_{1} = κ̂^{−1} · D_{x}^{−1}. Recall that an evolution representation for the Liouville equation (76) was obtained earlier in Eq. (86).

In order to correlate the resulting expressions with (58), we apply the transformation x ↔ y to the Hamiltonian equation (118), consider the Hamiltonian density (117), which is independent of y, and use Lemma 44. Then we obtain

D̄_{x}(H_{Toda}) = D̄_{y}(∑_{i=1}^{r} a_{i}u_{xx}^{i}) = D̄_{y}(\frac{1}{2} ∑_{i,j=1}^{r} κ_{ij}u_{x}^{i}u_{x}^{j}),  (119)

whence follows expression (58).

By using the Hamiltonian representation (118), we describe the negative elements φ_{k} of the sequence 𝔄 with k < 0:

Proposition 45. The right-hand side of the Hamiltonian evolution equation

u_{y} = κ̂^{−1} ∘ D_{x}^{−1} ∘ E_{u}(H_{Toda})  (118′)

is the inverse image of the translation u_{t_{0}} = φ_{0} with respect to the mapping R_{Toda}; the translation u_{y} is the element φ_{−1} of the sequence 𝔄.

Proof. By Lemma 34, we have

R_{Toda}(φ⃗_{−1}) = □ (�_{u_{y}}(s)) = □(D̄_{y}(s)) = □(1) = u_{x}.
□

Therefore, representations (118) and (118′) of the Toda equation provide the translations

u_{y} = φ_{−1} = κ̂^{−1} ∘ D̄_{x}^{−1} ∘ E_{u}(H_{Toda})

and

u_{x} = φ_{0} = κ̂^{−1} ∘ D̄_{y}^{−1} ∘ E_{u}(H_{Toda})

that are related by the recursion operator R_{Toda}:

φ_{−2} = ¯□(T̄) ←_{R′_{Toda}} u_{y} = φ_{−1} \overline{\frac{R_{Toda}}{→}} ; ← \underline{R′_{Toda}} φ_{0} = u_{x} \overline{→^{R_{Toda}}} φ_{1} = □(T)

The operator R′_{Toda} obtained from R_{Toda} by using the discrete symmetry x ↔ y generates the symmetries φ_{k} ∈ 𝔄 of the Toda equation with k ≤ −1.

Now we return to the discussion on the Hamiltonian formalism for the Euler equations E_{E−L} = {E_{u}(L) = 0}. We identify the symmetries φ of these equations with the evolution equations u_{t} = φ. Consider a symmetry φ(u_{x}, u_{xx}, …) of Eq. (15), i.e., the potential evolution equation

u_{t} = φ(u_{x}, u_{xx} …),  (120a)

then the induced evolution 𝔪_{t} of the momenta is described by the nonpotential equation

𝔪_{t} = −\frac{1}{2} κ̄ · D_{x}(^{t}φ(𝔪, 𝔪_{x}, …)).  (120b)

In addition, assume that the evolution φ is Hamiltonian:

u̇ = \frac{1}{2} \frac{δH}{δ𝔪} ,  𝔪̇ = −\frac{1}{2} \frac{δH}{δu} .

Then we have

\begin{aligned}u̇ & = \underbrace{κ̄^{−1} · D_{x}^{−1}}_{A_{1}} \frac{δH}{δu} , \\ 𝔪̇ & = −\frac{1}{2} \underbrace{D_{x} · κ̄}_{Â_{1}} \frac{δH}{δ𝔪} , \end{aligned}  (121)

i.e., both equations (120) are Hamitonian simultaneously and their Hamiltonian structures A_{1} and Â_{1} are mutually inverse.

We note two classical examples of the pairs of evolution equations that admit mutually inverse Hamiltonian operators.

Example 20 ([59]). The potential KdV equation (92) is Hamiltonian with respect to the operator B_{1} = D_{x}^{−1}, while B̂_{1} = D_{x} is the first Hamiltonian operator (see (10)) for the KdV equation (91). One can easily verify that Eq. (92) is compatible with the wave equation (75), i.e., the right-hand side ϕ_{1} in the potential KdV equation s_{t} = ϕ_{1} is a symmetry of s_{xy} = 0, while the nonpotential equation (91) describes the evolution of the momentum T = s_{x} (up to the constant factor −1/2).

The potential and nonpotential modified Korteweg–de Vries equations, Eq. (93) and (127), respectively, supply another example. In fact, the succeeding subsections 7.1 and 7.2 contain the detailed analysis of these two examples. We demonstrate that the first pair defines the symmetry hierarchy 𝔅 for wave equation (75) and the second pair is related with the hierarchy 𝔄 of symmetries of the Toda equation (55).

Now we return to Eq. (121) and note an important property of the Hamiltonian equations

u̇ = A_{1} ∘ E_{u}(H)

and

𝔪̇ = −\frac{1}{2} Â_{1} ∘ E_{𝔪}(H) = −\frac{1}{2}E_{u}(H).

It turns out that we have already met the expression in the right-hand side of the latter equation in Lemma 5 on page 13. Lemma 5 assigns the generating sections ψ_{η} = E_{u}(η_{0}) to the densities η_{0} of conservation laws [η] for evolution equations. Formulas (120) give an interpretation of Lemma 5: the generating sections ψ describe (up to the sign) the evolution 𝔪̇ of the momenta 𝔪 along the Hamiltonian symmetries φ of the initial equation E.

Moreover, we relate two pairs of mappings of different types. Assume that there are recursion operators R_{u} and R_{𝔪} for different evolution equations (120a) and (120b), respectively, and consider the mappings

R_{u}: κ → κ, T_{u}: κ̂ → κ̂

which produce symmetries and generating sections of conservation laws for the same equation E_{u}, respectively. The relation ([42])

T_{u} = R_{u}^{∗}  (122)

holds for evolution equations; see, for example, diagram (141) on page 83. In terms of the (120), Eq. (122) means that the skew–symmetric recursion operator

A_{R} = (\begin{matrix}0 & R_{u} \\ −R_{u}^{∗} & 0 \end{matrix})  (123)

is defined for the Hamiltonian representation (115) of the initial Euler equation E = {E_{u}(L) = 0}. This situation is realized for the above– mentioned pairs of equations, see (130) on page 72. We also note that the condition R_{v} = R_{u}^{∗} is valid in this case.

Now we study the problem of constructing a bi-Hamiltonian hierarchy (to be more precise, a pair of hierarchies with respect to u and 𝔪) by using the recursion operator (123). Namely, we find the conditions for the operator A_{R} to induce the second structure {u, 𝔪}_{A_{R}} on the Hamiltonians H_{i} in the diagram

H_{0}  H_{1}  H_{2}
\frac{δ}{δ𝔪} ↓  \frac{δ}{δ𝔪} ↓  \frac{δ}{δ𝔪} ↓
φ_{0} ⟶_{R} φ_{1} ⟶_{R} φ_{2} ⟶_{R} …

such that the Jacobi identity (7b) holds. The commutativity condition [φ_{i}, φ_{j}] = 0 is sufficient for the Jacobi identity to hold for bracket (6), supplied by the operator A_{R}; the sections 𝔪_{t_{i}} = ψ_{i} also commute in this case. We denote by 𝔘 the minimal Lie algebra generated by the sections u_{t_{i}} = φ_{i}. We emphasize that, in general, we require these operators (123) to provide the Lie algebra structure (7b) on the Hamiltonians H_{i} ∈ H̄^{n}(π) but not on the whole horizontal cohomology group H̄^{n}(π). Therefore, our concept extends Definition 7 of a Hamiltonian operator since we treat the particular case of a hyperbolic Euler equation E and a sequence of its Hamiltonian symmetries φ_{i} whose Hamiltonians are H_{i}, respectively. Further on, suppose 𝔘 is a Lie subalgebra of the Noether symmetries of the Lagrangian L. This assumption is sufficient for the existence of the Hamiltonians H_{i} such that

φ_{i} = R \frac{δH_{i−1}}{δ𝔪} = 1 · \frac{δH_{i}}{δ𝔪} ,  ψ_{i} = −R^{∗} \frac{δH_{i−1}}{δu} = −1 · \frac{δH_{i}}{δu} . (124)

Indeed, the existence of the conserved densities is prescribed by the Noether theorem (see Theorem 6 on page 14).

We obtain a more usual description of the pair of the Magri schemes (see [70]) by using the splitting (121):

[figure: ψ_{i+1}  …  ψ_{i+1}  …; A_{1}; Â_{2}; φ_{i+1}^{↑^{R}}  φ_{i}^{↑^{R}}←^{E_{𝔪}} H_{i}; A_{2}; Â_{1}; H_{i} \overline{→^{E_{u}}} ψ_{i}^{^{R^{∗}}↑}  ψ_{i}^{^{R^{∗}}↑}; A_{1}; Â_{2}; φ_{i}^{↑^{R,}}  φ_{i−1}^{↑^{R}} ←^{E_{𝔪}} H_{i−1}.; A_{2}; Â_{1}; ψ_{i−1}^{^{R^{∗}}↑}  …^{↑^{R}}  ψ_{i−1}^{^{R^{∗}}↑}  …^{↑^{R}}]  (125)

Here the operators A_{1} and Â_{1} are defined by constraint (113) between u and 𝔪, while the second operators A_{2} and Â_{2} originate from the relations

R = A_{2} ∘ A_{1}^{−1}, R^{∗} = Â_{2} ∘ Â_{1}^{−1}

respectively. In the sequel, we preserve the notation and mark the operators Â_{1,2} and B̂_{1,2} for the nonpotential equations (127) and (91), respectively, with the ‘hat’ sign.

The notion of the ℓ^{∗}-covering in the category of differential equations was introduced in [42], and the unifying approach towards the recursion operators R: κ → κ, the conjugate recursion operators T : κ̂ → κ̂, the Hamiltonian structures A: κ̂ → κ, and the symplectic structures Â: κ → κ̂ was elaborated for the Magri scheme (125) technique. The return from diagram (125) to (124) provides new interpretation of these structures and their interrelations.

Remark 14. The correlation between the generating sections of conservation laws for the initial Euler equation E_{E−L} = {F ≡ κ̄ · E_{u}(L) = 0} (these sections are denoted by ψ_{L}) and for the same equation in its evolution representation (115) (we denote these sections by ψ) is given by the diagram

ψ ⟶^{−D_{x}^{−1}} ψ_{L} ⟶^{κ̄^{−1}} φ,  (126)

where the first arrow follows from the definition of a generating section:

d_{h}η = ∇(F) dx = ∇ ∘ D_{x}(D_{x}^{−1}(F)) dx ,

and, therefore,

ψ_{L} = ∇^{∗}(1), ψ = −D_{x} ∘ ∇^{∗}(1),

while the second arrow is induced by Lemma 9.

7. Properties of the Korteweg–de Vries hierarchies

In this section, we illustrate the concept ([77, 59, 60]) formulated in the preceding section. In what follows, we consider the bi–Hamiltonian equations (91), (92), (93), and the r-component analogs

E_{mKdV} = {ϑ_{t_{1}} = D_{x} ∘ κ̂ ∘ □(T)}  (127)

of the modified Korteweg–de Vries equation (84). We also introduce the new variable

ϑ = κ · u_{x}  (128)

that is subject to the latter equation. This dependent variable differs from the momenta

𝔪_{Toda} = −\frac{1}{2}(κ u_{x})^{∗}

by the constant factor (−2). This gauge simplifies the calculations and resulting expressions.

Now we study the relations between the equations E_{pKdV} and E_{KdV}, and between E_{pmKdV} and E_{mKdV}, respectively. We claim that the potentials u and s for ϑ and T are such that the linearizations

ℓ_{ϑ}^{(u)} = D_{x} ∘ κ̂, ℓ_{T}^{(s)} = D_{x}

are equal to the first Hamiltonian structures Â_{1} and B̂_{1} for the equations E_{mKdV} and E_{KdV}, respectively. The first Hamiltonian structures A_{1} and B_{1} for E_{p(m)KdV} are inverse with respect to Â_{1} and B̂_{1}. The Hamiltonians for the equations E_{(p)KdV} are well-known. The Hamiltonians H_{k} for the equations E_{(p)mKdV} are described in final Theorem 52 on page 78.

7.1. The Korteweg–de Vries equation. Let L_{s} be the Lagrangian

L_{s} = −[\frac{1}{2}s_{x}s_{y} dx ∧ dy].

Assign the wave equation

E_{s} = {s_{xy} = 0},

see Eq. (75) on page 46, to L_{s} and consider the commutative Lie subalgebra

𝔅 = span_{ℝ}⟨R_{pKdV}^{k+1}(ϕ_{−1}), ϕ_{−1} = 1, k ≥ 0⟩ ⊂ sym E_{s}

of its symmetries. We identify this subalgebra with the hierarchy of the higher potential Korteweg–de Vries evolution equations, see Eq. (92). We continue using the variable T = s_{x}, whose evolution is described by the elements of the nonpotential Korteweg–de Vries hierarchy for Eq. (91), instead of the canonical momentum 𝔪_{s} = −\frac{1}{2}s_{x} for the wave equation. By (121), these two hierarchies admit the pair of mutually inverse Hamiltonian operators B̂_{1} = B_{1}^{−1} = D_{x} for the equations E_{KdV} and E_{pKdV}, respectively. The coinciding elements within the initial part of the Magri schemes (see Eq. (125)) for E_{KdV} and E_{pKdV} are underlined in the diagram below:

…
[figure: B_{2}]
h_{1}[s] \overline{^{E_{s}} →} \underline{\underline{−βs_{4} + 3s_{1}s_{2}}}  …
[figure: − id; B_{1}=D_{x}^{−1}; B̂_{2}]
h_{1}[T] = [\frac{1}{2}(βT_{1}^{2} + T^{3}) dx] →^{E_{T}} −βs_{3} + \frac{3}{2}s_{1}^{2}^{↑^{R_{pKdV}}}
[figure: B_{2}; B̂_{1}=D_{x}]
h_{0}[s] \overline{^{E_{s}}  →}  \underline{s_{2}}  \underline{\underline{−βT_{3} + 3T · T_{1}}}^{^{R_{KdV}}↑}
[figure: − id; B_{1}; B̂_{2}]
h_{0}[T] = [\frac{1}{2}T^{2} dx]  \overline{^{E_{T}} →} s_{1}^{^{R_{pKdV}}↑}
[figure: B̂_{1}]
\underline{T_{1}}^{^{R_{KdV}}↑}
[figure: B̂_{2}=−β D_{x}^{3}+D_{x}∘T +T ·D_{x}]
h_{−1}[T] = [T dx]  \overline{^{E_{T}} →} 1^{^{R_{pKdV}=R_{KdV}^{∗}}↑}

(129)

The initial terms of this diagram are given by the Hamiltonians whose densities are known from the vastest literature:

h_{−1} = [T dx],  h_{0} = [\frac{1}{2}T^{2} dx],  h_{1} = [\frac{1}{2}(βT_{x}^{2} + T^{3}) dx], etc.

Now we consider an example to the relation (122). Namely, the identity

R_{pKdV} = R_{KdV}^{∗}

is satisfied by the recursion operator R_{pKdV} for the potential Korteweg– de Vries equation (92) and the recursion operator

R_{KdV} = −βD_{x}^{2} + 2T + T_{1} · D_{x}^{−1}

for the Korteweg–de Vries equation (91). The recursion R_{pKdV} is factorized by the Hamiltonian structures as follows:

\overbrace{(−β D_{x} + s_{1} · D_{x}^{−1} + D_{x}^{−1} ∘ s_{1})}^{B_{2}} ∘D_{x} =

D_{x}^{−1} ∘ \underbrace{(−β D_{x}^{3} + D_{x} ∘ T + T · D_{x})}_{B̂_{2}} . (130)

The second Hamiltonian structure B̂_{2} for Eq. (91) is a factor in the right-hand side of Eq. (130). Recall that this structure appeared also in Remark 15 on page 49. Another important property of this structure is the correlation between the equation E_{KdV}, the Toda equation E_{Toda}, and the Virasoro algebra ([39]). We analyse the latter property in Remark 17, see the end of this subsection.

The Hamiltonian operators B_{1} and B̂_{1} are inverse, therefore the Magri schemes (125) for Eq. (92) and (91) are correlated.

…

[figure: B_{2}]
h_{k}[s] \overline{^{E_{s}} →} \underline{\underline{ψ_{k}}}  …
[figure: − id; B_{1}=D_{x}^{−1}; B̂_{2}]
h_{k}[T] \overline{^{E_{T}} →} s_{t_{k}} = ϕ_{k} = Ψ_{k}^{↑^{R_{pKdV}=R_{KdV}^{∗}}}
[figure: B_{2}; B̂_{1}=B_{1}^{−1}]
h_{k−1}[s] \overline{^{E_{s}}→} \underline{ψ_{k−1}}  \underline{\underline{T_{t_{k}} = Φ_{k} = ψ_{k}}}^{↑^{R_{KdV}=R_{pKdV}^{∗}}}
[figure: − id; B_{1}; B̂_{2}]
h_{k−1}[T] →^{E_{T}} s_{t_{k−1}} = ϕ_{k−1} = Ψ_{k−1}^{^{R_{pKdV}}↑}
[figure: B̂_{1}]
\underline{T_{t_{k−1}} = Φ_{k−1} = ψ_{k−1}}.^{↑^{R_{KdV}}}
[figure: B̂_{2}]
…^{^{R_{pKdV}=R_{KdV}^{∗}}↑}  …^{↑^{R_{KdV}}}

(131)

Here ϕ_{k} and Φ_{k} are symmetries and ψ_{k} and Ψ_{k} are the generating functions of conservation laws for the potential Korteweg–de Vries equation

s_{t_{1}} = ϕ_{1}

and the Korteweg–de Vries equation

T_{t_{1}} = Φ_{1},

respectively. We conclude that the opposite sides of diagram (131) are identified with the one-step vertical shift.

Corollary 46. We conclude from diagram (131) that the symmetries Φ_{k} of the bi-Hamiltonian hierarchy for the Korteweg–de Vries equation are the gradients ψ_{k} of the Hamiltonians h_{k}[s] for the potential Korteweg– de Vries equation (92), and vice versa.

The equations E_{KdV} and E_{pKdV} share the same set of the Hamiltonians with the densities h_{k}[T] and h_{k}[s], respectively. Thence we obtain the following property of symmetry subalgebra 𝔄 of the Toda equation:

Theorem 47. The generators φ_{k} of the commutative Lie algebra 𝔄 are the Noether symmetries of the Toda equation:

φ_{k} ∈ sym L_{Toda}.

Proof. Indeed, we have

𝔄 ∋ φ_{k} = □(ϕ_{k−1}) = □ ∘ E_{T} (h_{k}[T]) ∈ sym L_{Toda} ⊂ sym E_{Toda}

by Remark 6 on page 42. □

Remark 15. The conservation laws [η_{k}] associated with the generating sections

ψ_{k+1}^{Toda} = κ̂ · □ ∘ E_{T} (h_{k})

(see Eq. (69) on page 42 and diagram (126)) are exactly the Hamiltonians h_{k} dx of the higher Korteweg–de Vries equations s_{t_{k}} = ϕ_{k}.

This fact, [h_{k} dx] ∈ H̄^{1}(E_{Toda}), was noted in a similar situation in [22, §10] but required a nontrivial proof there.

By Lemma 44, the densities h_{k} of the Hamiltonians for both Korteweg–de Vries equations (91) and (92) are conserved on the corresponding higher analogs T_{t_{k}} = D_{x}(ϕ_{k}) and s_{t_{k}} = ϕ_{k} of these equations. We have

D̄_{t_{k}}(h_{k}) = D̄_{x}(Ω_{k}^{KdV}).

We claim that the hierarchy 𝔅 is composed by conserved densities for the hierarchy 𝔄 of potential modified equation (93). We need the following useful lemma to prove this claim.

Lemma 48 ([63]). The relation

E(�_{φ}(L)) = �_{φ}(E(L)) + ℓ_{φ}^{∗}(E(L))

holds for any φ ∈ κ and L ∈ Λ̄^{n}(π).

Proof. Suppose Δ ∈ CDiff(κ, Λ̄^{n}(π)). By multiple integrating by parts, we transform the expression ℓ_{Δ(φ)} to

ℓ_{Δ_{0}(φ)} + D_{x} ∘ Δ′(φ),

where the order of Δ_{0} is zero and Δ′(φ) ∈ CDiff(κ, Λ̄^{n}(π)); then by using Eq. (14) we obtain

E(Δ(φ)) = ℓ_{φ}^{∗}(Δ^{∗}(1)) + ℓ_{Δ^{∗}(1)}^{∗}(φ)

for any section φ ∈ κ. Now, let L ∈ Λ̄^{n}(π) be a form and put Δ = ℓ_{L}: κ → Λ̄^{n}(π). The linearization ℓ_{E(L)} = ℓ_{E(L)}^{∗} of the image of the Euler operator is self–adjoint, hence we obtain the equality

E(ℓ_{L}(φ)) = ℓ_{φ}^{∗}(E(L)) + ℓ_{E(L)}(φ),

whence follows Lemma 48. □

Remark 16. In particular, from this lemma it follows that any Noether symmetry φ_{L} of a Lagrangian L (we recall that the condition �_{φ}(L) = 0 holds on J^{∞}(π) in this case) is also a symmetry of the Euler equation (15) that is assigned to L, i.e.,

sym L ⊆ sym E.

Lemma 48 gives an explanation why the inverse statement is wrong.

Proposition 49 ([51]). Suppose k ≥ 0 is arbitrary. Then the kth term ϕ_{k} = E_{T} (h_{k} dx) of the hierarchy 𝔅 is a conserved density for the kth higher potential modified Korteweg–de Vries equation.

Proof. First, we recall Theorem 35 that describes the correlation of the times within the hierarchies 𝔄 and 𝔅. Second, we apply Lemma 48 that provides the rule to commutate the Euler operator and an evolutionary derivation. Third, we recall that the higher symmetries ϕ_{k} are independent of the jet variable s. Thence we obtain

D̄_{t_{k}}(ϕ_{k}) = �_{ϕ_{k}}(E_{T} (h_{k})) = \underbrace{E_{T} (\overbrace{�_{ϕ_{k}}(h_{k})}^{D̄_{t_{k}} (h_{k})=D̄_{x}(Ω_{k}^{KdV})} )}_{≡0} −ℓ_{ϕ_{k}}^{∗}(E_{T} (h_{k})) =

= − ∑_{j>0}(−1)^{j}D̄_{x}^{j} ∘ (\frac{∂ϕ_{k}}{∂s_{j}} · E_{T} (h_{k})) ∈ im D̄_{x}.

Therefore the density ϕ_{k} is conserved on the equation E_{(k)} = {u_{t_{k}} = φ_{k}}. □

Remark 17. The second Hamiltonian structure

B̂_{2} = −β D̄_{x}^{3} + D̄_{x} ∘ T + T · D̄_{x},

for the equation E_{KdV} endowes the Fourier coefficients 𝔱_{k} of the energy– momentum component (see Eq. (58))

T = ∑_{k∈ℤ} \frac{𝔱_{k}}{x^{k+2}}

for the Toda equation (55) with the structure of the Virasoro algebra ([39]) with the central charge c = −\frac{3}{4}β. We have

\begin{aligned}2πi [𝔱_{n}, 𝔱_{m}] & = 2(n − m) 𝔱_{n+m} − β · (n^{3} − n) δ_{n+m,0}, \\ [𝔱_{k}, β] & = 0. \end{aligned}  (132)

We emphasize that the known property [8, 78] of the integral T is adapted to the case of the Toda equation (55) associated with a nondegenerate symmetrizable matrix K that is not necessarily the Cartan matrix of a semisimple Lie algebra. The central charge c is expressed by using the constant β = ∑_{i} a_{i}Δ^{i} that depends on the symmetrizator a⃗. See Remark 5 on page 38 for a similar situation.

In the paper [52], we considered a class of generalizations of the Virasoro algebra, see Eq. (132), such that the determining relations for those generalizations are given by an N-ary skew–symmetric bracket and N ≥ 2.

7.2. The analogs of the modified Korteweg–de Vries equation. In this subsection, we investigate the relation between the potential and nonpotential modified Korteweg–de Vries equations that correspond to the symmetry subalgebra 𝔄 ⊂ sym L_{Toda} ⊂ sym E_{Toda} of the Euler type Toda equations (55). We analyse the properties of the Miura transformation T = T (ϑ, ϑ_{x}) that maps solutions of the modified Korteweg–de Vries equation (127) to solutions of Eq. (91) and thence we establish the invariant nature of the operator □, see Eq. (62) on page 38, that appeared in description of the structure of symmetries for the Toda equations. Also, we construct the Hamiltonian structures for the hierarchy 𝔄 and prove that modified equations (93) and (127) share the Hamiltonians [h_{k} dx] with the Korteweg–de Vries equations (91) and (92).

We recall that the dependent variable ϑ introduced in Eq. (128) satisfies the modified Korteweg–de Vries equation (127) and its higher analogs E_{mKdV(k)}. By Corollary 38 on page 58, all right-hand sides in the equations E_{mKdV(k)} are local with respect to ϑ.

Similarly to Sec. 7.1, we establish identifications between the higher symmetries of the potential equation E_{pmKdV} and the generating sections of conservation laws for the equation E_{mKdV}, and vice versa:

…  …
[figure: A_{2}]
H_{k}[u] \overline{^{E_{u}} →} \underline{\underline{ψ⃗_{k}}}  ϑ_{t_{k+1}} = φ_{k+1}^{mKdV} = ψ⃗_{k+1}^{↑^{R_{mKdV}}}
[figure: − id; A_{1}=κ̂^{−1}∘D_{x}^{−1}; Â_{2}]
H_{k}[ϑ] \overline{^{E_{ϑ}} →} u_{t_{k}} = φ⃗_{k} = ψ⃗_{k}^{mKdV}^{↑^{R_{pmKdV}}}
[figure: A_{2}; Â_{1}=A_{1}^{−1}]
H_{k−1}[u] \overline{^{E_{u}}→} \underline{ψ⃗_{k−1}}  \underline{\underline{ϑ_{t_{k}} = φ_{k}^{mKdV} = ψ⃗_{k}}}^{↑^{R_{mKdV}=R_{pmKdV}^{∗}}}
H_{k−1}[ϑ] →^{E_{ϑ}} u_{t_{k−1}} = φ⃗_{k−1} = ψ⃗_{k−1}^{mKdV}^{^{R_{pmKdV}}↑}
\underline{ϑ_{t_{k−1}} = φ_{k−1}^{mKdV} = ψ⃗_{k−1}}.^{↑^{R_{mKdV}}}
…^{^{R_{pmKdV}=R_{mKdV}^{∗}}↑}  …^{↑^{R_{mKdV}}}

[figure: − id; A_{1}; Â_{2}]

[figure: Â_{1}] [figure: Â_{2}]

The Lie algebras 𝔄 and 𝔅 are commutative, see Lemma 39 and Theorem 41. Therefore, Jacobi’s identities (7b) are valid for the operators A_{1,2} and B_{1,2} that are either induced by relations (121) for the subscript 1 or obtained from the recursions (123) by using the Magri scheme (125) for the subscript 2. By construction, the recursion operators R_{p(m)KdV} admit the standard decompositions

R_{pmKdV} = A_{2} ∘ A_{1}^{−1} = □ ∘ D_{x}^{−1} ∘ ℓ_{T} ,
R_{pKdV} = B_{2} ∘ B_{1}^{−1},

where the skew–symmetric factors A_{1,2} and B_{1,2} are

\{ \begin{matrix}A_{1} & = κ̂^{−1} ∘ D_{x}^{−1}, \\ A_{2} & = □ ∘ D_{x}^{−1} ∘ □^{∗}, \end{matrix}  \{ \begin{matrix}B_{1} & = D_{x}^{−1}, \\ B_{2} & = −β D_{x} + s_{1} · D_{x}^{−1} + D_{x}^{−1} ∘ s_{1}. \end{matrix}

We denote by ℓ_{T}^{u} the linearization of functional (58) with respect to the dependent variables u and by ℓ_{T}^{ϑ} its linearization with respect to ϑ. The operator ℓ_{T}^{u} is given in Eq. (74) on page 46, and we have

ℓ_{T}^{ϑ} = (…, ∑_{σ} \frac{∂T}{∂ϑ_{σ}^{i}} D_{σ}, …).

Lemma 50. The relation □^{∗} = ℓ_{T}^{ϑ} holds. Moreover,

ℓ_{T}^{u} = □^{∗} ∘ ℓ_{ϑ}^{u}.

Proof. The verification of the first statement is straightforward. Expressing u_{x} via ϑ, u_{x} = κ^{−1}ϑ, we write down functional (58) in terms of ϑ: we have

T = \frac{1}{2} ∑_{l,m} κ^{lm}ϑ^{l}ϑ^{m} − ∑_{l} Δ^{l} · ϑ_{x}^{l},

where κ^{−1} = ∥κ^{lm}∥ and the identity

∑_{i=1}^{r} a_{i} · κ^{ij} = Δ^{j}

is used. Consequently, we obtain

□^{∗} = ^{t}(κ^{−1} · ϑ − Δ⃗ · D̄_{x}).

The second statement follows from the definition of the linearization on page 8. □

Several decompositions follow from this lemma. For example, we obtain

ℓ_{T} = □^{∗} ∘ A_{1}^{−1} = □^{∗} ∘ Â_{1} = □^{∗} ∘ D_{x} ∘ κ̂ = □^{∗} ∘ ℓ_{ϑ}^{u}.

The recursion operator R_{pKdV} is factorized to the product

R_{pKdV}(ϕ_{k−1}) = �_{□(ϕ_{k−1})}(s) = ℓ_{s}(□(ϕ_{k−1})) =

= D_{x}^{−1} ∘ ℓ_{T} ∘ □(ϕ_{k−1}) = B_{1} ∘ □^{∗} ∘ Â_{1} ∘ □(ϕ_{k−1}) of the Hamiltonian operators glued together by the operators □ and □^{∗}.

Proposition 51 ([51, 60]). Each Noether symmetry

φ_{L} = □ ∘ E_{T} (Q(x, T)) ∈ sym L_{Toda}

of the Toda equation associated with a nondegenerate symmetrizable matrix K is Hamiltonian with respect to the Hamiltonian structure A_{1} = κ^{−1} · D_{x}^{−1} and the Hamiltonian H = [Q(x, T)]:

φ_{L} = A_{1} ∘ E_{u}(H).

The proof of Proposition 51 follows from the definition of the Euler operator, E(H dx) = ℓ_{H}^{∗}(1), and Lemma 50.

Remark 18. The propetry of the Noether point symmetries

φ_{0}^{f} = □ ∘ E_{T} (T · f(x) dx)

of the Toda equation to be Hamiltonian,

φ_{0}^{f} = A_{1} ∘ E_{u}(T · f(x) dx),

was established in [77]. One easily verifies that equation (93) is also Hamiltonian with respect to the cohomology class h_{0}, see diagram (131) on page 73:

φ_{1} = A_{1} ∘ E_{u}(h_{0} dx).

Now we extend Remark 18 onto the whole hierarchy 𝔄. Here we formulate the most remarkable relation between the hierarchy 𝔄 for the potential modified Korteweg–de Vries equation (93) and the hierarchy 𝔅 for scalar equation (91).

Theorem 52 ([51, 60]). The Hamiltonian [h_{k} dx] for the kth higher Korteweg–de Vries equation is the Hamiltonian for the Noether symmetry φ_{k} ∈ 𝔄 for any integer k ≥ 0.

This theorem follows from Proposition 51 and Remark 15 on page 74 that assign the Noether symmetries φ_{k} ∈ 𝔄 to the conserved densities h_{k}.

Remark 19. The Lagrangian representation of the Hamiltonian equations E_{(k)} within the hierarchy 𝔄 is

E_{(k)} = \{A_{1} ∘ E_{u}([−\frac{1}{2} ∑_{i,j} κ_{ij}u_{x}^{i}u_{t_{k}}^{j} − h_{k−1}] dx) = 0\}, k ≥ 0.

Here we complete the description of the Magri scheme for the equation (93). The Toda equation (118) is the first nonlocal term of the symmetry tower below:

…  …
…  [figure: ℓ_{s}]  …

u_{t_{k+1}} = φ⃗_{k+1}^{↑^{R_{pmKdV}}}←^{□} s_{t_{k}} = ϕ_{k}^{↑^{R_{pKdV}}}
[figure: A_{2}; B_{2}]
h_{k} \overline{^{E_{u}} →} ψ⃗_{k}^{^{R_{pmKdV}^{∗}}↑}  [figure: ℓ_{s}]  ψ_{k−1}^{↑^{R_{pKdV}^{∗}}} ←^{E_{s}} h_{k−1}
[figure: A_{1}; B_{1}]
u_{t_{k}} = φ⃗_{k}^{↑^{R_{pmKdV}}} ← ^{□} s_{t_{k−1}} = ϕ_{k−1}^{↑^{R_{pKdV}}}
[figure: A_{2}; B_{2}]
h_{k−1} \overline{^{E_{u}}→} ψ⃗_{k−1}^{^{R_{pmKdV}^{∗}}↑}  [figure: ℓ_{s}]  ψ_{k−2}^{↑^{R_{pKdV}^{∗}}} ←^{E_{s}} h_{k−2}.
[figure: A_{1}; B_{1}]
u_{t_{k−1}} = φ⃗_{k+1}^{↑^{R_{pmKdV}}}←^{□} s_{t_{k−2}} = ϕ_{k−2}^{↑^{R_{pKdV}}}

Finally, we have the following assertion

Proposition 53 ([51, 60]). (1) The substitution

T = T (ϑ, ϑ_{x}): E_{mKdV} ⟶^{T (ϑ,ϑ_{x})} E_{KdV}  (133)

is the Miura transformation between the higher equations

E_{mKdV(k)} = {ϑ_{t_{k}} = D_{x} · κ̂(φ_{k})}

and

E_{KdV(k)} = {T_{t_{k}} = D_{x}(ϕ_{k})}.

(2) The operator

□^{∗} = ℓ_{T}^{ϑ}: φ ↦ �_{φ}(T)

maps the Lie algebra sym E_{mKdV} ∋ φ of symmetries of the equation E_{mKdV} into the symmetry Lie algebra sym E_{KdV} for the Korteweg–de Vries equation (91), see Example 15.

(3) The operator □ = □^{∗∗} maps the generating sections of conservation laws which are dual to the symmetries,

□: ϕ_{k} = E_{T} (h_{k} dx) ↦ φ_{k} ∈ 𝔄,

in the opposite direction.

Remark 20. From diagrams (104) and (131) we see that the successive application of the mappings □^{∗} = ℓ_{T}^{ϑ} and □ = □^{∗∗} does not preserve the index of a higher symmetry:

ψ⃗_{k} = φ_{k}^{mKdV} →^{□^{∗}} ψ_{k} = Φ_{k} \overline{^{B_{1}}→} ϕ_{k}

φ⃗_{k} = ψ⃗_{k}^{mKdV}^{^{Â_{1}}↑}  ← ^{□=□^{∗∗}}  ϕ_{k−1}^{↑^{R_{pKdV}}}

(134)

and the recursion operator R_{pKdV} measures the difference in their action.

Part II. Group–theoretic properties of the mathematical physics equations: methods and applications

Within Part II of the present paper, we illustrate the cohomological concepts and algorithms in the geometry of differential equations ([10, 62, 63, 94]). Namely, we apply these methods to the study of the properties of the dispersionless Toda equation, the multi–component nonlinear Schrödinger equation, the Liouville equation, and related systems. Invariant solutions, Noether’s symmetries, local and nonlocal conservation laws, weakly nonlocal recursion operators, parametric families of Bäcklund (auto)transformations, and zero–curvature representations are the structures, which are obtained for these equations of mathematical physics.

Chapter 3. Symmetries, solutions, and conservation laws for differential equations

In this chapter, we consider two examples of applications of the PDE geometry methods; namely, we analyse the geometric properties of the dispersionless Toda equation ([50, 56]) and obtain the structures ([55]) for the multi-component analog of the nonlinear Schrödinger equation, which is related with the former one (see [16]).

8. Nonlinear Schrödinger equation

In this section, we study the properties of the m-component analogs of the nonlinear Schrödinger equation ([98, 1, 92])

Ψ_{t} = iΨ_{xx} + if(|Ψ|) Ψ,  (135)

where Ψ is an m-component vector (m ≥ 1), i = \sqrt{−1}, and f ∈ C^{1}(ℝ) is a smooth function.

The scalar (m = 1) nonlinear Schrödinger equation describes the propagation of light pulses or wave packets in the linear dissipation and nonlinear self-focusing media, e.g., in optical fibers, nonlinear cristals and gases, multicomponent Bose–Einstein condensates at zero temperature, etc. The collective evolution of a complex of space– incoherent solitons in a nonlinear media with a Kerr–like nonlinearity is described by system (135) of the nonlinear Schrödinger equations, where Ψ = ^{t}(Ψ^{1}, …, Ψ^{m}) are the amplitudes of the light wave,

I = ∑_{i=1}^{m} |Ψ^{i}|^{2}

is the density of energy, t is the coordinate along the direction of the wave propagation, and x is the coordinate along the front of the wave.

Suppose f = id, then this equation is known ([98]) to admit a commutative bi-Hamiltonian hierarchy of higher symmetries and an infinite number of conserved densities in involution. Still, for an arbitrary f this is not so. In the sequel, we obtain the symmetry algebra of the latter equation in the physically actual case of a homogeneous function f of weight Δ; also, we desribe the set of m^{2} conserved currents, which generalize the previously known m separate conservation laws of energy of an ith mode and the conservation law of total momentum of the system. Those conservation laws correspond to two types of Hamiltonian symmetries of the Schrödinger equation, the scaling transformation and the translation.

So, consider the m-component analog of the nonlinear Schrödinger equation, the multi–soliton complex ([1])

F^{k} ≡ Ψ_{t}^{k} − iΨ_{xx}^{k} − if(I) · Ψ^{k} = 0,
F̄^{k} ≡ Ψ̄_{t}^{k} + i Ψ̄_{xx}^{k} + if(I) · Ψ̄^{k} = 0, 1 ≤ k ≤ m, I ≡ ΨΨ̄. (136)

This equation is Hamiltonian for an arbitrary function f:

(Ψ Ψ̄)_{t} = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) (\begin{matrix}δH_{NLS}/δΨ \\ δH_{NLS}/δΨ̄ \end{matrix}) ,  (137)

where the density H_{NLS} of the Hamiltonian H_{NLS} = [H_{NLS} dx] is

H_{NLS} = −iΨ_{x}Ψ̄_{x} + i ∫ ^{I} f(I) dI.

We also note that H_{NLS} is a conserved density for Eq. (136), quite similarly to Eq. (11) for the dispersionless Toda equation or to Eq. (119) on page 66 for the Toda equation (55). From the Hamiltonian representation (137) we deduce that the complex conjugate variables Ψ̄ are the canonically conjugate variables (the momenta) for the dynamical coordinates Ψ; in Sec. 5 we investigated a generalization of this situation.

Equation (135) is known to possess the following property (see [1] and references therein): in addition to the conservation of the total energy ΨΨ̄, there are m separately conserved densities

Q_{i} = Ψ^{i} · Ψ̄^{i},  D̄_{t}(Q_{i}) = 0.  (138)

Physically speaking, there is no energy transfer between the modes Ψ^{i}. Still, this observation about the properties of the nonlinear Schrödinger equation is uncomplete, since these m integrals of motion (138) are only particular instances within the set of m^{2} currents ([55])

η_{ij} = Ψ^{i}Ψ̄^{j} dx + i (Ψ_{x}^{i} Ψ̄^{j} − Ψ^{i} Ψ̄_{x}^{j}) dt,  d̄_{h}η_{ij} = 0.  (139)

These currents are conserved on Eq. (135) with an arbitrary nonlinearity f(I). They reflect the correlation between the intensities of the light modes with different indexes i and j, 1 ≤ i, j ≤ m. The set of conservation laws (139) remained unnoticed in the papers [1, 92], and the arising conservation restrictions seem to have not been taken into consideration by the authors when making computer experiments.

The generating sections of the conservation laws [η_{ij}] defined in Eq. (139) are

ψ⃗_{(ij)} = ^{t}(ψ_{(ij)}, ψ̄_{(ij)}),

where

ψ_{(ij)}^{i} = Ψ̄^{j}, ψ̄_{(ij)}^{j} = Ψ^{i},

and

ψ_{(ij)}^{i′} = ψ̄_{(ij)}^{j′} = 0 for i′ ≠ i, j′ ≠ j.

The canonical Hamiltonian structure

Γ_{1} = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix})

assigns the point symmetries

φ⃗_{(ij)} = ^{t}(φ_{(ij)}, φ̄_{(ij)}) : φ_{(ij)}^{i} = Ψ^{j}, φ̄_{(ij)}^{j} = −Ψ̄^{i}

to these sections. Suppose i = j, then we get the gauge symmetry ([98])

A_{λ}(φ⃗_{(ii)}): Ψ^{i} ↦ exp(λ)Ψ^{i}, Ψ̄^{i} ↦ exp(−λ)Ψ̄^{i}.  (140)

The Hamiltonians of the symmetries φ⃗_{(ii)} are the conserved densities Q_{i} in Eq. (138).

In the paper [98], the case f(I) = I of cubic nonlinearity in Eq. (135) was analysed. Then, the cubic m-component equation (135) admits the recursion operator

R_{NLS} = (\begin{matrix}−D_{x} & 0 \\ 0 & D_{x} \end{matrix}) + \frac{1}{2} (−Ψ ; Ψ̄) · D_{x}^{−1} ∘ (Ψ̄, Ψ)−

\frac{1}{2} ∑_{i,j} φ⃗_{(ij)} · D_{x}^{−1} ∘ ψ⃗_{(ij)}.

This operator generates an infinite sequence of the commuting local higher symmetries of Eq. (135), starting with scaling symmetry (140):

Q_{i} dx  H_{NLS}  H_{4}
↓^{EΨ⃗}  ↓^{EΨ⃗}  ↓^{EΨ⃗}
ψ⃗_{(ii)} ⟶^{R_{NLS}^{∗}} ψ⃗_{1} ⟶^{R_{NLS}^{∗}}  ψ⃗_{2} ⟶^{R_{NLS}^{∗}} ψ⃗_{3} ⟶^{R_{NLS}^{∗}} ψ⃗_{4} ⟶^{R_{NLS}^{∗}} …
↓^{Γ_{1}}  ↓^{Γ_{1}}  ↓^{Γ_{1}}  ↓^{Γ_{1}}  ↓^{Γ_{1}}
φ⃗_{(ii)} ⟶^{R_{NLS}} Ψ⃗_{x} ⟶^{R_{NLS}}  Ψ⃗_{t} ⟶^{R_{NLS}} φ⃗_{3} ⟶^{R_{NLS}} φ⃗_{4} ⟶^{R_{NLS}} ….

(141)

This diagram is an example of a sequence of Hamiltonian symmetries such that there are no Hamiltonians for a half of these symmetries, φ⃗_{2k+1}: we have ψ⃗_{2k+1} ∉ im EΨ⃗.

Remark 21. We have obtained the formulae for the generating sections ψ⃗_{(ij)} of conservation laws (139) and the Hamiltonian symmetries φ⃗_{(ij)}. Now we conclude that the recursion operator R_{NLS} is weakly nonlocal ([61]), i.e., this operator is decomposable to the form

R = local part + ∑_{α} φ_{α} ∘ D_{x}^{−1} ∘ ψ_{α},  (142)

where φ_{α} are symmetries and ψ_{α} are generating sections of conservation laws. We recall that in the preceding chapter we used the recursion operators’ decompositions similar to (142), and then we generated the commuting symmetry sequences for the Toda equations and established nontrivial properties of these sequences.

We finally note that the symmetries φ⃗ that are independent on x and t (only such symmetries were considered in [98]) do not exhaust the whole algebra of classical symmetries for the nonlinear Schrödinger equation (135). Moreover, the resulting symmetry algebra of this equation is noncommutative.

Example 21. Suppose there are no restrictions for the nonlinearity function f(I) in Eq. (136). Then the point symmetry algebra of the latter equation is generated by the fields whose sections are

φ⃗_{(ij)}, Ψ⃗_{x}, Ψ⃗_{t}, (\begin{matrix}φ^{i} \\ & φ̄^{i} \end{matrix}) = (\begin{matrix}2t Ψ_{x}^{i} − ix Ψ^{i} \\ 2t Ψ̄_{x}^{i} + ix Ψ^{i} \end{matrix}) .

Further on, suppose additionally that f is homogeneous,

f(λI) = λ^{Δ} · f(I).

Then equation (135) acquires the scaling symmetry

(\begin{matrix}φ^{i} \\ & φ̄^{i} \end{matrix}) = (\begin{matrix}2Ψ^{i} + Δ x Ψ_{x}^{i} + 2Δ t Ψ_{t}^{i} \\ 2Ψ̄^{i} + Δ x Ψ̄_{x}^{i} + 2Δ t Ψ̄_{t}^{i} \end{matrix}) .

9. The dispersionless Toda equation

The dispersionless Toda equation is an analog of Eq. (55) with the continuous variation of the index j that enumerates the dependent variables u^{j}. In this section, the classical symmetry algebra of the dispersionless Toda equation is computed and 5 classes of conservation laws for the latter equation are reconstructed; also, some questions of the Lagrangian formalism with higher-order derivatives are discussed.

Consider the hyperbolic Toda equations

u_{xy} = exp(Ku),  u ′_{xy} = K · exp(u′),  (143)

associated with the type A_{r−1} Lie algebras with the Cartan matrices K, introduce an additional independent variable z ∈ ℝ, and extend the values of the discrete index j ∈ [1, r], which enumerates the dependent variables u^{j}, onto the whole line ℝ. Also, let the rank r tend to infinity: r → ∞. Then for any section u of the bundle π we set

u^{j} = u(x, y, z)|_{z=jε},

where ε is the cell step.

As r → ∞ and ε → +0, the continuous limit equation, which is refered as either the dispersionless Toda equation, or the heavenly equation, or the Boyer–Finley equation ([11]), also appeares in many models, e.g., in gravity ([87, 11]) as a reduction of the antiselfdual vacuum Einstein equation (ASDVEE). The heavenly equation E_{heav} exists due to the special structure (see Eq. (45)) of the Cartan matrices K = ∥k_{ij}∥ of the type A_{r} algebras. As a matter of fact, instead of the Cartan (r × r)-matrices K one should consider ([87, 88]) the Cartan operator K̂ = −D_{z}^{2}. Then the scalar equations, which originate from Eq. (143), are ([2, 11])

Ê = {F̂ ≡ u_{xy} − exp(−u_{zz}) = 0}  (144a)
u′_{xy} = −D_{z}^{2} ∘ exp(u′);  (144b)

the Cartan operator K̂ defines the sign “−” in the one–dimensional equations

u_{ττ} = exp(−u_{zz}),  u′_{ττ} = −D_{z}^{2} ∘ exp(u′)  (145)

that appeared in the Introduction.

Following the papers [87, 16], we consider the limit

lim_{ε→+0} lim_{r→∞} L_{Toda} = D_{z}^{−1} (\frac{1}{2}u_{x}u_{yzz} − exp(−u_{zz}))

of the Lagrangian density L_{Toda}, defined in Eq. (56), as r → ∞ and ε → +0. Then the Lagrangian L_{Toda} itself is mapped to the functional

L̂ = ∫∫ dxdy ∫ Ł dz

with the density

Ł = −\frac{1}{2} u_{xz}u_{yz} − exp(−u_{zz})

that depends on the second derivatives of the sections u = u(x, y, z). Applying the Euler operator E_{u} to L̂, we obtain the equation

E_{heav} = {F_{heav} ≡ u_{xyzz} − D_{z}^{2} ∘ exp(−u_{zz}) = 0};  (146)

we see that double integrating in z maps Eq. (146) to Eq. (144a), while the substitution u′ = −u_{zz} maps it to Eq. (144b).

9.1. Symmetries and exact solutions. Computing the symmetries φ ∈ Sym Ê of Eq. (144a) by using the analytic transformations software Jet ([72]), we get the following solution:

Proposition 54. The point symmetries

φ(x, y, z, u, u_{x}, u_{y}, u_{z})

of Eq. (144a), which solve the determining equation

D̄_{xy}(φ) + exp(−u_{zz}) · D̄_{z}^{2}(φ) = 0,

are

φ_{1}[f] = (u_{x} − \frac{1}{2}z^{2} D̄_{x}) f(x), φ̄_{1}[g] = (u_{y} − \frac{1}{2}z^{2} D̄_{y}) g(y), (147a)
φ_{2} = −\frac{1}{2}zu_{z} + u − \frac{1}{2}z^{2},  (147b)
φ_{3} = u_{z},  (147c)
φ_{4}[q] = q(x) z,  φ̄_{4}[q̄] = q̄(y) z,  (147d)
φ_{5}[r] = r(x),  φ̄_{5}[r̄] = r̄(y),  (147e)

where f, q, and r are arbitrary smooth functions of x and g, q̄, and r̄ are functions of y. The commutation rules for symmetries (147) are given in the skew–symmetric table below:

φ_{1}[f] φ̄_{1}[g] φ_{2}  φ_{3}  φ_{4}[q]  φ̄_{4}[q̄]  φ_{5}[r]  φ̄_{5}[r̄]
φ_{1}[f]  0  0  0 φ_{4}[−f′] φ_{4}[−fq′]  0  φ_{5}[−fr′]  0
φ̄_{1}[g]  0  0 φ̄_{4}[−g′]  0  φ̄_{4}[−gq̄]  0  φ̄_{5}[−gr̄′]
φ_{2}  0 φ_{3}+φ_{4}[2] φ_{4}[q]  φ̄_{4}[q̄]  φ_{5}[2r]  φ̄_{5}[2r̄]
φ_{3}  0  φ_{5}[−q] φ̄_{5}[−q̄]  0  0
φ_{4}[q]  0  0  0  0
φ̄_{4}[q̄]  0  0  0
φ_{5}[r]  0  0
φ̄_{5}[r̄]  0

Remark 22. 1. The operator ˆ□ = u_{x} − \frac{1}{2}z^{2} D̄_{x} in Eq. (147a) is an analog of the operator □, which was introduced in Eq. (62) on page 38.

2. The symmetries of Eq. (144b) that correspond to (147a)–(147c) were obtained in [2]; the symmetries φ_{4}, φ_{5} ∈ ker D_{z}^{2} of the equation Ê demonstrate the (inessential) distinction between the geometries of Eq. (144a) and Eq. (144b), this distinction is provided by the substitution u′ = −u_{zz}.

Remark 23. In fact, equations (144) and (146) are huge inspite of an apparent compactness of the notation. Indeed, one can estimate the number of kth order internal coordinates for these equations or the total number of nontrivial relations that define the kth prolongation E_{heav}^{(k)}. Therefore the calculation of the higher symmetry algebra sym E_{heav}^{∞} for Eq. (146) is really difficult.

The passage from difference–differential equations (46) associated with Cartan’s matrix (45) to their dispersionless limit (144a) provides the additional constraint

u_{zzz} = 0.

The latter equation must be taken into consideration when describing the Lie algebra sym E_{heav}^{∞} composed by all symmetries of the limit dispersionless Toda equation.

9.1.1. Constructing exact solutions to the dispersionless Toda equation. By using the structure of the point symmetry Lie algebra Sym E for Eq. (144a) and applying various geometric methods ([10]), the problem of constructing exact solutions to the heavenly equation was considered in [2] and [71]. In the former paper, invariant solutions of the equation u_{xy} = ±(exp(u))_{zz} were obtained; they correspond to Eqs. (152) and (153) of the present article. A class of solutions that are noninvariant with respect to the whole Lie algebra Sym E was pointed out in [71]. The approach of this paper is wider than in [2] in the following sence: any time an equation under consideration being reduced to an auxiliary equation for a function that depends on fewer arguments than the initial one, we construct the point symmetry Lie algebra for the new equation and obtain its invariant solutions that are parameterized by arbitrary functions. Therefore, what we get is much more than a list of particular solutions. Also, we note that the method of solving Eq. (144a) together with the constraint φ_{i} = 0 is similar to the scheme ([71]) used to construct non-invariant solutions of differential equations. Indeed, we treat the variable z in the constraints φ_{i} = 0 as a formal parameter in order to write down these additional equations φ_{i} = 0 in total differentials.

The symmetry φ_{1} ∈ Sym Ê. Consider the generator

φ_{1} = φ_{1}[f] + φ̄_{1}[g] = u_{x} f(x) − \frac{1}{2}z^{2} f′(x) + u_{y} g(y) − \frac{1}{2}z^{2} g′(y)

of an infinitesimal conformal symmetry of Eq. (144a); this generator depends on two arbitrary smooth functions f and g. Now we represent the invariance condition φ_{1} = 0 in the characteristic form and obtain the first integral

t = ∫ ^{x} dx/f(x) − ∫ ^{y} dy/g(y)

that is independent on the variable u. In order to construct another integral C_{2} we treat the base coordinate z as a parameter:

\frac{1}{2}z^{2} log f(x) + \frac{1}{2}z^{2} log g(y) − u = C_{2}(z).

Then, a solution u(x, y, z) to the equation φ_{1} = 0 is defined by the condition

Π(t, C_{2}(z)) = 0,

where Π is an arbitrary (for a while) function of two “integrals”, one of which depends on z. Solving this relation with respect to u, we get

u = \frac{1}{2}z^{2} log f(x)g(y) + Φ(t, z).  (148)

Moreover, if u is a solution to Eq. (144a), then it is necessary that the function Φ satisfies the one–dimensional equation

Φ_{tt} + exp(−Φ_{zz}) = 0.  (145′)

The reduced equation is nothing else than the one–dimensional analog of Eq. (145). In a similar situation, the hyperbolic Liouville equation is reduced to solving its one–dimensional analog when one constructs solutions that are invariant with respect to the conformal symmetries of the initial Liouville equation. We emphasize that the second “integral” C_{2}(z) of the equation φ_{1} = 0 was obtained by using a nontrivial interpretation of the coordinate z as a parameter which is auxiliary with respect to the independent variables x and y. This approach seems to have not been used in [2]; in the sequel, we apply it to Eq. (145′) again. We use the Jet ([72]) software and compute the generators of the point symmetry Lie algebra of Eq. (145′); then, we obtain the invariant solutions of this equation. The result is given in

Lemma 55. The Lie algebra of classical symmetries of Eq. (145′) is generated by eight evolutionary vector fields whose generating functions are

ϕ_{1} = tΦ_{t} − z^{2}, ϕ_{2} = Φ_{t}, ϕ_{3} = zΦ_{z} + z^{2} − 2Φ, ϕ_{4} = Φ_{z},
ϕ_{5} = zt,  ϕ_{6} = t,  ϕ_{7} = z,  ϕ_{8} = 1.

Further on, we point out the families of solutions to Eq. (145′) and assign the solution class (148) for the dispersionless Toda equation (see Eq. (144a)) to each of them.

The symmetry ϕ_{1}. Consider the equation ϕ_{1} = 0; again, we require the variable z to be a formal parameter, then we have z^{2} log |t| − Φ = C(z). Hence we express the solution Φ, substitute it into Eq. (145′), and solve the latter with respect to C; we finally get

Φ(t, z) = z^{2} log |t| − \frac{1}{4}z^{2} log |z| − \frac{1}{8}z^{2} + C_{1}z + C_{2},

where C_{1} and C_{2} are arbitrary constants.

The symmetry ϕ_{2}. One easily checks that the real ϕ_{2}-invariant solutions of Eq. (145′) do not exist at all, since the equation exp(−Φ_{zz}) = 0 is insoluble over ℝ. Still, we consider a more general case: namely, we seek solutions to Eq. (145′) that are invariant with respect to the linear combination

ϕ_{(a:b)} = ϕ_{2} + (a : b)ϕ_{4}

of the symmetries, where (a : b) ∈ ℝℙ^{1}, i.e., we seek the propagating wave solutions. Substituting the function Φ(z−(a : b) t) into Eq. (145′), we get the equation

(a : b)^{2} Ψ = − exp(−Ψ).  (149)

Here by Ψ we denote the second derivative Φ′′ of Φ with respect to its argument

w = z − (a : b) t.

Suppose the condition

−\sqrt{e} < (a : b) < \sqrt{e}

is satisfied, then the obstacle for the low–velocity ϕ_{(a:b)}-invariant solutions to exist is clear from Eq. (149). Indeed, there is the critical (minimal) velocity |a : b| = \sqrt{e} and the wave solution

Φ = −\frac{1}{2}(z ± \sqrt{e} t)^{2} + α(z ± \sqrt{e} t) + β,

where α, β ∈ ℝ. Quite remarkably, if the velocity \sqrt{e} < |a : b| < ∞ is greater then the minimum velocity, then there are distinct roots Ψ_{1,2} of Eq. (149) and Eq. (145′) admits two solutions

Φ = \frac{1}{2}Ψ_{1,2} · w^{2} + αw + β,  w = z − (a : b) t,  α, β ∈ ℝ, (150)

similtaneously. This pair of roots Ψ_{1,2} appeares after the obvious bifurcation: no solutions of Eq. (149) and (145′) are assigned to a low inclination (a : b)^{2} of the straight line y = (a : b)^{2} Ψ on the 0Ψy-plane; then, at the inclination (a : b)^{2} = e this straight line is tangent to the graph of exponent y = − exp(−Ψ) at the point (−1, −e), while at a greater incline e < (a : b)^{2} < ∞ the point of tangency is split to the pair of distinct intersection points (Ψ_{1,2}, (a : b)^{2}Ψ_{1,2}), where −∞ < Ψ_{1} < −1 < Ψ_{2} < 0. Each admissible value of Ψ determines the second derivative of a polynomial solution (150) of Eq. (145′). A unique root Ψ = 0 and the ϕ_{4}-invariant solution of Eq. (145′) (see below) is assigned to the point a : b = ∞.

The symmetry ϕ_{3}. First, we solve the auxiliary ordinary differential equation:

xy′(x) − 2y = −x^{2} ⟹ y(x) = (γ − \frac{1}{2} log x^{2}) · x^{2}.  (151)

We shall use formulae (151) twice, in order to construct the solutions of Eq. (145′) that are invariant with respect to the symmetry ϕ_{3} and to get the φ_{2}-invariant solutions of the initial equation (144a). In these two cases, the function γ depends on two distinct sets of variables: the first case is γ = γ(t) and the second case is γ = γ(x, y).

Substituting the expression

Φ = (γ(t) − \frac{1}{2} log z^{2}) · z^{2}

that appeared in Eq. (151) into Eq. (145′) and paying attention to the sign of the constant of integration, we get the following expressions for the function γ(t):

γ_{1} = \frac{3}{2} − log[\sqrt{ϵ} sinh 2 artanh exp(±\sqrt{ϵ}(t − t_{0}))],
γ_{2} = \frac{3}{2} + log[±t − t_{0}],
γ_{3} = \frac{3}{2} − log[\sqrt{ϵ} cosh log tan(\sqrt{ϵ}(±t − t_{0}))],

where ϵ > 0 and t_{0} ∈ ℝ. Now we return to formula (148) and assign the solution class

u = \frac{1}{2}z^{2} log f(x)g(y) + (γ_{i}(t) − \frac{1}{2} log z^{2}) · z^{2}, i = 1, 2, 3, (152)

of the dispersionless Toda equation (144a) to each of the functions Φ(t, z).

The symmetry ϕ_{4}. Consider the constraint Φ_{z} = 0; then, a solution to the equation Φ_{tt} = 1 is a polynomial

Φ(t) = \frac{1}{2}t^{2} + C_{1}t + C_{2},

where C_{1} and C_{2} are constants of integration; the corresponding solutions of the dispersionless Toda equation (144a) are again defined in Eq. (148).

The symmetry φ_{2} ∈ Sym Ê. Now we construct the φ_{2}-invariant solutions of Eq. (144a). First, we consider the auxiliary ordinary differential equation in Eq. (151); then, we obtain the hyperbolic scal^{+}-Liouville equation (see Eq. (39) on page 30). Now, subsitute the expression (γ(x,

y) − \frac{1}{2} log z^{2}) · z^{2} for u in the equation u_{xy} = exp(−u_{zz}) and make the transformation

X = x exp(3/2), Y = y exp(3/2).

Hence we get the Liouville equation

γ_{X Y} = exp(−2γ)

whose solutions ([69]) are easily obtained from formula (87) on page 48 by a trivial coordinates change; we have

γ(X , Y) = −\frac{1}{2} log[−f′(X)g′(Y)\{Q([f(X) + g(Y)]^{2})\}^{−2}],

where the mapping Q is one of sin, id, or sinh, whence we finally obtain

u(x, y, z) = \frac{z^{2}}{2} log \frac{\{Q([f(e^{3/2}x) + g(e^{3/2}y)]^{2})\}^{2}}{−z^{2}f′(e^{3/2}x)g′(e^{3/2}y)} .  (153)

This class of solutions of Eq. (144a) is in fact present in paper [2], and their physical interpretation is known (ibid.): these expressions provide the instanton solutions ([11]) of the antiselfdual vacuum Einstein equations (ASDVEE).

The symmetry φ_{3} ∈ Sym Ê. In order to obtain the φ_{3}-invariant solution of Eq. (144a), we solve the equations u_{z} = 0 and u_{xy} = exp(−u_{zz}) together. The solution is

u(x, y, z) = xy + f(x) + g(y),

where f and g are arbitrary functions.

The symmetries φ_{4} and φ_{5} of Eq. (144a) depend neither on the unknow function u nor its derivatives. Therefore, if the equation u_{xy} = exp(−u_{zz}) is overdetermined by the condition φ_{4} = 0 or φ_{5} = 0, then the problem of search for exact solutions to the dispersionless Toda equation is not simplified.

9.2. The Noether symmetries and conservation laws. In this subsection, we consider the problem of constructing the conservation laws for the dispersionless Toda equations (144) and (146). First, we discuss the general correlation between an Euler equation (15) and the fixed set of conservation laws which reflect the conservation properties of the energy–momentum tensor.

9.2.1. On the Lagrangian formalism involving higher–order derivatives. Suppose that L = ∫ L dx is a Lagrangian with the second order derivatives and g_{μν} is the corresponding metric; then we can construct a conservative analog of the energy–momentum tensor T^{νμ} that satisfies the equation

∑_{μ} D̄_{μ}(T^{νμ}) = 0,  D̄_{μ} ≡ D_{μ}|_{E}.  (154)

The equations of motion E are

∑_{μ,ν} D_{μ}D_{ν} (\frac{∂L}{∂u_{;μ;ν}^{i}} ) = ∑_{ν} D_{ν} (\frac{∂L}{∂u_{;ν}^{i}} ) − \frac{∂L}{∂u^{i}}  (155)

and the tensor T^{νμ} coincides with its classical definition ([9]) if the higher–order term

∥\frac{∂L}{∂u_{;μ;ν}^{a}} ∥

is trivial. Again, we use the notation u_{;μ}^{i} ≡ D_{x^{μ}}(u^{i}) and u_{;μ;ν}^{i} ≡ D_{x^{μ}} · D_{x^{ν}}(u^{i}). We raise and lower the indexes ; μ, ; ν by using the metric g_{μν}. By a straighforward substitution into Eq. (154) one checks that the tensor

T^{νμ} = −g^{νμ}L+∑_{i} \frac{∂L}{∂u_{;μ}^{i}} u^{i;ν} +∑_{i,λ}[\frac{∂L}{∂u_{;μ;λ}^{i}} ·D_{λ} (u^{i;ν})−D_{λ} (\frac{∂L}{∂u_{;μ;λ}^{i}} )·u^{i;ν}]

satisfies the required restrictions ([50]), and therefore condition (154) provides conservation laws for system (155). Still, we note an important feature of Eq. (146): as r → ∞ and ε → +0 in the Toda equations, the metric g_{μν} becomes degenerate and the Lagrangian L̂ looses its covariance. Indeed, it is impossible to find a nondegenerate metric ĝ_{μν} such that

Ł ∼ \frac{1}{8}u_{;μ}^{;z}u^{;μ}_{;z} + exp(−u_{;z}^{;z}).

For the same reason we cannot define the energy–momentum tensor T^{νμ} for the equation E_{heav} as the variation of the Lagrangian L̂_{heav} with respect to the metric ĝ_{νμ} since the nondegenerate metric does not exist.

9.2.2. Constructing the conservation laws. From the preceding section it is clear that the search of conservation laws for Eq. (146) is less trivial than finding the symmetry Lie fields by their generating functions and requires additional reasonings. In the sequel, we find the symmetries sym E_{heav} of the dispersionless Toda equation, select its Noether symmetries sym L̂_{heav} in a fixed coordinate system, and reconstruct the conservation laws by using the homotopy method, which is described in [10, 94] and [45].

Lemma 56 ([50]). The generating functions (147a), (147c)–(147e) are the Noether symmetries of the equation E_{heav}, while (147b) is not.

SYMMETRIES AND CONSERVATION LAWS IN INTEGRABLE MODELS 93

Proposition 57 ([50]). The conserved currents η_{i} ∈ Λ̄^{2}(E_{heav}), assigned to the Noether symmetries φ_{i} of Eq. (146), are

η_{1} = \{\frac{3}{8}f(x) uu_{xyzz} + \frac{1}{12}f(x) u_{z}u_{xyz} − \frac{1}{24}f(x) u_{xy}u_{zz} + \frac{1}{24}f(x) u_{y}u_{xzz}

− \frac{1}{12}f′(x) u_{y} − \frac{1}{12} f(x)u_{xz}u_{yz} + \frac{z}{6}f′(x) u_{yz} + \frac{1}{8}f(x)u_{x}u_{yzz}

− \frac{z^{2}}{8} f′(x) u_{yzz} − f(x) ∫_{0}^{1} t^{2}uu_{zzz}^{2} exp(−tu_{zz}) dt

+ f(x) ∫_{0}^{1} tuu_{zzzz} exp(−tu_{zz}) dt} dy ∧ dz

+ \{−\frac{1}{8}f′(x) uu_{xzz} − \frac{1}{8}f(x) uu_{xxzz} + \frac{1}{4}f′′(x) u + \frac{1}{12}f′(x) u_{z}u_{xz}

+ \frac{1}{12}f(x) u_{z}u_{xxz} − \frac{z}{6}f′′(z) u_{z} − \frac{1}{24}f′(x) u_{x}u_{zz} − \frac{1}{24}f(x) u_{xx}u_{zz}

+ \frac{z^{2}}{24}f′′(x) u_{zz} + \frac{1}{6}f(x) u_{x}u_{xzz} − \frac{1}{12}f′(x) u_{x} − \frac{1}{12}f(x) u_{xz}^{2}

+ \frac{z}{6}f′(x) u_{xz} − \frac{z^{2}}{8} f′(x) u_{xzz}\} dz ∧ dx

+ \{−\frac{1}{6}f(x) u_{xy}u_{xz} + \frac{1}{3}f(x) u_{x}u_{xyz} − \frac{1}{12}f′(x) u_{x}u_{yz} − \frac{1}{4}f′(x) uu_{xyz}

− \frac{z}{6}f′′(x) u_{y} − \frac{z^{2}}{4} f′(x) u_{xyz} + \frac{1}{12}f(x) u_{y}u_{xxz} − \frac{1}{4}f(x) uu_{xxyz}

+ \frac{1}{12}f(x) u_{z}u_{xxy} − \frac{1}{12}f(x) u_{xx}u_{yz} + \frac{z}{6}f′(x) u_{xy} + \frac{1}{12}f′(x) u_{y}u_{xz}

+\frac{1}{12}f′(x) u_{z}u_{xy}+\frac{z^{2}}{12}f′′(x) u_{yz}+f(x) ·∫_{0}^{1}[−tuu_{xzzz}−tu_{xz}u_{zz}+tu_{x}u_{zzz}

+tu_{z}u_{xzz}+t^{2}uu_{xzz}u_{zzz}−t^{2}u_{x}u_{zz}u_{zzz}] exp(−tu_{zz}) dt + f′(x) ·∫_{0}^{1}[−u_{z}

+zu_{zz}−\frac{z^{2}}{2} u_{zzz}−tuu_{zzz}+\frac{z^{2}}{2} tu_{zz}u_{zzz}] exp(−tu_{zz}) dt\} dx ∧ dy,

η_{3} = \{\frac{5}{24}u_{z}u_{yzz} − \frac{1}{8}u_{yz}u_{zz} + \frac{1}{24}u_{y}u_{zzz} − \frac{1}{8}uu_{yzzz}\} dy ∧ dz

+ \{\frac{5}{24}u_{z}u_{xzz} − \frac{1}{8}u_{xz}u_{zz} + \frac{1}{24}u_{x}u_{zzz} − \frac{1}{8}uu_{xzzz}\} dz ∧ dx

+ \{\frac{1}{4}uu_{xyzz}+\frac{1}{3}u_{z}u_{xyz}−\frac{1}{12}u_{xy}u_{zz}−\frac{1}{6}u_{xz}u_{yz}+\frac{1}{12}u_{x}u_{yzz}+\frac{1}{12}u_{y}u_{xzz}

+ u_{zz} exp(−u_{zz}) + exp(−u_{zz}) + u_{z}u_{zzz} exp(−u_{zz})} dx ∧ dy,

η_{4} = \{−\frac{1}{6}q(x) u_{yz} + \frac{z}{4}q(x) u_{yzz}\} dy ∧ dz

+ \{\frac{1}{6}q′(x) u_{z} − \frac{z}{12}q′(x) u_{zz} − \frac{1}{6}q(x) u_{xz} + \frac{z}{4}q(x) u_{xzz}\} dz ∧ dx

+ \{\frac{z}{2}q(x) u_{xyz} − \frac{1}{6}q(x) u_{xy} − \frac{z}{6}q′(x) u_{yz} + \frac{1}{6}q′(x) u_{y}

− zq(x) u_{zzz} exp(−u_{zz}) + q(x) exp(−u_{zz})} dx ∧ dy,

η_{5} = \frac{r(x)}{4} u_{yzz} dy ∧ dz + \{−\frac{r′(x)}{12} u_{zz} + \frac{r(x)}{4} u_{xzz}\} dz ∧ dx

+ \{−\frac{r′(x)}{6} u_{yz} + \frac{r(x)}{2} u_{xyz} + r(x) u_{zzz} exp(−u_{zz})\} dx ∧ dy.

Any divergence d̄_{h}(η_{i}) equals 0 on the equation E_{heav}; calculating the divergences does not require precise evaluation of the integrals in the homotopy parameter t in rational functions, because differentiating with respect to x or z under the integral sign is allowed.

Proving Lemma 56 and Proposition 57 is based on Lemma 48 and involves the general scheme of reconstruction of conservation laws by their generating sections.

Method of reconstruction of conserved currents. Now we describe a method ([94]) for constructing the differential (n − 1)-forms that are exact on the equation E = {F = 0}; we follow the paper [45].

Consider the function f = u/τ and the evolutionary vector field �_{f} whose flow is

A_{τ}: (x^{i}, u_{σ}^{j}) ↦ (x^{i}, τu_{σ}^{j}),

then we have

\frac{dA_{τ}^{∗}(ω)}{dτ} = A_{τ}^{∗} (�_{f}(ω)) = A_{τ}^{∗} (ℓ_{ω}(f))  (156)

for any differential form ω. Let this form ω be

ω = ⟨F, ψ⟩ = F ψ dx^{1} ∧ … ∧ dx^{n}

and suppose that η is the desired current which is assigned to the generating function ψ on the equation {F = 0}: d_{h}(η) = ∇(F) and ψ = ∇^{∗}(1). We note that the right-hand side in Eq. (156) contains the term

ℓ_{ω}(f) = ⟨ℓ_{ω}^{∗}(1), f⟩ + d_{h}G(ℓ_{ω} ∘ f),

where the first summand is cohomological to 0 since the image of the Euler operator is trivial if the Lagrangian density is a total divergence,

ℓ_{ω}^{∗}(1) = ℓ_{⟨F,ψ⟩}^{∗}(1) = E(⟨F, ψ⟩) = E(d_{h}η) = 0.

Next, the mapping G: CDiff(F, Λ̄^{n}) → Λ̄^{n−1}, which is restricted onto the equation E, is defined by the rule

G(∑_{σ} a_{σ}D_{σ}) = ∑_{|σ|>0} ∑_{j∈σ} \frac{(−1)^{|σ|−1}}{|σ|} D_{σ−1_{j}}(a_{σ})ω_{(−j)},

where

ω_{(−j)} = (−1)^{j+1} dx^{1} ∧ … d̂x^{j} … ∧ dx^{n}

and σ − 1_{j} is the result of excluding an index j from a multiindex σ = (σ_{1}, …, σ_{n}).

Now we integrate Eq. (156) in τ from 0 to 1 and get

∫_{0}^{1} \frac{d}{dτ} A_{τ}^{∗}(ω) dτ = A_{1}^{∗}(ω) − A_{0}^{∗}(ω) = ⟨F, ψ⟩ − A_{0}^{∗} (⟨F, ψ⟩) =

= ∫_{0}^{1} A_{τ}^{∗} (ℓ_{ω}(f)) dτ = d_{h} ∫_{0}^{1} A_{τ}^{∗} (G(ℓ_{ω} ∘ f)) dτ = d_{h}η,

whence we obtain the required formula for the current η.

Proof of Lemma 56 and Proposition 57. First we note that symmetries (147) of Eq. (144a) are also symmetries of the Euler equation (146): Sym Ê ⊂ sym E_{heav}, because the relation ℓ_{F_{heav}} = D_{z}^{2} ∘ ℓF̂ holds due to Lemma 7 on page 18. By using Lemma 48, we check that symmetries (147a) and (147c)–(147e) of Eq. (146) are Noether, while for symmetry (147b) we have

ℓ_{F_{heav}}(φ_{2}) + ℓ_{φ_{2}}^{∗}(F_{heav}) = 2u_{xyzz} − \frac{3}{2}D_{z}^{2}(exp(−u_{zz})) ≠ 0.

From Theorem 6 it follows that the Noether symmetries of Eq. (146) are in one-to-one correspondence with the generating sections of its conservation laws. By Corollary 10 on page 36, this correspondence for the equation E_{heav} is provided by the identity mapping. Following the reasonings scheme above, to each of four Noether symmetries φ_{i} we assign the conserved current η_{i}; the arising expressions are rather large and therefore omitted. Nevertheless, the result is correct, for the conditions d̄_{h}(η_{i}) = 0 may be checked straightforwardly. □

Remark 24. Suppose that the arbitrary function f in the conserved current η_{1} is f ≡ 1. Then η_{1} is a countinuous in z analog of component (58) of the energy–momentum tensor Θ = T dx+T̄ dy for the Toda equations (55). We recall that the conserved current T dx for Eq. (55) corresponds to the Noether symmetry φ_{0}^{1} = □(E_{T} (T dx)) ∈ sym L_{Toda}. Another (nonlocal) conserved current

η = u_{xz} exp(−u_{zz}) dx ∧ dy + (\frac{1}{2}u_{xz}^{2} − u_{xx}) dx ∧ dz,

for Eq. (146) (see Eq. (11) on page 12) could be also treated as an analog of the integral (58) for Eq. (55) due to the following reason.

Consider the Hamiltonian representation

u_{y} = D_{x}^{−1} ∘ D_{z}^{−2} ∘ E_{u}(H_{heav} dx ∧ dz),  (157)

of Eq. (146), here

H_{heav} = − exp(−u_{zz})

is the continuous limit of the Hamiltonian, see Eq. (117), for the Toda equations. By Lemma 44, the density H_{heav} is conserved on the related Hamiltonian equation (157), therefore the current η is also conserved.

Chapter 4. Bäcklund transformations and zero–curvature representations

In this chapter, we investigate the relationship between Bäcklund transformations and zero–curvature representations for the hyperbolic Liouville equation, the wave equation, and the Liouville scal^{+}-equation. They are

u_{xy} = exp(2u),  v_{xy} = 0,  Υ_{xy} = exp(−2Υ),  (158)

respectively. In Sec. 1 on page 29 we we discussed a natural geometric scheme that provides Eqs. (158) and gives their interpretation. In what follows, we construct one–parametric families of Bäcklund transformations and zero–curvature representations and consider exmaples of integrating Bäcklund transformations and zero–curvature in nonlocal variables.

10. Bäcklund transformations and their deformations

In this section, we study the properties of one–parametric deformations of Bäcklund transformations for Eq. (158). Consider the covering τ_{t}: Ẽ_{t} → E_{Liou} defined by the extended total derivatives

D̃_{x} = D̄_{x} + ũ_{x} \frac{∂}{∂ũ}, D̃_{y} = D̄_{y} + ũ_{y} \frac{∂}{∂ũ},  [D̃_{x}, D̃_{y}] = 0. (159)

We assume that the partial derivatives of the nonlocal variable ũ with respect to x and y are

ũ_{x} = u_{x} + exp(−t) · exp(ũ + u),  (160a)
ũ_{y} = −u_{y} + 2 exp(t) · sinh(ũ − u),  (160b)

respectively. Recalling Remark 3 on page 19, let the diffeomorphism μ be the swapping u ↔ ũ of the fiber variable u and the nonlocal variable ũ combined with the mapping x ↦ −x and y ↦ −y. Then diagram (25) supplies Bäcklund autotransformation B(Ẽ_{t}, τ_{t}, τ_{t} ∘ μ, E_{Liou}) for Eq. (187), and for the Liouville equation. The equations Ẽ_{t} of this Bäcklund autotransformation ([14]) are, obviously,

(ũ − u)_{x} = exp(−t) · exp(ũ + u),  (161a)
(ũ + u)_{y} = 2 exp(t) · sinh(ũ − u).  (161b)

Put

u_{k} ≡ \frac{∂^{k}u}{∂x^{k}} and u\overline{k} ≡ \frac{∂^{k}u}{∂y^{k}}

for any k ∈ ℕ. Consider the scaling symmetry

X^{0} = −x \frac{∂}{∂x} + y \frac{∂}{∂y}

of the Liouville equation and extend X^{0} onto entire E_{Liou}^{∞}:

X̂ = −x \frac{∂}{∂x} + y \frac{∂}{∂y} + ∑_{k≥1} ku_{k} \frac{∂}{∂u_{k}} − ∑_{k≥1} ku\overline{k} \frac{∂}{∂u\overline{k}} .  (162)

Proposition 58. The symmetry X̂ cannot be extended to a symmetry of the covering equation Ẽ_{t}.

Proof. Assume the converse. By ˜�_{φ} denote the evolutionary vector field

∑_{σ} D̃_{σ}(φ) · \frac{∂}{∂u_{σ}}

on Ẽ_{t}, where φ ∈ C^{∞}(Ẽ_{t}). By definition, set

ℓ̃_{F} (φ) = ˜�_{φ} (F).

Suppose there is a smooth function a ∈ C^{∞}(Ẽ_{t}) such that the linearized system

ℓ̃_{F} (φ) = 0,  D̃_{x^{i}}(a) = ˜�_{φ,a} (ũ_{x^{i}}) ≡ ( ˜�_{φ} + a \frac{∂}{∂ũ}) (ũ_{x^{i}}) (163)

holds. This means that the field ˜�_{φ,a} is a local symmetry of the covering equation Ẽ_{t} and X̂ is extended onto Ẽ_{t} constructively. Nevertheless, system (163) is not compatible since

D̃_{x} ∘ D̃_{y}(a) ≠ D̃_{y} ∘ D̃_{x}(a).

Indeed, D̃_{x} ∘D̃_{y}(a)−D̃_{y} ∘D̃_{x}(a) does not depend on a at all and equals

xu_{x}^{2}e^{t+u−ũ} + u_{x}yu_{y}e^{t+ũ−u} − xu_{x}e^{2ũ} − u_{x}yu_{y}e^{t+u−ũ}
−2yu_{y}e^{2t+ũ+u} + 2xu_{x}e^{2t+ũ+u} − xu_{x}^{2}e^{t+ũ−u} + xe^{2u}u_{x}
−ye^{2u}u_{y} + 2e^{t}xu_{x}^{2} + yu_{y}e^{2ũ} − 2e^{t}yu_{y}u_{x} ≠ 0.

This contradiction concludes the proof. □

Therefore, the scaling symmetry X̂ is a τ_{t}-shadow only (i.e., a solution to the equation ℓ̃_{F} (φ) = 0) and generates the family of the covering equations Ẽ_{t} over E_{Liou}; these Ẽ_{t} are parametrized by t ∈ ℝ.

Consider the Cartan distribution C̃_{t} on the equation Ẽ_{t} and by U_{t} denote the Cartan’s connection form on Ẽ_{t}. In local coordinates, the form U_{t} is

U_{t} = ∑_{σ} d_{C}(u_{σ}) ⊗ \frac{∂}{∂u_{σ}} + (dũ − (u_{x} + exp(ũ + u − t)) dx +

+ (u_{y} − 2 exp(t) sinh(ũ − u)) dy)⊗\frac{∂}{∂ũ}, (164) where d_{C} is the Cartan differential. In coordinates, we have

d_{C}(u_{σ}) = du_{σ} − ∑_{i} D̄_{i}(u_{σ}) dx^{i}.

Let Ω ∈ D(Λ^{μ}(Ẽ)), then μ is the degree of the derivation Ω. By [[·, ·]]^{FN} we denote the Frölicher–Nijenhuis bracket ([37, 62]):

[[Ω, Θ]]^{FN}(f) = L_{Ω}(Θ(f)) − (−1)^{μν} · L_{Θ}(Ω(f)),  (165)

where Ω, Θ ∈ D(Λ^{∗}(E)), f ∈ C^{∞}(E), and the degrees μ, ν are μ = deg Ω, ν = deg Θ, respectively. We also assume that

L_{Ω} = [i_{Ω}, d]: Λ^{k}(E) → Λ^{k+deg Ω}(E)

is the Lie derivative and

i_{Ω} : Λ^{k}(E) → Λ^{k+deg Ω−1}(E)

is the inner product. The Frölicher–Nijenhuis bracket is a natural geometric structure in differential calculus. The de Rham differential d and the Richardson–Nijenhuis bracket [[·, ·]]^{RN} (see [52] for its applications) are other examples of natural structures.

Theorem 59 ([37]). Let τ: Ẽ → E be a covering and A_{t}: Ẽ → Ẽ be a smooth family of diffeomorphisms such that A_{0} = id and τ_{t} = τ ∘ A_{t}: Ẽ → E is the covering for any t ∈ ℝ. Then the evolution of the Cartan connection form U_{τ_{t}} is

\frac{dU_{τ_{t}}}{dt} = [[X̂_{t}, U_{τ_{t}}]]^{FN},  (166)

where X̂_{t} is a τ_{t}-shadow for any t ∈ ℝ.

Suppose Ẽ is a finite-dimensional manifold, then there is the isomorphism

D(Λ^{∗}(Ẽ)) ≃ Λ^{∗}(Ẽ) ⊗ D(Ẽ).

Thence, any derivation Ω ∈ D(Λ^{∗}(Ẽ)) is decomposable to the finite sum such that the summands are Ω = ω ⊗ X, where ω ∈ Λ^{∗}(Ẽ) and X ∈ D(Ẽ). The Frölicher–Nijenhuis bracket of such elements is

[[ω ⊗ X, θ ⊗ Y]]^{FN} = ω ∧ θ ⊗ [X, Y] + ω ∧ L_{X}(θ) ⊗ (Y) +

+ (−1)^{i} dω ∧ (X θ) ⊗ Y − (−1)^{ij} θ ∧ L_{Y} (ω) ⊗ X −

− (−1)^{(i+1)j} dθ ∧ (Y ω) ⊗ X, (167)

where X, Y ∈ D(Ẽ), ω ∈ Λ^{i}(Ẽ) and θ ∈ Λ^{j}(Ẽ). If the dimension of Ẽ is not necessarily finite, then there is the embedding

Λ^{∗}(Ẽ) ⊗ D(Ẽ) ⊂ D(Λ^{∗}(Ẽ))

defined by the rule

(ω ⊗ X)(f) = X(f)ω

for any function f ∈ C^{∞}(Ẽ).

Consider the coverings τ_{t}: Ẽ_{t} → E defined in Eq. (159). Then we have

\frac{dU_{t}}{dt} = (exp(ũ + u − t) dx − 2 exp(t) sinh(ũ − u) dy) ⊗ \frac{∂}{∂ũ}. (168)

We claim that the scaling symmetry X̂ is the τ_{t}-shadow such that the evolution of the connection form U_{t} (164) in the covering τ_{t}, see Eq. (159), is given by (168) in virtue of equation (166). We need Lemmas 60-65 to prove this.

Lemma 60. [[X̂, U_{t}]]^{FN} dũ = (dU_{t}/dt) dũ.

Lemma 61. [[X̂, U_{t}]]^{FN} dx = [[X̂, U_{t}]]^{FN} dy = [[X̂, U_{t}]]^{FN} du = 0.

Proof. The proof of Lemmas 60 and 61 is based on successive application of formula (167). The coefficient of ∂/∂x is

− ∑_{σ} d_{C}u_{σ} ∧ L_{∂/∂u_{σ}}(−x) − d_{C}ũ ∧ L_{∂/∂ũ}(−x) = 0;

the coefficient of ∂/∂y is calculated analogously. Now consider Eq. (162) and decompose X̂ in the coordinates ⟨x, y, u_{k}, u\overline{k}⟩. We get X̂ = ∑_{α} ω_{α} ⊗ X_{α}, where ω_{α} is a 0-form and X_{α} is a derivation for any α. We

have i = 0 and j = 1 in Eq. (167). Thence we get

∑_{α}(ω_{α} ∧ L_{X_{α}}(d_{C}u) + dω_{α} ∧ (X_{α} d_{C}u)) ⊗ \frac{∂}{∂u},

where the first summand is

∑_{α} ω_{α} ∧ d(X_{α} d_{C}u) + ∑_{α} ω_{α} ∧ (X_{α} d(d_{C}u)) = −u_{x} dx + u_{y} dy,

and the second summand equals u_{x} dx − u_{y} dy, whence their sum is also trivial. Finally, we calculate the coefficient

∑_{α}(ω_{α} ∧ L_{X_{α}}(d_{C}ũ) + dω_{α} ∧ (X_{α} d_{C}ũ))

of ∂/∂ũ by using the explicit formula for d_{C}ũ. The first summand is equal to −u_{x} dx − u_{y} dy, and the second summand is

(u_{x} + e^{−t} exp(ũ + u)) dx + (u_{y} − 2e^{t} sinh(ũ − u)) dy.

Consequently, we obtain the required expression

(e^{−t} exp(ũ + u) dx − 2e^{t} sinh(ũ − u) dy) ⊗ \frac{∂}{∂ũ}.

This concludes the proof. □

However, the calculation of coefficients of ∂/∂u_{k} or ∂/∂u\overline{k} in [[X̂, U_{t}]]^{FN} is untrivial for k ≥ 1. First, we have Lemma 62 ([47]). Let u(x) and f(u) be smooth functions and D_{x} be the total derivative with respect to x. Denote u_{k} ≡ D_{x}^{k}(u(x)), k ≥ 0, u_{0} ≡ u. Then the relation

n · D_{x}^{n}(f(u)) = ∑_{m=1}^{n} m u_{m} \frac{∂}{∂u_{m}} D_{x}^{n}(f(u))  (169)

holds for any integer n ≥ 1.

The proof of Lemma 62 is based on Lemma 63 and Corollary 64 below.

Lemma 63. Let the assumptions of Lemma 62 hold. Take an integer n > 0 and a positive integer l ≤ n − 1. Then the relation

D_{x} (\frac{∂}{∂u_{l}} D_{x}^{n−1}(f(u))) = \frac{∂}{∂u_{l}} D_{x}^{n}(f(u)) − \frac{∂}{∂u_{l−1}} D_{x}^{n−1}(f(u)) (170)

is valid.

Corollary 64. Let the assumptions of Lemma 63 hold. Then we also have

(n + 1)u_{n+1} \frac{∂}{∂u_{n+1}} D_{x}^{n+1}(f(u)) = (n + 1)u_{n+1} \frac{∂}{∂u_{n}} D_{x}^{n}(f(u)) =

= (n + 1)u_{n+1} · f′(u). (171)

Proof of Lemma 62 ([47]). We prove (169) by induction on n. For n = 1, relation (169) holds. For n ≥ 1, we have

(n + 1) D_{x}^{n+1}(f(u)) = D_{x}(n D_{x}^{n}(f(u)) + D_{x}^{n}(f(u))) =

by the inductive assumption,

= D_{x} (∑_{m=1}^{n} mu_{m} \frac{∂}{∂u_{m}} D_{x}^{n}(f(u)) + D_{x}^{n}(f(u))) =

by the Leibnitz rule,

= ∑_{m=1}^{n} mu_{m+1} \frac{∂}{∂u_{m}} D_{x}^{n}(f(u)) +

+ ∑_{m=1}^{n} mu_{m} D_{x} \frac{∂}{∂u_{m}} D_{x}^{n}(f(u)) + D_{x} D_{x}^{n}(f(u)) =

by Eq. (170) applied to the second sum,

= ∑_{m=1}^{n} mu_{m} \frac{∂}{∂u_{m}} D_{x}^{n+1}(f(u)) + ∑_{m=1}^{n} mu_{m+1} \frac{∂}{∂u_{m}} D_{x}^{n}(f(u)) −

− ∑_{m=1}^{n} mu_{m} \frac{∂}{∂u_{m−1}} D_{x}^{n}(f(u)) + D_{x}D_{x}^{n}(f(u)) =

by the definition of D_{x} and the subscript shift in the latter sum,

= ∑_{m=1}^{n} mu_{m} \frac{∂}{∂u_{m}} D_{x}^{n+1}(f(u)) + ∑_{m=0}^{n}(m + 1)u_{m+1} \frac{∂}{∂u_{m}} D_{x}^{n}(f(u)) −

− ∑_{m=0}^{n−1}(m + 1)u_{m+1} \frac{∂}{∂u_{m}} D_{x}^{n}(f(u)) =

since almost all summands in the latter two sums coincide,

= ∑_{m=1}^{n} mu_{m} \frac{∂}{∂u_{m}} D_{x}^{n+1}(f(u)) + (n + 1)u_{n+1} \frac{∂}{∂u_{n}} D_{x}^{n}(f(u)) =

by (171),

= ∑_{m=1}^{n} mu_{m} \frac{∂}{∂u_{m}} D_{x}^{n+1}(f(u)) + (n + 1)u_{n+1} \frac{∂}{∂u_{n+1}} D_{x}^{n+1}(f(u)) =

= ∑_{m=1}^{n+1} mu_{m} \frac{∂}{∂u_{m}} D_{x}^{n+1}(f(u)).

The proof is complete. □

Here we offer another proof of Lemma 62. This proof is based on the weights technique. The idea of this proof was communicated by V. V. Trushkov.

Proof of Lemma 62 ([58]). By definition, put the weight wt(u_{k}) = k, wt(u_{k} · u_{l}) = k + l, and wt(u_{k_{1}} + u_{k_{2}}) = k_{1} if k_{1} = k_{2}. We have

D_{x}^{n}(f(u)) = ∑_{m=1}^{n} P_{n,m} · f^{(m)}(u),  (172)

where P_{n,m} = ∑_{j⃗} const(n, m) · u_{j_{1}} · . . . · u_{j_{l(n,m)}}. We claim that P_{n,m} is a

differential polynomial such that

j_{1} + … + j_{l(n,m)} = n ∀j⃗, ∀ n, m.  (173)

We prove this fact by induction on n. Indeed, if wt(P_{n,m}) = n, then wt(D_{x}(P_{n,m}))) = n + 1 due to the Leibnitz rule. Besides,

D_{x}^{n+1}(f(u)) = ∑_{m=1}^{n} (D_{x}(P_{n,m}) · f^{(m)}(u) + P_{n,m} · u_{1} · f^{(m+1)}(u))

and therefore the weight wt(D_{x}^{n+1}(f(u))) is well defined and equals n + 1.

Now, consider the weight counting operator

W ≡ ∑_{m≥1} m · u_{m} \frac{∂}{∂u_{m}}  (174)

that acts on the right-hand side in Eq. (172):

∑_{m≥1} m · u_{m} \frac{∂}{∂u_{m}} ∘ ∑_{k=1}^{n} ∑_{j⃗} const(n, k) · u_{j_{1}} · . . . · u_{j_{l(n,k)}} · f^{(k)}(u) =

= ∑_{k=1}^{n} ∑_{j⃗} const(n, k) · n · u_{j_{1}} · . . . · u_{j_{l(n,k)}} · f^{(k)}(u) = n · D_{x}^{n}(f(u)),

since condition (173) holds for all multiindexes j⃗. We finally conclude that the functions D_{x}^{n}(f(u)) are eigenfunctions for operator (174) and the integers n ∈ ℕ are the eigenvalues. Therefore, relation (169) is a solution to the problem λ · φ = W(φ). The proof is complete. □

Lemma 65. [[X̂, U_{t}]]^{FN} du_{k} = [[X̂, U_{t}]]^{FN} du\overline{k} = 0 for any k ≥ 1.

Proof. Let k ∈ ℕ. Taking into account Eq. (167), consider the 1-form

[[X̂, U_{t}]]^{FN} du_{k} = ((k − 1)D̄_{y}u_{k} − ∑_{l=1}^{k−1} lu_{l} \frac{∂}{∂u_{l}} D̄_{x}^{k−1}(exp(2u))) · dy.

We see that all coefficients of dx, du, du_{l}, du\overline{l} vanish for all l ≥ 1. Finally, note that D̄_{y}u_{k} = D̄_{x}^{k−1}(exp(2u)). By Lemma 62, the coefficient of dy is trivial. Arguing as above, we conclude that [[X̂, U_{t}]]^{FN} du\overline{k} = 0. This completes the proof. □

Theorem 66. The τ_{t}-shadow in Eq. (162) satisfies the relation

[[X̂, U_{t}]]^{FN} = (exp(ũ + u − t) dx − 2 exp(t) sinh(ũ − u) dy) ⊗ \frac{∂}{∂ũ},

i.e., the abelian group of diffeomorfisms A_{t} = exp(tX̂) provides smooth one-parametric family (160) of one-dimensional coverings over Liouville’s equation E_{Liou}.These coverings correspond to Bäcklund autotransformations (161) for the Liouville equation, see diagram (25). Evolution of the connection form is given by (168).

Proof. Decompose the bracket [[X̂, U_{t}]]^{FN} with respect to the basis ⟨\frac{∂}{∂x}, \frac{∂}{∂y}, \frac{∂}{∂u_{k}}, \frac{∂}{∂uk̄}, \frac{∂}{∂ũ}⟩. By Lemmas 61 and 65, all coefficients of the derivations \frac{∂}{∂x}, \frac{∂}{∂y}, \frac{∂}{∂u_{k}}, \frac{∂}{∂uk̄} equal 0 for k ≥ 0. Thence, Lemma 60 supplies the required expression, which is given in Eq. (168). □

Remark 25. Consider Bäcklund transformation ([14])

(v − u)_{x} = e^{−t} exp(u + v), (v + u)_{y} = −e^{t} exp(u − v),  (175)

t ∈ ℝ, between the Liouville equation (76) and the wave equation v_{xy} = 0, as well as Bäcklund transformation ([14])

(Υ − u)_{x} = 2e^{−t} cosh(Υ + u), (Υ + u)_{y} = −e^{t} exp(u − Υ), (176)

t ∈ ℝ, between E_{Liou} and the Liouville scal^{+}-equation Υ_{xy} = exp(−2Υ). Then these transformations possess the property similar to one described in Theorem 66. Namely, scaling symmetry (162) of E_{Liou} is the required τ_{t}-shadow in both cases. The proof of these theorems is quite analogous to the proof of Theorem 66 above and appeals to useful identity (169) in total derivatives.

11. Integrating Bäcklund transformations in nonlocal variables

In this section, we study nonlocal aspects of integrating Bäcklund transformations between PDE. The aim of this section is to illustrate a natural scheme that provides nonlocal variables associated with a certain PDE and to obtain nonlocal structures for the hyperbolic Liouville equation. We reconstruct the coverings τ_{j} from Eq. (161) and (175)– (176) and demonstrate the nonlocal variables to be potentials for the fiber variables u, v, and Υ. In what follows, we use the notation E_{u} as a synonim of E_{Liou}, meaning that the Liouville equation u_{xy} = exp(2u) is imposed on the fiber variable u.

Now we construct the one-dimensional non-abelian coverings such that we can integrate Bäcklund transformations (161), (175)–(176) in the corresponding nonlocal variables. We fix an arbitrary t ∈ ℝ and consider the extended total derivatives

D̃_{x}^{E_{u}} = D̄_{x}^{E_{u}} − e^{2u} \frac{∂}{∂Ξ_{t}} ,  D̃_{y}^{E_{u}} = D̄_{y}^{E_{u}} + (Ξ_{t}^{2} + 2u_{y}Ξ_{t} − e^{2t})\frac{∂}{∂Ξ_{t}} ,
D̃_{x}^{E_{u}} = D̄_{x}^{E_{u}} − e^{2u} \frac{∂}{∂Ξ_{∞}} , D̃_{y}^{E_{u}} = D̄_{y}^{E_{u}} + (Ξ_{∞}^{2} + 2u_{y}Ξ_{∞})\frac{∂}{∂Ξ_{∞}} ,
D̃_{y}^{E_{u}} = D̄_{y}^{E_{u}} − e^{2u} \frac{∂}{∂Ξ′_{t}} ,  D̃_{x}^{E_{u}} = D̄_{x}^{E_{u}} + (Ξ′_{t}^{2} + 2u_{x}Ξ′_{t} + e^{−2t})\frac{∂}{∂Ξ′_{t}} ,
D̃_{x}^{E_{v}} = D̄_{x}^{E_{v}} + e^{2v} \frac{∂}{∂Ξ_{t}^{v}} ,  D̃_{y}^{E_{v}} = D̄_{y}^{E_{v}} + (2v_{y}Ξ_{t}^{v} + e^{2t})\frac{∂}{∂Ξ_{t}^{v}} ,
D̃_{y}^{E_{Υ}} = D̄_{y}^{E_{Υ}} + e^{−2Υ} \frac{∂}{∂Ξ_{t}^{Υ}} , D̃_{x}^{E_{Υ}} = D̄_{x}^{E_{Υ}} + ((Ξ_{t}^{Υ})^{2} − 2Υ_{x}Ξ_{t}^{Υ} + e^{−2t})\frac{∂}{∂Ξ_{t}^{Υ}} . (177)

We see that the extended derivatives commute in all cases:

[D̃_{x}, D̃_{y}] = 0.

Therefore the coverings

τ_{t}: Ẽ_{t} → E_{u}^{∞}, τ_{∞}: Ẽ_{∞} → E_{u}^{∞}, τ′_{t}: Ẽ′_{t} → E_{u}^{∞},

τ_{t}^{v}: Ẽ_{t}^{v} → E_{v}^{∞}, τ_{t}^{Υ} : Ẽ_{t}^{Υ} → E_{Υ}^{∞} (178)

are well defined. The explicit form of the covering equations Ẽ_{t}, Ẽ_{∞}, Ẽ′_{t}, Ẽ_{t}^{v}, and Ẽ_{t}^{Υ} is discussed in Remark 27.

Remark 26. Coverings (178) with nonlocal variables (177) cannot be reduced to local conservation laws for the underlying equations E_{u}, E_{v}, and E_{Υ}; these coverings are called non–abelian. We also claim that the t-parameterized coverings (e.g. τ_{t} at the points t_{1} and t_{2}) are equivalent (τ_{t_{1}} ≃ τ_{t_{2}}), i.e., there is a functional dependence between the nonlocal variables (Ξ_{t_{1}} and Ξ_{t_{2}} in our case). We have

Ξ_{t_{1}} + y · exp(2t_{1}) = Ξ_{t_{2}} + y · exp(2t_{2}) = Ξ_{t=−∞} ∀t_{1}, t_{2} ∈ ℝ;

similar relations hold for other coverings (178).

Remark 27. The covering equations can be obtained explicitly since each nonlocal variable in (177) is a potential for at least one of the dependent variables u, v, and Υ. For example,

u = \frac{1}{2} log(−\frac{∂Ξ_{∞}}{∂x} ).

Now we derive the covering equations imposed on the fiber variable Ξ_{t} and its limit Ξ_{∞} at the point t = −∞:

Ẽ_{t} = \{\frac{∂Ξ_{t}}{∂y} = Ξ_{t}^{2} + \frac{Ξ_{t} · \frac{∂^{2}Ξ_{t}}{∂x∂y}}{\frac{∂Ξ_{t}}{∂x}} − exp(2t)\} ,  (179)

where t ∈ ℝ ∪ {−∞}. The equations Ẽ′_{t}, Ẽ_{t}^{v}, and Ẽ_{t}^{Υ} are obtained analogously.

11.1. Integrating in nonlocal variables. We emphasize that transformations (161), (175), and (176) cannot be integrated in local variables. Therefore we consider one-dimensional non-abelian coverings (178) and extend the sets of (local) variables with new nonlocal ones, see (177), and then we integrate the transformations successfully. The results can be summarized as follows.

Theorem 67 ([58]). Consider Bäcklund (auto)transformations (161), (175)–(176) for the equations E_{u}, E_{v}, and E_{Υ}. These transformations are integrated in nonlocal variables explicitly:

(1) Bäcklund autotransformation (161) for E_{u}:

ũ = u + t − log Ξ_{t} � u = t + ũ − log Ξ_{t}[ũ](−x, −y),

i.e., to inverse the transformation and obtain u[ũ], the inversion x ↦ −x and y ↦ −y is required.

(2) Bäcklund transformation (175) between E_{u} and the wave equation v_{xy} = 0:

v = u + t − log Ξ_{∞} and, conversely, u = v + t − log Ξ_{t}^{v}.

(3) Bäcklund transformation (176) between E_{u} and the scal^{+}–Liouville equation Υ_{xy} = exp(−2Υ):

Υ = −u + t + log Ξ′_{t} and, conversely, u = −Υ − t − log Ξ_{t}^{Υ}.

Proof. We consider the case ũ[u](x, y) within Bäcklund autotransformation (161). By definition, put U = exp(ũ) and T = exp(−ũ). From Eq. (161) we obtain the Bernoulli equation

U_{x} = u_{x} · U + exp(u − t) U^{2},

whence U^{−1} = T = exp(−u − t) · Ξ, where the nonlocal variable Ξ is such that D̃_{x}(Ξ) = − exp(2u), and also get the Riccati equation

T_{y} = u_{y} · T + exp(u + t) T^{2} − exp(t − u).  (180)

Substituting exp(−u − t) · Ξ for T in (180), we get D̃_{y}(Ξ) = Ξ^{2} + 2u_{y}Ξ − exp(2t). Now we refer (177) and compare the result with the derivatives D̃_{x}(Ξ_{t}) and D̃_{y}(Ξ_{t}).

The proof of other five cases is quite analogous. Assume f(x, y) ∈ {u, ũ, v, Υ} is a known solution to the PDE E_{f}. Then we obtain either two Bernoulli equations for Eq. (175) or one Bernoulli equation and one Riccati equation for Eq. (161) and Eq. (176) after a proper change of variables. Solving these ordinary differential equations for the solution g(x, y) ∈ {u, ũ, v, Υ} to the PDE E_{g} that is related with E_{f} by one of Bäcklund (auto)transformations (175)–(176), we finally obtain the rules to differentiate the nonlocal variable, which appears in one of the coverings (178). □

Consider diagram (24) that arises in the definition of a Bäcklund transformation and apply Eq. (25) to Theorem 67. Consider the coverings τ_{1} and τ_{2} in Eq. (177). We see that in all cases one of these two mappings is a first order differential operator that depends on the nonlocal variable only, while another projector is a zero order morphism. Thence we have

Theorem 68. Consider equations (179). Then the following relations hold:

u = \frac{1}{2} log (−Ξ_{t})_{x},  ũ = t + log[Ξ_{t}^{−1}\sqrt{(−Ξ_{t})_{x}}],
u = \frac{1}{2} log (−Ξ_{∞})_{x},  v = t + log[Ξ_{∞}^{−1}\sqrt{(−Ξ_{∞})_{x}}],
u = \frac{1}{2} log (−Ξ′_{t})_{y},  Υ = t + log[\frac{Ξ′_{t}}{\sqrt{(−Ξ′_{t})_{y}}} ],

v = \frac{1}{2} log (Ξ_{t}^{v})_{x},  u = t + log[(Ξ_{t}^{v})^{−1}\sqrt{(Ξ_{t}^{v})_{x}}],
Υ = −\frac{1}{2} log (Ξ_{t}^{Υ})_{y},  u = −t + log[(Ξ_{t}^{Υ})^{−1}\sqrt{(Ξ_{t}^{Υ})_{y}}].

In other words, the nonlocal variables that satisfy Eq. (179) are potentials for both solutions f, g to the equations E_{f}, E_{g} within the list E_{u}, E_{v}, and E_{Υ}. The property of all the coverings in (178) to be nonlinear differential operators of order not greater than 1 is a special feature of these equations.

11.2. On nonlocal symmetries. Let E be an equation and τ: Ẽ → E be a covering. By definition, a τ-shadow φ is a solution of the linearized equation ℓ̃_{E}(φ) = 0. The shadow fields ˜�_{φ} ∈ D̄_{C}(Ẽ) are not true nonlocal symmetries since they do not describe the evolution of the nonlocal variable and, generally, not all of them can be extended up to nonlocal symmetries.

We have proved that Bäcklund transformations (161), (175), and (176) themselves do contain certain information about nonlocal variables such that these transformations can be integrated successfully. The corresponding non-abelian coverings in (178) provide nonlocal conservation laws for the underlying differential equations. Still, the structures on the covering equations are “too close” to the initial ones, so that the point symmetries of the initial equations and the classical symmetries of the covering equations are in one-to-one correspondence such that we get no nonlocal symmetries except the liftings of local transformations.

Now we introduce new nonlocal variables such that the symmetries we are in search of depend on them. Let Σ^{↔}_{t}= Ξ_{t} + u_{y} be the new nonlocal variable such that

D̃_{x}^{E_{u}}(Σ^{↔}_{t}) = 0,
D̃_{y}^{E_{u}}(Σ^{↔}_{t}) = (Σ^{↔}_{t})^{2} + u_{yy} − u_{y}^{2} − exp(2t).

Consider the limit of Σ^{↔}_{t} as t → −∞. We see that at the point t = −∞ there appeares the automodel variable

Σ_{∞} = u_{x} + \frac{exp(2u)}{Ξ_{∞}}

such that D̃_{y}^{E_{u}}(Σ_{∞}) = 0. We claim that

Σ^{↔}_{∞}= lim_{t→−∞} Σ^{↔}_{t}

and Σ_{∞} differ by the discrete symmetry x ↔ y. Indeed, consider the derivatives D̃_{x} and D̃_{y} of Σ_{∞} and Σ^{↔}_{∞}. We get

D̃_{x}(Σ_{∞}) = Σ_{∞}^{2} + u_{xx} − u_{x}^{2}, D̃_{y}(Σ_{∞}) = 0.

Also, we have

D̃_{x}(Σ^{↔}_{∞}) = 0, D̃_{y}(Σ^{↔}_{t}) = (Σ^{↔}_{∞})^{2} + u_{yy} − u_{y}^{2}.

Therefore we use the nonlocal variable Σ_{∞} only and treat all relations up to the symmetry transformation x ↔ y for the Liouville equation. By definition, put Σ_{t} = (x ↔ y) · (Σ^{↔}_{t}): we have

D̃_{x}(Σ_{t}) = Σ_{t}^{2} + u_{xx} − u_{x}^{2} − exp(2t) and D̃_{y}(Σ_{t}) = 0.

Nonlocal variables enable us to find shadows of nonlocal symmetries for Eq. (76) and to extend these shadows up to true nonlocal symmetries of the Liouville equation.

Proposition 69. (1) Let f(t, x, Σ_{t}) be a smooth function. Then the generating function

φ = \frac{1}{2}(Σ_{t}^{2} + u_{xx} − u_{x}^{2} − exp(2t)) · \frac{∂f}{∂Σ_{t}} + \frac{1}{2} \frac{∂f}{∂x} + u_{x} · f  (181)

is a τ_{t}-shadow of a nonlocal symmetry of the Liouville equation.

(2) Let f(x, Σ_{∞}) be a smooth function. Then the second order τshadow φ(x, Σ_{∞}, u, u_{x}, u_{xx}) for the Liouville equation is

φ = \frac{1}{2}(Σ_{∞}^{2} + u_{xx} − u_{x}^{2}) · \frac{∂f}{∂Σ_{∞}} + \frac{1}{2} \frac{∂f}{∂x} + u_{x}f = ˜□(f(x, Σ_{∞})). (182)

Nonlocal shadows (181) and (182) belong to the class (this class was considered in the paper [99]) of solutions (65) to the equation ℓ̃_{F} (φ) = 0. This class is now provided by the operator

˜□ = u_{x} + \frac{1}{2} D̃_{x}

that contains the extended total derivative D̃_{x}.

Recontruction of nonlocal symmetries. In order to extend the τ_{t}-shadows ˜�_{φ} up to true nonlocal symmetries

˜�_{φ,a} = ˜�_{φ} + a · \frac{∂}{∂Σ_{t}} ,

where a ∈ C^{∞}(Ẽ) and t ∈ ℝ ∪ {−∞}, we solve the equations

D̃_{x}(a) = ˜�_{φ,a} (D̃_{x}(Σ_{t})), D̃_{y}(a) = ˜�_{φ,a} (D̃_{y}(Σ_{t}))

for the function a.

Proposition 70. (1) Let f(t) be a smooth function and the functions φ and a(t, Σ_{t}, u_{x}, u_{xx}) be defined by the relations

φ = u_{x} · f(t), a = (Σ_{t}^{2} + u_{xx} − u_{x}^{2} − exp(2t)) · f(t).  (183)

Then the field ˜�_{φ} + a · ∂/∂Σ_{t} is a true nonlocal symmetry of Eq. (76).

(2) Let f(x) be a smooth function and the functions φ and a(Σ_{∞}, u_{x}, u_{xx}) be given in

φ = u_{x} f(x) + \frac{1}{2} \frac{df}{dx}, a = (Σ_{∞}^{2} + u_{xx} − u_{x}^{2}) f(x) + Σ_{∞} \frac{df}{dx} + \frac{1}{2} \frac{d^{2}f}{dx^{2}} . (184)

Then the field ˜�_{φ} + a · ∂/∂Σ_{∞} is a true nonlocal symmetry of Eq. (76).

The proof of Propositions 69 and 70 is very extensive and cannot be completed without application of the Jet ([72]) software for analytic transformations. The Jet environment allows to set the determining equations in a dialogue mode, select the simplest differential consequences to these equations, and then specify the expressions for the unknown symmetries.

Nonlocal symmetry (183) is defined up to elements of CD(Ẽ) ∋ g·D̃_{x}, g ∈ C^{∞}(Ẽ). Thence, the nonlocal symmetry class (183) is [ ˜�_{φ,a} ] = [−f(t) · ∂/∂x], where f(t) · ∂/∂x is the translation. Symmetry (184) is the lifting of a classical point symmetry φ_{0}^{f}, see Proposition 11 on page 38. As usual, a class of nonlocal symmetries ˜�_{φ,a} can be obtained from Eq. (183) and (184) by using the discrete transformation x ↔ y.

11.3. On permutability of Bäcklund transformations. Now we illustrate the permutability property of Bäcklund (auto)transformations (161), (175), and (176).

Proposition 71. (1) Let u^{j}, j = i, ii, be solutions to Eq. (76) such that B_{u}(u, u^{j}; t_{j}) = 0, t_{j} ∈ ℝ. Then there is a unique solution u′′′(x, y) to the system

\{ \begin{matrix}B_{u}(u′, u′′′; t_{2}) & = & 0, \\ B_{u}(u′′, u′′′; t_{1}) & = & 0. \end{matrix}  (185)

Namely, the solution u′′′ satisfies the relation

exp(u′′′) = exp(u) · \frac{k_{2} exp(u′) − k_{1} exp(u′′)}{k_{2} exp(u′′) − k_{1} exp(u′)},  (186)

where k_{j} ≡ exp(t_{j}).

(2) Let j = i, ii and t_{j} ∈ ℝ. Suppose both v^{j} are solutions to the wave equation v_{xy} = 0 such that B_{uv}(u, v^{j}; t_{j}) = 0. Also, suppose that u^{j} are solutions to the Liouville equation such that B_{uv}(u^{j}, v; t_{j}) = 0. Then there are unique solutions u′′′ and v′′′ to the systems

\{ \begin{matrix}B_{uv}(u′′′, v′; t_{2}) & = & 0 \\ B_{uv}(u′′′, v′′; t_{1}) & = & 0 \end{matrix}  and  \{ \begin{matrix}B_{uv}(u′, v′′′; t_{2}) & = & 0 \\ B_{uv}(u′′, v′′′; t_{1}) & = & 0, \end{matrix}

respectively. Denote k_{j} ≡ exp(t_{j}), then the following relations hold:

exp(u′′′) = exp(u) · \frac{k_{2} exp(v′) − k_{1} exp(v′′)}{k_{2} exp(v′′) − k_{1} exp(v′)},
exp(v′′′) = exp(v) · \frac{k_{1} exp(u′′) − k_{2} exp(u′)}{k_{2} exp(u′′) − k_{1} exp(u′)}.

(3) Let j = i, ii and t_{j} ∈ ℝ. Assume that Υ^{j} are solutions to the scal^{+}-equation E_{Υ} such that B_{uΥ}(u, Υ^{j}; t_{j}) = 0, and suppose that u^{j} are solutions to the Liouville equation such that B_{uΥ}(u^{j}, Υ; t_{j}) = 0. Then there are unique solutions u′′′ and Υ′′′ to the systems

\{ \begin{matrix}B_{uΥ}(u′′′, Υ′; t_{2}) & = & 0 \\ B_{uΥ}(u′′′, Υ′′; t_{1}) & = & 0 \end{matrix}  and  \{ \begin{matrix}B_{uΥ}(u′, Υ′′′; t_{2}) & = & 0 \\ B_{uΥ}(u′′, Υ′′′; t_{1}) & = & 0, \end{matrix}

respectively. We also have

exp(u′′′) = exp(u) · \frac{k_{2} exp(Υ′) − k_{1} exp(Υ′′)}{k_{2} exp(Υ′′) − k_{1} exp(Υ′)},
exp(Υ′′′) = exp(Υ) · \frac{k_{1} exp(u′′) − k_{2} exp(u′)}{k_{2} exp(u′′) − k_{1} exp(u′)},

where k_{j} ≡ exp(t_{j}).

Proof. We consider the case of Bäcklund autotransformation (161) only; cases 2 and 3 are treated analogously. Consider the subsystem in Eq. (185) that consists of relations (161) which contain the derivatives with respect to x only. Then the solution u′′′ defined in Eq. (186) follows from the linear dependence of the left-hand side within Eq. (185) and is unique. Consider another subsystem composed by the relations in Eq. (161) that contain the derivatives with respect to y. Then there are two solutions to this subsystem; we denote them by u′′′ and ū′′′. The solution u′′′ is defined in Eq. (186) and ū′′′ is obtained from the relation

exp(ū′′′) = exp(−u) · \frac{k_{1} exp(u′) − k_{2} exp(u′′)}{k_{2} exp(−u′) − k_{1} exp(−u′′)}.

The solution ū′′′ is irrelevant. Therefore, the function u′′′ is a unique solution to the whole system (185). □

Remark 28. Proposition 71 means that the diagrams

\begin{matrix}u & ⟶^{t_{1}} & u′ \\ ^{t_{2}}↓ & & ↓^{t_{2}} \\ u′′ & ⟶_{t_{1}} & u′′′ \end{matrix} , \begin{matrix}u & ⟶^{t_{1}} & v′ & ⟶^{t_{3}} & u′′ \\ ^{t_{2}}↓ & & ↓^{t_{2}} & ↓^{t_{2}} \\ v′′ & ⟶_{t_{1}} & u′ & ⟶_{t_{3}} & v′′′ \end{matrix} ,

and the diagram

u ⟶^{t_{1}} Υ′ ⟶^{t_{3}} u′′
t_{2}↓  ↓^{t_{2}}  ↓^{t_{2}}
Υ′′ ⟶_{t_{1}} u′ ⟶_{t_{3}} Υ′′′

are commutative for any t_{1}, t_{2}, t_{3} ∈ ℝ.

12. Zero–curvature representations

In this section, we illustrate the relationship between the parametric families of zero–curvature representations and Bäcklund transformations for Eq. (158). In Sec. 1 we assigned the Liouville equation to the Lie algebra 𝔤 = 𝔰𝔩_{2}(ℂ), and now we benefit from the use of different representations of this algebra.

There is a natural equivalence ([13]) between 𝔤-valued zero–curvature representations of a PDE E and the special type coverings over the equation E. Further on, we study the case 𝔤 ≃ 𝔰𝔩_{2}(ℂ). It is essential that there is a representation of 𝔰𝔩_{2}(ℂ) in vector fields. We use it in order to construct the required coverings over the hyperbolic Liouville equation

E_{Liou} = {u_{xy} = exp(2u)}.  (187)

Here ‘2’ is the Cartan 1 × 1-matrix of the Lie algebra A_{1} while x and y are the coordinates in the standard two–dimensional extension (z, z̄) ↪ ℂ^{2} ∋ (x, y). By ⟨e, h, f⟩ we denote the canonical basis such that

[h, e] = 2e,  [h, f] = −2f,  [e, f] = h.  (50′)

Consider the representation ρ: 𝔰𝔩_{2}(ℂ) → D(ℂ_{2}[Ξ]) of the Lie algebra 𝔤 in the space of polynomial-valued derivations:

ρ(e) = 1 · \frac{∂}{∂Ξ},  ρ(h) = −2Ξ · \frac{∂}{∂Ξ}, ρ(f) = −Ξ^{2} · \frac{∂}{∂Ξ} (188a)

such that the Lie bracket is the commutator of vector fields: [A, B] = A∘B −B ∘A. This representation of the Lie algebra 𝔰𝔩_{2}(ℂ) was used in the paper [52], where a class of N-ary analogs for the Lie algebraswas constructed. Also, consider the matrix representation

ϱ(e) = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) , ϱ(h) = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) , ϱ(f) = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix})  (188b)

such that the Lie bracket is the matrix commutator: [A, B] = A · B − B · A.

Given an equation E, consider the flat connection form (47) in the bundle C^{∞}(E^{∞}) ⊗ G → ℂ^{2}, where G is the Lie group of 𝔤. Suppose A, B ∈ C^{∞}(E^{∞}) ⊗ 𝔤. The zero-curvature condition (48), which is equivalent to the relation

[D̄_{x} + A, D̄_{y} + B] = 0,

is satisfied on the differential equation E. Finally, we obtain the matrix equation

D̄_{y}A − D̄_{x}B − [A, B] = 0.  (189)

Now, decompose the matrices A and B with respect to the basis in the representation ϱ: 𝔤 → {M ∈ Mat(2, 2) | tr M = 0}:

A = a_{e}⊗ϱ(e)+a_{h}⊗ϱ(h)+a_{f}⊗ϱ(f), B = b_{e}⊗ϱ(e)+b_{h}⊗ϱ(h)+b_{f}⊗ϱ(f), where a_{μ}, b_{ν} ∈ C^{∞}(E^{∞}), and construct the one-dimensional covering τ over E such that Ξ is the nonlocal variable, the extended total derivatives D̃_{x} and D̃_{y} are

D̃_{x} = D̄_{x} + a_{e} ⊗ ρ(e) + a_{h} ⊗ ρ(h) + a_{f} ⊗ ρ(f),
D̃_{y} = D̄_{y} + b_{e} ⊗ ρ(e) + b_{h} ⊗ ρ(h) + b_{f} ⊗ ρ(f),

respectively, and the rules to differentiate the variable Ξ are

\begin{aligned}D̃_{x}(Ξ) & = dx (a_{e} ⊗ ρ(e) + a_{h} ⊗ ρ(h) + a_{f} ⊗ ρ(f)), \\ D̃_{y}(Ξ) & = dy (b_{e} ⊗ ρ(e) + b_{h} ⊗ ρ(h) + b_{f} ⊗ ρ(f)). \end{aligned}  (190)

We see that the Maurer–Cartan condition (48), which is satisfied on E, is equivalent to the compatibility condition [D̃_{x}, D̃_{y}] = 0 that holds in virtue of the equation E^{∞}.

Example 22. First, we obtain Bäcklund transformation between the Liouville and the wave equations. Consider Eq. (53) on page 34 and choose the gauge

a_{e} ≡ a_{e}^{1} = exp(κu), b_{f} ≡ b_{f}^{1} = exp((2 − κ)u),

for an arbitrary constant κ. Then the covering equation Ẽ is

\{ \begin{matrix}v_{x} & = (κ − 2)u_{x} + exp(κu − v) \\ v_{y} & = κu_{y} − exp((2 − κ)u + v) \end{matrix}\} .  (191)

Here the variable v = log Ξ is a transformation of the nonlocal variable Ξ, see Eq. (190). The compatibility condition for Eq. (191) is

v_{xy} = (κ − 1) exp(2u).

Finally, suppose κ = 1, then Eq. (191) is the Bäcklund transformation ([14])

\begin{aligned}(v + u)_{x} & = exp(u − v), \\ (v − u)_{y} & = − exp(u + v) \end{aligned}  (191_{1})

between the Liouville equation Eq. (187) and the wave equation

v_{xy} = 0,  (192)

while the coordinate Ξ is exactly the one that allows integrating equation (191_{1}) in nonlocal variables, see Sec. 11.

Remark 29. Transformation (191_{1}) is the particular case t = 0, k ≡ exp(t) = 1 within the family of Bäcklund transformations (175) between Eq. (187) and Eq. (192). We note that the mapping k ↦ −k is the replacing of representation (188a) with the representation ρ̄: 𝔰𝔩_{2}(ℂ) → D(ℂ_{2}[Ξ]) such that

ρ̄(e) = −1,  ρ̄(h) = −2Ξ,  ρ̄(f) = Ξ^{2}.

Zero-curvature representations constructed by using Bäcklund transformations. Consider Bäcklund autotransformation (161) for the Liouville equation (187) and Bäcklund transformation (176) between the Liouville equation and the Liouville scal^{+}-equation

E_{Υ} = {Υ_{xy} = exp(−2Υ)}

(we recall that the latter equation is the hyperbolic representation of the Gauss equation for the conformal metric of constant curvature +1, see Example 10 on page 29). Next, consider the coverings over the Liouville equation that provide these transformations. Then, these coverings exceed ansatz (51).

Namely, the flat connection form for Bäcklund autotransformation (161) is

θ = (\begin{matrix}−\frac{1}{2}u_{x} & 0 \\ − exp(u − t) & \frac{1}{2}u_{x} \end{matrix}) dx + (\begin{matrix}\frac{1}{2}u_{y} & − exp(t + u) \\ − exp(t − u) & −\frac{1}{2}u_{y} \end{matrix}) dy.

The 𝔰𝔩_{2}-valued flat connection form θ for Bäcklund transformation (176) is

θ = (\begin{matrix}−\frac{1}{2}u_{x} & exp(−t − u) \\ − exp(u − t) & \frac{1}{2}u_{x} \end{matrix}) dx + (\begin{matrix}\frac{1}{2}u_{y} & − exp(t + u) \\ 0 & −\frac{1}{2}u_{y} \end{matrix}) dy.

By construction, these 𝔰𝔩_{2}-valued forms are zero–curvature representations for the Liouville equation.

Bäcklund transformations constructed by using zero–curvature representations. The problem of constructing multi–parametric families of Bäcklund transformations by using known zero–curvature representations for Eq. (187) was discussed in the paper [86]; later, this problem was studied thoroughly by V. Golovko in [32].

Now the exposition follows [86] and [32]. Let us find three 𝔰𝔩_{2}-valued classes of zero–curvature representations θ ∈ Λ̄^{1}(E_{Liou}^{∞}) ⊗ 𝔰𝔩_{2}(ℂ) for the hyperbolic Liouvile equation (187). Assume that A = A(u_{x}), B = B(u), and suppose [A, B] ≠ 0. Then Eq. (49′) on page 111 is reduced to

u_{x}^{−1} \frac{∂A}{∂u_{x}} − exp(−2u)\frac{∂B}{∂u} − [u_{x}^{−1}A, exp(−2u)B] = 0.

Applying ∂^{2}/∂u ∂u_{x} to this identity, we get the equation [M, N] = 0, where

M = \frac{∂(A/u_{x})}{∂u_{x}}, N = \frac{∂(B/ exp(2u))}{∂u}.

There are three possible cases:

(1) M = 0,

(2) N = 0, and

(3) M = r(u_{x}) · C, N = s(u) · C, where C ≠ 0 is a constant 𝔰𝔩_{2}(ℝ)- valued matrix �nd r, s ∈ C^{∞}(ℝ) are smooth functions.

Hence we get three gauge non-equivalent clases of zero–curvature representations:

Case 1 (M = 0). There are no nontrivial solutions to Eq. (49′).

Case 2 (N = 0). There are two classes of zero–curvature representations. They are

A = (\begin{matrix}2αu_{x} + 2β & 2α \\ u_{x}^{2}(1 − 2α) − 4βu_{x} + 2γ & −2αu_{x} − 2β \end{matrix}) ,
B = (\begin{matrix}0 & 0 \\ exp(2u) & 0 \end{matrix}) ,  (193a)
A = (\begin{matrix}αu_{x}^{2} + 2β & 2δ exp(−2αu_{x}) \\ 2γ exp(2αu_{x}) & −2αu_{x}^{2} − 2β \end{matrix}) ,
B = (\begin{matrix}α exp(2u) & 0 \\ 0 & −α exp(2u) \end{matrix}) ,  (193b)

Case 3 (M ≠ 0, N ≠ 0). Another solution to Eq. (49′) is

A = (\begin{matrix}u_{x} & 1 \\ 0 & −u_{x} \end{matrix}) , B = (\begin{matrix}0 & α exp(−2u) \\ exp(2u) & 0 \end{matrix}) ,  (194)

where α, β, γ, and δ are arbitrary constants.

By [13], any 𝔰𝔩_{2}-valued zero–curvature representation for E provides some special type covering over the equation E. Consider the representation ρ′: 𝔰𝔩_{2}(ℂ) → D(ℂ[[v]]):

ρ′(e) = exp(−v)\frac{∂}{∂v} ,  ρ′(f) = − exp(v)\frac{∂}{∂v} ,  ρ′(h) = −2\frac{∂}{∂v} ,

of the Lie algebra 𝔰𝔩_{2}(ℂ) in the space of differential operators on ℂ, i.e., v ∈ ℂ is the nonlocal variable and ρ′: 𝔰𝔩_{2}(ℂ) → D(ℂ[[v]]). Then we construct one-dimensional coverings over the Liouvile equation E_{Liou} that provide Bäcklund transformations between E_{Liou} and some differential equations that depend on the initial zero–curvature representation.

Proposition 72 ([32]). Representations (193a), (193b), and (194) correspond to Bäcklund transformations between E_{Liou} and the equations

v′_{x} = \frac{1}{4}(2α − 1) exp(v′) (\frac{v′_{xy}}{v′_{y}} − v′_{x})^{2} + 2(β exp(v′) − α) (\frac{v′_{xy}}{v′_{y}} − v′_{x}) + 2α exp(−v′) − 4β − 2γ exp(v′),

v′′_{x} = −\frac{αv′′_{xy}^{2}}{2v′′_{y}^{2}} − 4β + 2δ exp (−\frac{αv′′_{xy}}{v′′_{y}} − v′′) − 2γ exp (\frac{αv′′_{xy}}{v′′_{y}} + v′′) ,
v′′′_{xy}^{2} = exp(−2v′′′)(v′′′_{y}^{2} + 4α).

If α = 0 in representations (194), then we get Bäcklund transformation between E_{Liou} and the wave equation Eq. (192).

Finally, we analyse the removability of the parameters α, β, γ, and δ in zero–curvature representations (193)–(194) with respect to the gauge transformations

A ↦ SAS^{−1} − (D_{x}S)S^{−1}, B ↦ SBS^{−1} − (D_{y}S)S^{−1}.

The result is described in

Remark 30 ([32]). The parameter β in Eq. (193a) is removable by the gauge transformation

S = a · (\begin{matrix}1 & 0 \\ β/α & 1 \end{matrix})

that depends on an arbitrary constant a ∈ ℂ; under this transformation γ ↦ γ + β^{2}/α. All other parameters α, β, γ, and δ in zero–curvature representations (193)–(194) are non-removable.

Appendix A. Geometric methods of solving boundary–value problems

In this appendix, we consider several methods for solving boundary– value problems (mainly, the Dirichlet problem) for nonlinear equations of mathematical physics. Then we compare and analyse the results of a computer experiment in applying the described algorithms, see also [49]. Again, the methods which we study are based on the geometry of the jet spaces ([10, 63, 81]) and treating differential equations E (and their prolongations E^{∞} as well) as submanifolds E ⊂ J^{k}(π) of the jet space of order k for a certain fibre bundle π (respectively, we have E^{∞} ⊂ J^{∞}(π)). By using this approach, we pay attention to the analysis of the following objects: they are

• differential equations and sections s such that the jet j_{k}(s) ⊂ E defines a solution of E,

• boundary–value problems P = (E, D, f = s|_{∂D}) in a domain D,

• and their deformations Ṗ. We emphasize that the geometrically motivated technology of solving the boundary–value problems is discussed within this appendix. We do not stop on particular properties of some solutions in practical situations. Basic definitions and concepts were formulated in the Introduction, see also [10, 63, 79, 81]. We use the elliptic Liouville equation

E_{Liou} = {u_{zz̄} = exp(2u)}

as a basic example.

The appendix is organized as follows. In Sec. A.1, the method of monotonous iterations for the solutions u^{t} of the boundary–value problem P is described. Here t ∈ ℕ. This method is based of the theory of differential inequalities ([79]). In Sec. A.2, we construct a method for solving the boundary–value problems that involves the evolution representation of the equation E under study. We shall see that this interpretation is admissible in a sufficiently general situation ([10, 81]). Then, in Sec. A.3 we consider various methods based on the deformations ḟ of the boundary conditions f and simultaneous invariance of the equation E:

Ṗ = (E, D, ḟ).

In Sec. A.4 we describe the relaxation method based on the substitution

P ↦ P′ = (E′, D × ℝ_{+}, f ⊗ id_{t})

of the boundary–value problem such that solutions u of the problem P are stable stationary solutions for the new problem P′ with respect to the evolution equation E′. The deformations

Ṗ = (Ė, D, f),

which are considered in Sec. A.5, are in some sence antipodal to ones described in Sec. A.3. Now, the boundary condition f is invariant and the equation E(t) moves. The “planting” of a nonlinearity could be an example. In the final section of this appendix, we discuss the results of a computer experiment in practical application of all these methods.

Remark 31. In what follows, we assume that there is a unique classical solution to the boundary–value problem at hand (see [34]). Also, we recall that the transformation laws of solutions with respect to transformations of the independent variables are known for some equations, e.g., the conformally invariant Toda equations, see Proposition 11 on page 38). Therefore, the domain D, where the problem P is solved, can be chosen relatively regular (we set D ∼ B_{0}^{1} for Eq. (55)).

Remark 32. First, we point out a practically useful way to construct a set of solutions (that can be large enough) of a boundary–value problem P = (E, D, f). Assume that the equation E is invariant with respect to a vector field X. Denote its generating section by φ_{X} and suppose that A_{t} is the flow of the field X such that

A_{t}(∂D) = ∂D and A_{t}(j_{k}(f)) = j_{k}(f),

that is, the boundary condition is unvariant with respect to the symmetry X of the equation E at hand. Then problem P can be reduced to finding solutions of the boundary–value problem

P′ = (E ∩ {φ_{X} = 0}, D, f)

that are invariant with respect to X at any point in D. If the problem P′ has a solution, then, in general, this solution may be not unique.

Example 23. Consider the Dirichlet boundary–value problem

{u_{xx} + u_{yy} = exp(2u),  u|_{r=1} = f, f ∈ C^{0,λ}}  (195)

in the unit disc B_{0}^{1} and assume that the boundary condition is homogeneous and trivial, f ≡ 0. The solutions for the Liouville equation E_{Liou} that are invariant with respect to its point symmetries are

u = \frac{1}{2} log \frac{v_{x}^{2} + v_{y}^{2}}{sinh^{2}v} ,  (196a)
u = \frac{1}{2} log \frac{v_{x}^{2} + v_{y}^{2}}{v^{2}} ,  (196b)
u = \frac{1}{2} log \frac{v_{x}^{2} + v_{y}^{2}}{sin^{2} v} ,  (196c)

where v is a harmonic function,

Δv = 0.

Consider the radial symmetry

X = −y \frac{∂}{∂x} + x \frac{∂}{∂y} .

Then the solutions u(r) = u(\sqrt{x^{2} + y^{2}}) of problem (195) are the following:

u_{1}(r) = log \frac{2(1 + \sqrt{2})}{(1 + \sqrt{2})^{2} − r^{2}} ,  (197a)
u_{2}(r) = − log r − log(1 − log r),  (197b)
u_{3}(r) = \frac{1}{2} log \frac{α^{2}r^{−2}}{sin^{2}(αβ − α log r)},  β = \frac{±1}{α} arcsin α, α ≠ 0. (197c)

A unique classical solution u_{1} has no singularity at the point 0. The solution u_{2} has an integrable singularity at r = 0, and its gradient on the border vanishes. The solution u_{3} has the cardinal set of logarithmic singularities along the radius r. These singularities accumulate as r → 0.

Remark 33. Suppose that the equation E at hand is Euler. Then, obviously, one can search solutions of the Dirichlet boundary–value problem by using the direct minimization of the action functional. We emphasize that the projective methods can be used in addition to the lattice discretization methods.

A.1. The monotonous iterations method. The monotonous iterations method is a useful instrument in the theory of differential inequalities ([79]). This method allows to contruct solutions of the Dirichlet boundary–value problems

{Δu = h(u, x), u|_{∂D} = f,  ∂D ∈ C^{1+ε}, f, h ∈ C^{0,λ}} (198)

and then check the local uniqueness of these solutions.

Definition 13. A function α ∈ C(\overline{D}) ∩ C^{2}(D) (resp, β) is a lower (upper) solution of problem (198) is the following two conditions hold:

(1) Δα − h(α, x) ≥ 0 in D and

(2) f(x) ≥ α(x) on ∂D (respectively, ≤). Suppose further α(x) ≤ β(x) for each x ∈ \overline{D}. Then we set α ⪯ β.

Proposition 73 ([79]). Assume that a lower and an upper solutions of problem (198) are ordered, α ⪯ β. Suppose there is a nonnegative constant C ∈ ℝ such that

h(u_{1}, x) − h(u_{2}, x) ≤ C · (u_{1} − u_{2})

for any x ∈ \overline{D}, u_{1}, and u_{2} provided that α ≤ u_{2} ≤ u_{1} ≤ β. Consider two sequences \underline{U} = \{\underline{u}^{k}\} and \overline{U} = \{\overline{u}^{k}\} of solutions of the problem

Δu^{k} − C u^{k} = h(u^{k−1}, x) − C u^{k−1}, u^{k}|_{∂D} = f, x ∈ D, k ∈ ℕ.

(199)

Here the inital values are \underline{u}^{0} = α and \overline{u}^{0} = β, respectively. Then the monotonously nondecreasing (nonincreasing) sequence \underline{U} (\overline{U}) converges to a solution \underline{u} (resp., \overline{u}) for problem (198). Moreover, we have \underline{u} ≤ \overline{u} in \overline{D} and the bound \underline{u} ≤ u_{∗} ≤ \overline{u} holds for any solution u_{∗} ∈ [α, β].

Suppose further that h is monotonous with respect to u: h(u_{1}, x) − h(u_{2}, x) ≥ 0 provided that α ⪯ u_{2} ⪯ u_{1} ⪯ β. Then the solution for problem (198) is unique on [α, β]: \underline{u} = \overline{u}.

The notions of a lower and upper solution, which were introduced in Sec. A.1, and the method of proving the local uniqueness for solutions of problem (198) will be used in Sec. A.4. The relaxation method will be described there. In that case, the evolution of the superscript k is continuous unlike in problems (199).

A.2. Evolutionary representation of differential equations. The following remarkable result was in fact obtained in the papers concerning the formal theory of differential equations (see [10, 81] and references therein): any equation that satisfies the assumtions of the ‘2-line theorem’ for the Vinogradov’s C-spectral sequence ([10, 63]) admits a representation in the form of an evolution equation. An illustration is given in Example 3 on page 12. In practice, this means that any equation that does not have gauge symmetries (unlike the Maxwell, the Yang–Mills, and the Einstein equations and similar systems) admits a set of coordinate transformations and differential substitutions that map it to an evolution equation (possibly, the new equation is imposed on a larger set of dependent variables). The class of equations subject to the assumptions of the ‘2-line theorem’ is really wide. Of course, fixedprecision integrating of evolution equations is simpler that integrating of arbitrarily chosen equations of mathematical physics. Therefore, solving the initial boundary–value problem is divided to two stages. First, we find an evolution representation of the equation E. Then we reconstruct the boundary conditions for the auxiliary dependent variables if there appeares a necessity to introduce them.

Example 24. Consider the boundary–value problem

\begin{matrix}u_{xx} ± u_{yy} = exp(2u), D = \{(x, y), |x| < 1, |y| < 1\}, \\ u(1, y) = f_{1}(y), u(x, 1) = f_{2}(x), u(−1, y) = f_{3}(y), u(x, −1) = f_{4}(x) \end{matrix}

(200)

for the elliptic (resp., hyperbolic) Liouville equation E_{Liou}. Introduce the additional dependent variable v = u_{y}. Then the equation E_{Liou} has the form

\frac{∂}{∂y} (\begin{matrix}u \\ v \end{matrix}) = (\begin{matrix}v \\ ∓u_{xx} ± exp(2u) \end{matrix})

and the conditions on the bound ∂D are split to the initial and the boundary conditions:

u(x, −1) = f_{4}(x), u(1, y) = f_{1}(y),  u(−1, y) = f_{3}(y)
v(x, −1) = v[f_{2}], v(1, y) = df_{1}(y)/dy, v(−1, y) = df_{3}(y)/dy.

Thence, solving boundary–value problem (200) is reduced to reconstruction of the initial section v(x, −1) by the terminal section u(x, 1) = f_{2}(x). In practice, this can be done by using the conjugated gradient method.

A.3. Evolution of boundary conditions. Recall that any symmetry φ ∈ sym E^{∞} of an equation E = {F⃗ = 0} is an element of the kernel ker ℓ̄_{F} of the linearization

ℓ_{F} = ∥∑_{σ} \frac{∂F_{i}}{∂u_{σ}^{j}} D_{σ} · 1_{ij}∥

restricted onto E. The evolutionary vector field

�_{φ} = ∑_{j,σ} D̄_{σ}(φ^{j}) \frac{∂}{∂u_{σ}^{j}}

commutes with the total derivatives D_{i} and is tangent to the infinite prolongation

E^{∞} = {D_{σ}(F) = 0, |σ| ≥ 0} ⊂ J^{∞}(π).

The latter is the projective limit of smooth epimorphisms by definition. The manifold E^{∞} that is defined by the closed algebra of smooth functions is in fact infinite–dimensional, therefore, formally, the field �_{φ} has no flow. Indeed, there is no Cauchy theorem for the initial–value problem

u̇_{σ} = D̄_{σ}(φ), |σ| ≥ 0, u(t = 0) = u (201)

composed by the cardinal set of equations. The integral trajectories for problem (201) exist provided that φ(x, u, D_{i}(u^{j})) is a contact symmetry. In particular, the point summetries that are linear with respect to the derivatives suit well. From Eq. (201) it follows that the functions φ describe the correlated evolution of the dependent variables and their derivatives. This evolution preserves solutions of the equation E if φ ∈ ker ℓ̄_{F}.

A.3.1. Suppose a solution u_{0} of some boundary–value problem P_{0} = (E, D, f_{0}) for an equation E is known. Then, deform the problem P = (E, D, f) such that f(t = 0) = f_{0} and f(t = 1) = f. Here by ḟ(t) we denote the deformation velocity of the boundary condition on ∂D and by φ(t) the corresponding deformation of the solution. One easily checks that φ ∈ ker ℓ̄_{F}. Suppose further that the problem

φ ∈ ker ℓ̄_{F} ,  φ|_{∂D} = ḟ  (202)

is soluble at each t ∈ [0, 1). Then,

u = u_{0} + ∫_{0}^{1} φ(t) dt

is the required solution for the problem P.

The condition φ ∈ ker ℓ̄_{F} is the decomposition of the deformation u̇ = φ to a power series in t. The analysis of the deformation equations that are coefficiets of the higher powers of t is interesting by itself, see [64]. For example, insolubility of the equation at t^{2} implies impossibility to solve the problem P by using this deformation method.

Remark 34. The Liouville equation E_{Liou} has the following peculiar feature. Its general solution, see Eq. (87), that depends on an arbitrary holomorphic function v(z) is always invariant with respect to a point symmetry X whose coefficients are related with v(z) by the Abel transformation. The curve X(t) in the space sym E_{Liou} is assigned to the deformation ḟ of the boundary condition. This curve is such that the solution u(t′) is invariant with respect to X(t′) for any t′ ∈ [0, 1]. Therefore, the solution of boundary–value problem (195) can be reduced to analysis of the equations that define the curve X(t).

A.3.2. Consider a particular covering strucuture Ẽ^{∞} → E^{∞} over the equation E at hand, that is, suppose that the substitution u = u[v] maps solutions v ∈ Sol Ẽ to solutions u ∈ Sol E. Several examples are well known: the Cole–Hopf substitution

u = \frac{v_{x}}{v} ,  (203)

the Miura transformation u = v_{x} − v^{2}, and the formula

u = \frac{1}{2} log[4∂v · ¯∂v̄/(1 − vv̄)^{2}]  (204)

by Liouville ([69]) that relate the Burgers and the heat equations, the Korteweg–de Vries equation and the modified Korteweg–de Vries equation, and the Liouville and the Cauchy–Riemann equations, respectively. We recall that all three expressions (196) are transformed to Eq. (204) by an appropriate change of variables. Also, suppose the condition f is fixed. Then there can be several substitutions u[v] that solve the boundary–value problem P. Therefore, the reconstruction problem for the function v in the whole domain D is incorrect by Hadamars. Still, assume that a class of the substitutions v ↦ u is fixed. Then one easily obtains the coordinate expressions for the equations

ψ ≡ v̇ ∈ ker ℓ̃Ẽ,

which are analogous to Eq. (202), plus the condition

˜�_{ψ} (u[v]) = ḟ

defined by the boundary functions f and f_{0}, and the quadrature

v(1) = v(0) + ∫_{0}^{1} ψ(t) dt.

Thence we construct the solution u of the initial boundary–value problem P in the whole domain D by using the exact formula u = u[v] and the solution v(1) of the covering equation Ẽ. Still, we see that the condition ˜�_{ψ} (u[v]) = ḟ may not even be defined by an operator with directional derivative if the function u depends on the gradient grad v explicitly.

Nawadays, there exist regular algorithmic methods ([10], see also Chapter 4) that allow obtaining and classification of the coverings over equations of mathematical physics. These algoriths are already available as environments ([72]) for the symbolic transformations software. We also recall that the concept of coverings over differential equations is closely related with the theory of Bäcklund transformations and zero– curvature representations (see the preceding section) for PDE. The latter structures also provide some classes of the substitutions u[v] for a given equation E.

A.3.3. Now we consider in more details solving the Dirichlet boundary–value problem, see Eq. (195), for the Liouville equation E_{Liou} within the class of substitutions (196b) by using the methods described above. Recall that the harmonic functions v(t) admit the representations P [g] via the Poisson kernel P (see [34]) by their boundary value g ≡ v|_{∂D}. Therefore, the initial boundary–value problem P which is solved by using the homotopy ḟ of the solution u such that f(t) are its values on ∂D is reduced to the integral equation

\frac{d}{dx}P [g] · \frac{d}{dx}P [ġ] + \frac{d}{dy} P [g] · \frac{d}{dy} P [ġ] = (gġ + g^{2} · (f − f_{0})) · exp(2f)

with respect to the deformation ġ = ψ|_{∂D} of the bounder value for the harmonic function v. Suppose the boundary value for v is obtained, then an approximation of v on a lattice in D can be obtained by using multiple averaging of the values of v in the neighbouring interpolation points (see also Sec. A.4 below).

In Remark A.6.2 on page 124, the results of a computer experiment in application of this method for solving boundary–value problem (195) are discussed.

A.3.4. Finally, we note that solutions of some other equation E′ (in general, distinct from the equation E at hand) can be used for construction of the initial sections u_{0} that correspond to the boundary values f_{0} on ∂D. Here we assume that the solutions u(t) satisfy a degenerate equation E′ as t → 0.

Example 25. Suppose f ⇉ −∞ such that max f −min f ≤ const < ∞. Then a solution u of boundary–value problem (195) in the disc B_{0}^{1} approximated with respect to the norm ∥ · ∥_{C^{0}(B̄_{0}^{1})} by the harmonic function P [f] as closely as desired. Let

Δu(t) = exp(2u(t)), f(t) = f + log t, 0 < t ≤ 1,

then in the same notation we have

Δφ(t)−2 exp(2u(t))φ(t) = 0, ḟ(t) = t^{−1}, u(t) = u_{t_{min}} +∫_{t_{min}}^{t}φ(τ)dτ,

where t_{min} → +0 and u_{t_{min}} = P [f + log t_{min} ] is the initial approximation. Now suppose t is small. We expand the solution u(t) by using the Hadamars lemma and obtain u(t) = log t+P [f]+t^{2} ·U. If t → +0, then the function U satisfies the homogeneous problem

ΔU = exp(2P [f]), U|_{∂B_{0}^{1}} = 0

for the Poisson equation. Its solution is

U(p) = ∫_{B_{0}^{1}} exp(2P [f])(q) · G(p, q) dq,

where G is the Green function ([34]) for the Laplace operator in the unit disc.

In fact, the method described in Sec. A.3.4 is based on the simultaneous deformation Ṗ of the boundary condition f and the equation E itself. Indeed, we have E(0) = E′ and E(t) = E for t ∈ (0, 1] by construction. In Sec. A.4, we consider an opposite situation: an auxiliary equation E′ is used at any value of the parameter t. Here we assume that solutions of E′ tend to solutions of the problem P as t increases.

A.4. The relaxation method. Complement the mixed boundary– value problem, which is a generalization of problem (198), by the relaxation term ∂u/∂t. Assume that α and β are the lower and the upper solutions of the initial stationary problem, respectively. Then, extend the boundary value f onto ∂D × ℝ_{+} and fix an initial approximation u_{0} such that α ⪯ u_{0} ⪯ β: hence we obtain

\frac{∂u}{∂t} − Δu = −h(u, x), a \frac{∂u}{∂n⃗} + b u = f, u(x, 0) = u_{0}(x). (205)

Proposition 74 ([79]). Let the above assumptions hold. Assume further that h ∈ C_{u}^{1}[α, β]. Then the following two statements are equivalent:

(1) the stationary solution u_{s} of problem (205) is unique in [α, β];

(2) the solution u_{s} ∈ [α, β] is asymptotically stable such that the stability domain is [α, β].

Here we discussed a method based on the replacement of the elliptic equation E in problem (198) by the evolution equation E′. This method is an alternative to the evolutionary representation method for the initial equation E (see Sec. A.2).

A.5. Deformation of the equation. Now we consider the deformations Ṗ of the boundary–value problem P(t) that are induced by the evolution Ė of the equation E(t) = {F (t) = 0} at hand. Here we assume that the boundary value f remains invariant. Namely. consider the homotopy

E(t) = (1 − r(t)) · E_{0} + r(t) · E,

where r(0) = 0 and r(1) = 1. Then the velocity φ of deformation of the solution u(t) is subject to the boundary–value problem

ℓ̄_{F (t)}(φ) + r′(t) · (F − F_{0}) = 0, φ|_{∂D} = 0.

Therefore, the solution of the problem P is defined by the quadrature u = u_{0} + ∫_{0}^{1} φ(t) dt. We emphasize that this method provides reliable solution approximations in computations.

A.6. Discussion and practical hints.

A.6.1. Consider the deformation (1−r(t))·P_{0}+r(t)·P of the boundary values (see Sec. A.3) or of the equation itself (see Sec. A.5). Then the smooth step-like homotopy function

r(t) = exp(−ctg^{2}(πt/2)), 0 < t < 1,

is more preferable than the linear motion Ṗ = P − P_{0}. Then, the final stage of solving the boundary–value problem P is in fact equivalent to the Newton method that starts with the approximation obtained in the initial computations.

A.6.2. The comparative analysis of practical computations by using the algorithms described in this appendix was carried out in 2002 in the dimploma papers by A. V. Punina (Chair of Higher Mathematics, ISPU; Master thesis “Comparative analysis of the methods for solving the elliptic Liouville equation by using the homotopies”) and N. P. Cheluhoeva (Chair of Higher Mathematics, ISPU; Master thesis “Integral equations on the border method studied for solving the elliptic Liouville equation”). The following assertion is in order.

• The method of deforming the equation E (see Sec.A.5) is twice more precise with respect to the absolute deviation from known exact solutions than the method based on the deformation of the boundary function, see Sec. A.3, although the latter method converges 3-5 times faster.

• The relaxation method is the simplest among the algorithms such that the evolutions of t is continuous. The interative method is also preferable for solving the Laplace equation that appeared in Sec. A.3 since the use of the Poisson kernel requires greater time and the Gauss method cannot be speeded up owing to the strongly sparse matrix of the discrete Laplace operator Δ. In Sec. A.3, we described the reduction of the Dirichlet border–value problem to the one–dimensional problem and the integral equation. This method is very sensitive with respect to smoothness of the boundary conditions f(t) since the Poisson kernel has a singularity on ∂B_{0}^{1} and the harmonic functions are less smooth on ∂B_{0}^{1} than in D (where they are infinitely smooth). We also see that a solution of the Liouville equation, when moving along t, can be “attracted” to a nonclassical (of type (197b) or (197c)) solution which satisfies the same boundary conditions. This can happen if the time step is suffucuently large.

Summarizing, we conclude that all these methods demonstrated comparable precisions. Therefore, the choice of an algorithm should be based on the actual properties of the problem at hand.

A.6.3. Conservation laws for the analysed equation E serve an important instrument for the precision control in computations. Nowadays, there exist regular methods ([10, 63, 94]) of reconstruction of the exhaustive set of conservation laws for the equations subject to the ‘2-line theorem’. These methods are realized in the form of the computer software ([72]) for systems of analytic transformations.

A.6.4. Application of the methods of soving boundary–value problems described in this appendix does neither neglect nor underesteeme the standard check of continuity of the resulting solutions, their Hölder or the Sobolev space properties an so on. Meanwhile, we hope that these practical ideas will compliment the everyday set of instruments for numerical analysis of nonlinear equations of mathematical physics.

Final remarks

1. Recently, Demskoi and Startsev ([19]) analysed the correlation between the integrals Ω and symmetries φ for the Liouvillean hyperbolic systems. They assigned the operators ¯□ that factor symmetries of these systems to the product

φ = ¯□(ϕ(x, Ω))

to the linearizations ℓ_{Ω^{i}} of the integrals Ω. These results are reported in the note [19], which is found in this issue. In fact, they generalize the statements of Lemma 13 on page 39 and Lemma 50 on page 77 to the case of arbitrary integrals Ω^{i}, i ≥ 1 (we recall that Ω^{1} ≡ T).

2. The commutative Hamiltonian hierarchy 𝔄 of the local Noether symmetries φ_{k} ∈ sym L_{Toda}, where k ≥ 0, was constructed in Part I. We identified this hierarchy with the sequence of higher r-component analogs for the potential modified Korteweg–de Vries equation and related 𝔄 with the bi–Hamiltonian hierarchy 𝔅 for the scalar potential Korteweg–de Vries equation, see Eq. (93) on page 50. The degeneracy of the matrix coefficients of the higher-order derivtives is an immanent feature of these evolution equations.

A comment is in order. In the fundamental paper [22], the integrable bi–Hamiltonian hierarchy ℭ of the Drinfel’d–Sokolov equations was assigned to the hyperbolic Toda system associated with a semisimple Lie algebra 𝔤 or a Kac–Moody algebra 𝔤̂. The hierarchy ℭ was related with the sequence 𝔇 of the multi–component Kortewegde Vries equation’s analogs. Unlike in the case of the hierarchy 𝔄, the symbols of the Drinfel’d–Sokolov equations are always nondegenerate. Therefore, it is quite reasonable to check whether there is an interrelation between these two systems 𝔄 and ℭ, as well as 𝔅 and 𝔇. Also, the necessary and sufficient conditions for 𝔄 to be bi–Hamiltonian remain unclear.

Suppose the hierarchy 𝔄 is assigned to the Toda equation (55) which is associated with a nondegenerate symmetrizable matrix K. We conjecture that the hierarchy 𝔄 is bi-Hamiltonian if and only if K is the Cartan matrix of a semisimple Lie algebra. We also conjecture that the pair (A_{1}, A_{2}) of the operators is Hamiltonian in the sence of Definition 7 on page 10 under the same assumptions.

3. Chapter 3 contains the exposition of the geometric structures for the scalar heavenly equation, which is a continuous dispersionless limit of the r-component Toda systems as r tends to infinity. We observed that local structures for the limit equation are relatively few, and therefore a nonlocal setting must be introduced. Meanwhile, local Noether’s symmetries, conservation laws, and recursion operators for the Toda equations themselves were considered in Part I. Therefore it is logical to describe the permutability properties of the diagram

E_{Toda} ⟶ local structures for E_{Toda}
\begin{matrix}r→∞, \\ ε→+0, \\ u_{zzz}=0 \end{matrix}↓  ↓^{?}
E_{heav} ⟶ nonlocal structures for E_{heav},

that links together the local geometry of E_{Toda} and (yet undiscovered) nonlocal geometry of E_{heav}.

4. In Chapter 4, we applied the cohomological schemes developed by I. S. Krasil’shchik and constructed one–parametric families of Bäcklund transformations for Eq. (158) only, leaving apart the general case ([3]) of Bäcklund transformations for the Toda equations associated with the semisimple Lie algebras. Still, from the resulting expressions in [3] it is clear that the scaling symmetry is the required generator of deformations for any r ≥ 1 and any 𝔤 of rank r.

The author hopes that the reasonings of the present paper demonstrate the profits one obtains by using the invariant coordinate–free approach towards the mathematical physics equations. The reader will enjoy an excursion to the world of the PDE-related algebraic structures in the recent paper [52], where a natural class of N-ary generalizations for the Lie–algebra structures (in particular, of the symmetry algebra sym E_{Toda} for the Toda equations) was considered.

Acknowledgements. The author thanks I. S. Krasil’shchik for fruitful discussions and contructive criticism, and also thanks A. V. Ovchinnikov, V. V. Sokolov, and A. M. Verbovetsky, for their remarks and advice. The author is grateful to V. M. Buchstaber, E. V. Ferapontov, V. A. Golovko, P. Kersten, B. G. Konopel’chenko, V. G. Marikhin, A. K. Pogrebkov, A. V. Samokhin, A. B. Shabat, R. Vitolo, V. A. Yumaguzhin, and to the participants of the research seminar in algebra and geometry of differential equations (Independent University of Moscow) for useful discussions. The author thanks M. Marvan, who created the Jet ([72]) analytic transformations software, for a version of the program and practical hints.

The major part of this research was done at the Moscow State University. Also, the author is grateful to the universities of Twente, Lecce, and Salerno, where a part of the work was completed, for warm hospitality.

The research was partially supported by the scholarship of the Government of the Russian Federation, the INTAS grant YS 2001/2-33, and the Lecce University grant n. 650 CP/D.

References

[1] Akhmediev N., Ankiewicz A. Multi–soliton complexes // Chaos. — 2000. — 10, n. 3. — P. 600–612.

[2] Alfinito E., Soliani G., Solombrino L. The symmetry structure of the heavenly equation // Lett. Math. Phys. — 1997. — 41. — P. 379–389.

[3] Andreev V. A. Bäcklund transformations of the Toda lattices // Teor. matem. fizika. — 1988. — 75, 3. — P. 340–352 (in Russian).

[4] Arnol’d V. I. Mathematical methods in classical mechanics. — Nauka: Moscow, 1979. — 432 p. (in Russian).

[5] Arnol’d V. I. Geometric methods in the theory of ordinary differential equations. Izhevsk. resp. publ.: Izhevsk, 2000. — 400 p. (in Russian).

[6] Barnich G., Brandt F., Henneaux M. Local BRST cohomology in the antifield formalism: I. General theorems // Commun. Math. Phys. — 1995. — 174. — P. 57–92.

[7] Bilal A., Gervais J.-L. Extended C = ∞ conformal systems from classical Toda field theories // Nucl. Phys. B. — 1989. — 314, 3. — P. 646–686.

[8] Bilal A., Gervais J.-L. Systematic construction of conformal theories with higher–spin Virasoro symmetries // Nucl. Phys. B. — 1989. — 318, 3. — P. 579–630.

[9] Bogolyubov N. N., Shirkov D. V. Quantum fields. — Fizmatlit, Moscow, 1993. — 335 p. (in Russian).

[10] Bocharov, A. V., Chetverikov, V. N., Duzhin, S. V., et al.: Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Amer. Math. Soc., Providence, RI, 1999. Edited and with a preface by I. Krasil’shchik and A. Vinogradov.

[11] Boyer C. P., Finley J. D. Killing vectors in self–dual Euclidean Einstein spaces // J. Math. Phys. — 1982. — 23. — P. 1126–1130.

[12] Boyer C. P, Plebański J. F. An infinite hierarchy of conservation laws and nonlinear superposition principles for self–dual Einstein spaces // J. Math. Phys. — 1985. — 26, 2. — P. 229–234.

[13] Brandt F. Bäcklund transformations and zero curvature representations of systems of partial differential equations // J. Math. Phys. — 1994. — 35. — P. 2463–2484.

[14] Bullough R. K., Dodd R. K. Bäcklund transformations for the sine–Gordon equations // Proc. Roy. Soc., London. — 1076. — A351, 1667. — P. 499– 523.

[15] Bullough R. K., Dodd, R. K. Polynomial conserved densities for the sine– Gordon equations // Proc. Roy. Soc. London. — 1977. — A352. — P. 481– 503.

[16] Carlet G., Dubrovin B., Zhang Y. The extended Toda hierarchy.

arXiv:nlin.SI/0306060.

[17] Case K. M., Roos A. M Sine-Gordon and modified Korteweg-de Vries charges // J. Math. Phys. — 1982. — 23, 3. — P. 392–395.

[18] Cieśliński J. A generalized formula for integrable classes of surfaces in Lie algebras // J. Math. Phys. — 1997. — 38, 8. — C. 4255–4272.

[19] Demskoi D. K., Startsev S. Ya. On constructing the symmetries from the integrals of hyperbolic partial differential systems // Fundam. Appl. Math. (English transl.: J. Math. Sci.) — This issue. — 9 p. (in Russian).

[20] Dorfman I. Dirac structures and integrability of nonlinear evolution equations. Nonlinear Science: Theory and Applications. — John Wiley & Sons, Ltd., Chichester, 1993. — 176 p.

[21] Drinfel’d V. G., Sokolov V. V. Equations of the Korteweg–de Vries type and simple Lie algebras // Doklady AN SSSR — 1981. — 258, 1. — P. 11–16 (in Russian).

[22] Drinfel’d V. G., Sokolov V. V. Lie algebras and Korteweg–de Vries type equations / Sovrem. probl. matematiki. Nov. dostizhenija. 24. — VINITI: Moscow, 1984. — P. 81–180 (in Russian).

[23] Dubrovin B. A., Novikov S. P., Fomenko A. T. Modern geometry: Methods and applications. — Nauka, Moscow, 1979. — 760 p. (in Russian).

[24] Dunajski M., Mason L. J. Hyper–Kähler hierarchies and their twistor theory // Commun. Math. Phys. — 2000. — 213. — P. 641–672.

[25] Fehér L., O’Raifeartaigh L., Ruelle P., Tsutsui I., Wipf A. On Hamiltonian reductions of the Wess–Zumino–Novikov–Witten theories //Phys. Rep. — 1992. — 222, 1. — P. 1–64.

[26] Gel’fand I. M., Dorfman I. Ya. Hamiltonian operators and related algebraic structures // Funct. Anal. Appl. — 1979. — 13, 4. — P. 13–30 (in Russian).

[27] Gervais J.-L., Matsuo Y. W –geometries // Phys. Letters. — 1992. — B274. — P. 309–316.

[28] Gervais J.-L., Matsuo Y. Classical A_{n}–W –geometries // Commun. Math. Phys. — 1993. — 152. — P. 317–368.

[29] Gervais J.-L., Saveliev M. V. W –geometry of the Toda systems associated with non-exceptional Lie algebras // Commun. Math. Phys. — 1996. — 180,

2. — P. 265–296.

[30] Geurts M. L., Martini R., Post G. F. Symmetries of the WDVV equation // Acta Appl. Math. — 2002. — 72, 1-2. — P. 67–75.

[31] Golovko V. A. On conservation laws for the Toda systems / Proc. X Int. conf. “Lomonosov–2003”, sec. “Physics”. Moscow, MSU. — 2003. — P. 53–55 (in Russian).

[32] Golovko V. A. On zero–curvature representations and Bäcklund transformations for the Liouville equation / Proc. XXV conf. of Young Scientists. Faculty of Mathematics and Mechanics. Moscow, MSU. — 2003. — P. 20–22 (in Russian).

[33] Gusyatnikova V. N., Samokhin A. V., Titov V. S. et al. Symmetries and conservation laws of Kadomtsev–Pogutse equations // Acta Appl. Math. — 1989. — 15, 1. — P. 23–64.

[34] Hilbarg D., Troedinher N. S. Elliptic differential equations with partial derivatives of second order. Mir, Moscow, 1989. — 463 p. (in Russian).

[35] Ibragimov N. Kh., Shabat A. B. The Korteweg–de Vries equation from a group–theoretic standpoint // Doklady AN SSSR. — 1979. — 244, 1. — P. 57–61 (in Russian).

[36] Ibragimov N. Kh., Shabat A. B. Evolution equations admitting a nontrivial Lie–Bäcklund group // Funct. Anal. Appl. — 1980. — 14, 1. — P. 25–36 (in Russian).

[37] Igonin S., Krasil’shchik I. S. On one-parametric families of Bäcklund transformations // Advanced Studies in Pure Mathematics. — 2003. — 37. — P. 99–114.

[38] Jet Nestruev Smooth manifolds and observables. — MCCME, Moscow, 2000. — 300 p (in Russian).

[39] Kac V. G., Raina A. K. Bombai lectures on highest wieght representation of infinite dimensional Lie algebras. — Singapore etc.: World Scientific, 1987. — ix, 145 p.

[40] Kaliappan P., Lakshmanan M. Connection between the infinite sequence of Lie-Bäcklund symmetries of the Korteweg-de Vries and sine-Gordon equations // J. Math. Phys. — 1982. — 23, 3. — P. 456–459.

[41] Kazdan J. L., Warner F. W. Curvature functions for open 2-manifolds // Ann. Math., (2). — 1974. — 99, 2. — P. 203–219.

[42] Kersten P., Krasil’shchik I., Verbovetsky A. Hamiltonian operators and ℓ^{∗}- coverings // J. Geom. Phys. — 2004. — 50, 1-4. — P. 273–302.

[43] Khor’kova N. G. Conservation laws and nonlocal symmetries // Matem. zametki. — 1988. — 44, 1. — P. 134–144 (in Russian).

[44] Kiselev A. V. Methods of the differential equations’ geometry in analysis of the integrable field theory models. — Cand. sci. dissertation. — Moscow, MSU, 2004. — 137 p. (in Russian).

[45] Kiselev A. V. Classical conservation laws for the elliptic Liouville equation // Moscow Univ. Phys. Bull. — 6 (2000). — P. 11–13.

[46] Kiselev A. V. On the geometry of Liouville equation: symmetries, conservation laws, and Bäcklund transformations // Acta Appl. Math. — 2002. — 72, 1-2. — P. 33–49.

[47] Kiselev A. V. On Bäcklund autotransformation for the Liouville equation // Moscow Univ. Phys. Bull. — 6 (2002). — P. 22–26.

[48] Kiselev A. V. On some properties of the reucrsion operator for the Liouville equation / Proc. XXV conf. of Young Scientists. Faculty of Mathematics and Mechanics. — Moscow, MSU, 2003. — P. 74–77 (in Russian).

[49] Kiselev A. V. Methods of the PDE geometry applied to solving the boundary problems // Matem. i eyo prilozh. — 2004. — 1, 1. — P. 59–68 (in Russian).

[50] Kiselev A. V. On a continuous analog of the two–dimensional Toda systems // Matem. i eyo prilozh. — 2004. — 1, 1. — P. 69–74 (in Russian).

[51] Kiselev A. V. On the Korteweg–de Vries equations associated with the Toda systems. Deposition VINITI 10.03.2004, 412-B2004, 86 p. (in Russian).

[52] Kiselev A. V. On the associative Schlessinger==Stasheff algebras and the Wronskian determinants // Fund. Appl. Math. — 2003. — 9, 4. — 20 p. (to appear) (in Russian).

[53] Kiselev A. V. On homotopy Lie algebra structures in the rings of differential operators // Note di Matematica (2003). 22, n. 1-2. — 30 p. (to appear).

[54] Kiselev A. V. On the Noether symmetries of the Toda equations // Moscow Univ. Phys. Bull. — 2004. 2. — 3 p. (to appear).

[55] Kiselev A. V. On conservation laws in the soliton complexes / Proc. XXVI conf. of Young Scientists. Faculty of Mathematics and Mechanics. — Moscow, MSU, 2004. — P. 62–63 (in Russian).

[56] Kiselev A. V. On constructing exact solutions to the dispersionless Toda equation // Matem. i eyo prilozh. — 2004. — 1, 2. — 6 p. (to appear) (in Russian).

[57] Kiselev A. V. On conservation laws for the Toda equations // Acta Appl. Math. (2004). — 8 p. (to appear).

[58] Kiselev A. V., Golovko V. A. Non–abelian coverings over the Liouville equation // Acta Appl. Math. (2004). — 16 p. (to appear).

[59] Kiselev A. V., Ovchinnikov A. V. On some Hamiltonian hierarchies associated with the Toda equations // Proc. conf. “Lomonosov’s readings–2004. Physics.” — 2004. — Moscow, MSU. — P. 102–105 (in Russian).

[60] Kiselev A. V., Ovchinnikov A. V. On the Hamiltonian hierarchies associated with the hyperbolic Euler equations // J. Dynamical and Control Systems. — 10 (2004) n. 3 (July 2004). — P. 431–451.

[61] Krasil’shchik I. A simple method to prove locality of symmetry hierarchies. — 2002. — Preprint DIPS-9/2002. — 4 p.

Internet: http://diffiety.org/preprint/2002/09_02.pdf

[62] Krasil’shchik I. S., Kersten P. H. M. Symmetries and recursion operators for classical and supersymmetric differential equations. — Kluwer Acad. Publ., Dordrecht etc., 2000. — 380 p.

[63] Krasil’shchik J., Verbovetsky A. Homological methods in equations of mathematical physics, Advanced Texts in Mathematics. — Open Education and Sciences, Opava, 1998. — 150 p.

arXiv.math.DG/9808130.

[64] Kumei S. Invariance transformations, invariance group transformations, and invariance groups of the sine–Gordon equations // J. Math. Phys. — 1975. — 16, 12. — C. 2461–2468.

[65] Leznov A. N. On complete integrability of a nonlinear system of partial differential equations in the two–dimensional space // Teor. matem. fizika. — 1980. — 42, 3. — P. 343–349 (in Russian).

[66] Leznov, A. N. and Saveliev, M. V.: Group–Theoretical Methods for Integration of Nonlinear Dynamical Systems. Birkhäuser Verlag, Basel, 1992.

[67] Leznov A. N., Saveliev M. V. Spherically symmetric equations in gauge theories for an arbitrary semisimple compact Lie group // Phys. Lett. B. — 1978. — 79, 3. — P. 294–296.

[68] Leznov A. N., Smirnov V. G., Shabat A. B. The internal symmetry group and the integrability conditions of two–dimensional dynamical systems // Teor. matem. fizika. — 1982. — 51, 1. — P. 10–21 (in Russian).

[69] Liouville J. Sur l’equation aux différences partielles d^{2} log λ/du dv±λ/(2a^{2}) = 0 // J. de math. pure et appliquée. — 1853. — 18, 1. — P. 71–72.

[70] Magri F. A simple model of the integrable equation // J. Math. Phys. — 1978. — 19, 5. — P. 1156–1162.

[71] Martina L., Sheftel M. B., Winternitz P. Group foliation and non-invariant solutions of the heavenly equation // J. Phys. A: Math. Gen. — 2001. — 34. — P. 9243–9263.

[72] Marvan M. Jets. A software for diferential calculus on jet spaces and diffieties, ver. 4.9 (December 2003) for Maple V Release 4. — Opava. — 1997.

Internet: http://diffiety.ac.ru/soft/soft.htm.

[73] Marvan M. On the horizontal gauge cohomology and nonremovability of the spectral parameter // Acta Appl. Math. — 2002. — 72, 1-2. — P. 51–65.

[74] Marvan M. Another look on recursion operators / Proc. Conf. Differential Geometry and Applications. — 1995. — Masaryk Univ., Brno, Czech Republic. — P. 393-402.

[75] Meshkov A. G. Symmetries of scalar fields. III. Two–dimensional integrable models // Teor. matem. fizika. — 1985. — 63, 3. — P. 323–332 (in Russian).

[76] Miura R. M. Korteweg-de Vries equation and generalizations. I. // J. Math. Phys. — 1968. — 9, 8. — P. 1202–1204.

[77] Ovchinnikov A. V. Toda systems associated with the Lie algebras and the W - algebras in some problems of mathematical physics. Cand. sci. dissertation. — 1996. — Moscow, MSU. — 96 p. (in Russian).

[78] Ovchinnikov A. Toda systems and W -algebras / Proc. 1st Non-orthodox School on Nonlinearity and Geometry. Ed. D. Wójcik, J. Cieśliński. — Polish Sc. Publ. PWN, Warsawa, 1998. — P. 348–358.

[79] Pao C. V. Nonlinear parabolic and elliptic equations. Plenum Press: New York etc., 1992. — 658 p.

[80] Poincaré H. Les fonctions fuchsiennes et l’equation Δu = exp(u) // J. math. pures et appl., 5^{e} ser. — 1898. — 4. — P. 157–230.

[81] Pommaret J.-F. Partial differential equations and group theory: new perspectives for applications. Kluwer: Dordrecht, 1994. — 473 p.

[82] Polyakov A. M. Gauge fields and strings, Harwood Acad. Publs., Chur, Switzerland, 1987.

[83] Razumov A. V., Saveliev M. V. Lie algebras, geometry, and Toda–type systems. Cambridge Lecture Notes in Physics 8. — Cambridge Univ. Press, 1997. — 327 p.

[84] Rogers C., Shadwick W. F. Bäcklund transformations and their applications. — Academic press, NY etc., 1982. — 334 p.

[85] Sakovich S. Yu. On special Bäcklund autotransformations // J. Phys. A: Math. Gen. — 1991. — 24. — P. 401-405.

[86] Sakovich S. Yu. On conservation laws and zero–curvature representations of the Liouville equation // J. Phys. A: Math. Gen. — 1994. — 27. — P. L125–L129.

[87] Saveliev M. V. On the integrability problem of the continuous Toda system // Teor. matem. fizika. — 1992. — 92, 3. — P. 457–465 (in Russian).

[88] Saveliev M. V., Vershik A. M. On the continuous Lie algebras and the Cartan operators // Commun. Math. Phys. — 1989. — 126. — P. 367–381.

[89] Shabat A. B. Higher symmetries of two–dimensional lattices // Phys. Lett. A. — 1995. — 200. — P. 121–133.

[90] Shabat A. B., Yamilov R. I. Exponential systems of type I and the Cartan matrices. — Preprint. Ufa, Bashkir fil. AN SSSR, 1981. — 22 p. (in Russian).

[91] Shadwick W. F. The Bäcklund problem for the equation ∂^{2}z/∂x^{1}∂x^{2} = f(z) // J. Math. Phys. — 1978. — 19, 11. — P. 2312–2317.

[92] Sukhorukov A. A., Akhmediev N. N. Intensity limits for stationary and interacting multi–soliton complexes. — 2001. — Preprint

arXiv:nlin.PS/0103026.

[93] Toda M. Theory of nonlinear lattices. — Mir, Moscow, 1984. — 264 p. (in Russian).

[94] Vinogradov A. M. The C-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory. II. The nonlinear theory // J. Math. Anal. Appl. — 1984. — 100, 1. — P. 1–129.

[95] Wahlquist H. D., Estabrook F. B. Bäcklund transformation for solutions of the Korteweg–de Vries equation // Phys. Rev. Lett. — 1973. — 31, 23. — P. 1386–1390.

[96] Wang J. P. Symmetries and conservation laws of evolution equations. — PhD thesis, Vrije Universiteit, Amsterdam, 1998. — 166 p.

[97] Witten E. Some exact multipseudoparticle solutions of classical Yang–Mills theory // Phys. Rev. Lett. — 1977. — 38, 3. — P. 121-124.

[98] Zhiber A. V. Equation of n-waves and the system of nonlinear Schrödinger equations from a group–theoretic viewpoint // Teor. matem. fizika. — 1982. — 52, 3. — P. 405–413 (in Russian).

[99] Zhiber A. V., Shabat A. B. Klein–Gordon equations admitting a nontrivial group // Doklady AN SSSR. — 1979. — 247, 5. — P. 1103–1107 (in Russian).

[100] Zhiber A. V., Sokolov V. V. Exactly integrable hyperbolic equations of Liouvillean type // Uspekhi matem. nauk. — 56, 1. — 2001. — P. 63–106 (in Russian).

[101] Zograph P. G., Takhtajan L. A. On the Liouville equation, accessor parameters, and geometry of the Teichmüller space for Riemannian surfaces of genus 0 // Matem. sbornik. — 1987. — 132(174), 2. — P. 147–166 (in Russian).

153003 Russia, Ivanovo, Rabfakovskaya str. 34, Ivanovo State Power University, Chair of Mathematics.

Current address: Dipartimento di Matematica ‘Ennio De Giorgi’, Università degli Studi di Lecce, Via per Arnesano, 73100 Lecce (LE), Italy.

E-mail address: arthemy@poincare.unile.it