The Diffiety Institute Preprint Series

Preprint DIPS–4/2003 June 16, 2003

Revised May 12, 2005

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

The Diffiety Institute

Coverings and the fundamental group for partial differential equations

by

Sergei IGONIN

Coverings and the fundamental group for partial differential equations

SERGEI IGONIN

Abstract. Following I. S. Krasilshchik and A. M. Vinogradov [8], we regard PDEs as infinite-dimensional manifolds with involutive distributions and consider their special morphisms called differential coverings, which include constructions like Lax pairs and Bäcklund transformations. We show that, similarly to usual coverings in topology, at least for some PDEs differential coverings are determined by actions of a sort of fundamental group. This is not a group, but a certain system of Lie algebras, which generalize Wahlquist-Estabrook algebras. From this we deduce an algebraic necessary condition for two PDEs to be connected by a Bäcklund transformation. We compute these infinite-dimensional Lie algebras for several KdV type equations and prove nonexistence of Bäcklund transformations.

As a by-product, for some class of Lie algebras 𝔤 we prove that any subalgebra of 𝔤 of finite codimension contains an ideal of 𝔤 of finite codimension.

Contents

Introduction 2 1. Basics 4 1.1. Some terminology 5 1.2. Infinite-dimensional manifolds 5 1.3. PDEs as manifolds with distributions 7 1.4. Coordinate description 9 1.5. Differential coverings 10 1.6. Coverings as transformations of PDEs 12 1.7. Actions of Lie algebras on manifolds 13 1.8. Zero-curvature representations 15 1.9. Translation-invariant coverings 16 1.10. Wahlquist-Estabrook coverings 17 2. Analogs of the fundamental group for differential coverings 19 2.1. An instructive example 19 2.2. The definition of the fundamental algebras 20 2.3. Regular coverings and their symmetry algebras 24 2.4. Quasi-finite Lie algebras 26 2.5. Local structure of irreducible coverings 27 2.6. Necessary conditions for existence of Bäcklund transformations 29 3. Coverings of scalar evolution equations 31

1991 Mathematics Subject Classification. 37K30, 37K35, 35Q53. Key words and phrases. Coverings of PDEs, Bäcklund transformations, Wahlquist-Estabrook struc-

tures, actions of Lie algebras on manifolds, the KdV equation, the Krichever-Novikov equation.

4. Coverings of the KdV equation 36 4.1. The canonical form of coverings 36 4.2. The fundamental algebras 39 5. Coverings of the Krichever-Novikov equation 41 5.1. The canonical form of coverings 42 5.2. The fundamental algebras 45 6. Coverings of the equation u_{t} = u_{xxx} 48 7. Non-existence results for Bäcklund transformations 51 Acknowledgements 52 References 52

Introduction

In this paper we study special correspondences called (differential) coverings between systems of PDEs. Roughly speaking, a covering E_{1} → E_{2} is a differential mapping from one system E_{1} to another system E_{2} such that the preimage of each local solution of E_{2} is a family of E_{1} solutions dependent on a finite number m of parameters.

For example, if v(x, t) is a solution of the modified KdV equation

(1)  v_{t} = v_{xxx} − 6v^{2}v_{x}

then the function

(2)  u = v_{x} − v^{2}

satisfies the KdV equation u_{t} = u_{xxx} + 6uu_{x}. This is the famous Miura transformation, which determines a covering from the modified KdV equation to the KdV equation. For a given local solution u(x, t) of the KdV equation, a one-parameter family of functions v(x, t) is recovered from equations (2) and (1). That is, we have m = 1 for this covering. In general, systems E_{1} and E_{2} may be overdetermined, but must be consistent.

More precisely, following [1, 8, 9], we regard E_{1}, E_{2} as submanifolds in infinite jet spaces. The (usually infinite-dimensional) submanifold of infinite jets satisfying a system of PDEs is called the infinite prolongation of the system and possesses a canonical involutive distribution called the Cartan distribution. This distribution is spanned by the total derivative operators (regarded as commuting vector fields on the infinite jet space) with respect to the independent variables. A (differential) covering τ: E_{1} → E_{2} is a bundle of finite rank^{1} m such that the differential τ_{∗} maps the Cartan tangent subspaces of E_{1} isomorphically onto the ones of E_{2}. Note that even local classification of coverings is highly nontrivial due to different possible configurations of the distributions.

It was shown in [8] that all kinds of Lax pairs, zero-curvature representations, Wahlquist-Estabrook prolongation structures, and Bäcklund transformations in soliton theory are special types of coverings. In particular, a Bäcklund transformation between two systems E_{1} and E_{2} is given by another system E_{3} and a pair of coverings E_{1} ← E_{3} → E_{2}.

The name ‘coverings’ for such bundles is used because they include usual topological coverings of finite-dimensional manifolds, see Example 3 below.

1One can consider also coverings of infinite rank [1, 8], but we study only the case of finite rank

Recall that for a finite-dimensional manifold M its topological coverings are in oneto-one correspondence with actions of the fundamental group π_{1}(M) on (discrete) sets. The main result of this paper is that at least for some PDEs E differential coverings are also determined by actions of a sort of fundamental group. However, this is not a group, but a certain system of Lie algebras that we call the fundamental algebras of E. They are arranged in a sequence of epimorphisms

(3)  ⋯ → 𝔣_{k+1} → 𝔣_{k} → ⋯ → 𝔣_{1} → 𝔣_{0}.

Differential coverings of rank m are determined by actions of these Lie algebras on m-dimensional manifolds W, that is, homomorphisms from 𝔣_{k} to the algebra D(W) of vector fields on W. Two coverings are isomorphic if and only if the corresponding actions are isomorphic.

More precisely, the following facts hold:

• for each action ρ: 𝔣_{k} → D(W) we introduce an involutive distribution on the manifold E × W such that the trivial bundle E × W → E becomes a covering denoted τ(ρ) (here E is endowed with the fixed Cartan distribution),

• for any covering τ: Ẽ → E we define an action ρ(τ): 𝔣_{k} → D(Ẽ) for some k such that τ_{∗}ρ(τ) = 0,

• for an action ρ_{0} : 𝔣_{k} → D(W) and the covering τ = τ(ρ_{0}), the action ρ(τ) is equal to the composition of the natural embedding D(W) ⊂ D(E ×W) with the action ρ_{0},

• a morphism of coverings τ_{1} and τ_{2} of E induces a morphism of the actions ρ(τ_{1}) and ρ(τ_{2}),

• a covering Ẽ → E on a neighborhood of each point of Ẽ is isomorphic to the covering τ(ρ) for some action ρ of 𝔣_{k} and some k.

The algebra 𝔣_{0} is equal to the Wahlquist-Estabrook prolongation algebra of E [8, 20, 22]. To obtain algebras 𝔣_{k} for k ≥ 1, we replace the Wahlquist-Estabrook ansatz by jets of arbitrary order and find a canonical form of coverings with respect to the local gauge equivalence.

Note that some similarity between Wahlquist-Estabrook algebras and the topological fundamental group was noticed in [9]. However, before the present paper this idea was not developed and did not lead to any applications.

We prove that all finite-dimensional quotients of the fundamental algebras are coordinateindependent invariants of the system of PDEs. Namely, recall that quotients of the topological fundamental group π_{1}(M) occur as automorphism groups of regular topological coverings of M. Similarly, finite-dimensional quotients of the fundamental algebras occur as Lie algebras of infinitesimal automorphisms of certain coverings of E.

We conjecture that the fundamental algebras themselves are also coordinate-independent invariants and hope to prove this elsewhere using the homological techniques of [5, 12, 21].

We formulate some conditions for a system of PDEs to possess fundamental algebras. We check these conditions and compute algebras (3) for three PDEs: the KdV equation, the nonsingular Krichever-Novikov equation, and the linear equation

(4)  u_{t} = u_{xxx}.

In all three cases each 𝔣_{k} is obtained from a single Lie algebra 𝔎 applying several times the operation of one-dimensional central extension.

For the KdV equation we have 𝔎 = 𝔰𝔩_{2}(ℂ) ⊗_{ℂ} ℂ[λ].

For the nonsingular Krichever-Novikov equation the algebra 𝔎 is isomorphic to a certain subalgebra of the tensor product of 𝔰𝔩_{2}(ℂ) with the algebra of regular functions on an affine elliptic curve. Note that in this case 𝔣_{0} = 0, that is, the Wahlquist-Estabrook ansatz gives no nontrivial coverings.

For equation (4) the algebra 𝔎 possesses a filtration by solvable ideals

𝔎_{0} ⊂ 𝔎_{1} ⊂ ⋯ ⊂ 𝔎_{k} ⊂ ⋯ ⊂ 𝔎

such that the quotient 𝔎/ ∪_{k=0}^{∞} 𝔎_{k} is solvable as well.

The described methods to compute fundamental algebras can be applied to other evolution equations as well.

In order to develop this theory, we obtain the following results on Lie algebras, which may be of independent interest. A Lie algebra 𝔤 is said to be quasi-finite if any subalgebra of 𝔤 of finite codimension contains an ideal of 𝔤 of finite codimension. We prove that

• a central extension of a quasi-finite algebra is quasi-finite,

• for a finite-dimensional semisimple Lie algebra 𝔤 and a commutative associative algebra A the tensor product 𝔤 ⊗ A regarded as a Lie algebra is quasi-finite,

• the algebra 𝔎 of the nonsingular Krichever-Novikov equation is quasi-finite. Recall that for a connected topological covering M̃ → M one has π_{1}(M̃) ⊂ π_{1}(M).

It turns out that some analog of this property is also valid for differential coverings, see Theorems 12 and 13.

We obtain also a necessary condition for two systems of PDEs possessing fundamental algebras to be connected by a Bäcklund transformation: their fundamental algebras have to be similar in a certain sense, see Theorem 14. As an example of using this necessary condition, we prove that equation (4) is not connected by any Bäcklund transformation neither with the KdV equation nor with the nonsingular Krichever-Novikov equation. Note that this is apparently the first rigorous non-existence result for Bäcklund transformations.

In this paper we consider only complex-analytic PDEs. Generalization of this theory to smooth PDEs is possible, but is a little more technical, since the analogs of Proposition 3 and Theorem 7 for smooth manifolds do not hold. However, practically all results will remain valid in the smooth case if one excludes from considered manifolds a thin subset of degenerate points. Detailed exposition for smooth PDEs will be done elsewhere.

1. Basics

In this section we review some notions of PDE geometry, actions of Lie algebras on manifolds and prove auxiliary lemmas needed for further theory.

In Subsections 1.2–1.6 we mainly follow [1, 8, 10]. However, there are certain modifications because of the fact that we deal with complex-analytic manifolds, while in [1, 8, 10] only smooth manifolds are considered. In particular, we have to use sheaves instead of globally defined functions. Besides, the notions of subequations and irreducible equations are new.

Most of the notions of Subsection 1.7 are studied in more detail in [4].

In order to be more readable, all concepts of PDE geometry are introduced in two ways: invariant and coordinate.

1.1. Some terminology. In this paper all manifolds, functions, vector fields, and mappings are supposed to be complex-analytic.

For a manifold M we denote by D(M) the Lie algebra of vector fields on M. For a function f on M and a point a ∈ M, the differential of f at a is denoted by d_{a}f.

The differential of a mapping φ: M_{1} → M_{2} of manifolds is denoted by φ_{∗}.

ℤ_{+} is the set of nonnegative integers.

For subspaces V_{1}, …, V_{k} of a linear space, the space ⟨V_{1}, …, V_{k}⟩ is the linear span of V_{1}, …, V_{k}.

In this paper a surjective submersion is called a bundle. To emphasize its properties that in Section 1.2 will be extended to infinite-dimensional manifolds, we give the following definition.

Definition 1. A mapping φ: M_{1} → M_{2} of manifolds is called a bundle if • the mapping φ is surjective,

• for any point a ∈ M_{1} there is a neighborhood a ∈ U ⊂ M_{1} and a manifold W such that φ(U) is open in M_{2} and one has the commutative diagram

U  \overline{ξ →} φ(U) × W
[figure: φ; p]
φ(U)

where ξ is a complex-analytic diffeomorphism and p is the projection to the first factor.

In this case the preimages φ^{−1}(b) of points b ∈ M_{2} are submanifolds in M_{1} and are called the fibres of φ. They are not necessarily isomorphic to each other, but have the same dimension called the rank of φ.

For a bundle φ: M_{1} → M_{2}, a vector field V ∈ D(M_{1}) is said to be φ-vertical if φ_{∗}(V) = 0.

In what follows we say that a certain property holds locally if it holds on a neighborhood of each point of the manifold under consideration.

1.2. Infinite-dimensional manifolds. We want to extend the category of finite-di-mensional manifolds in order to include certain type of infinite-dimensional manifolds that occur in PDE geometry.

Definition 2. Define a category INF as follows.

• First, an elementary object of INF is an infinite chain of bundles

(5)  ⟶^{φ_{i+2,i+1}} M^{i+1} ⟶^{φ_{i+1,i}} M^{i} ⟶^{φ_{i,i−1}} … ⟶^{φ_{1,0}} M^{0},

where M^{i} are finite-dimensional manifolds.

Two elementary objects

{M_{1}^{i}, φ_{i+1,i}^{1}}, {M_{2}^{i}, φ_{i+1,i}^{2}}

such that

∃ p, q ∈ ℤ M_{1}^{i+q} = M_{2}^{i}, φ_{i+q+1,i+q}^{1} = φ_{i+1,i}^{2} ∀ i ≥ p

are regarded to be identical.

Denote by M elementary object (5). A point of M is a sequence

(6)  (a_{0}, a_{1}, …, a_{i}, …), a_{i} ∈ M^{i}, φ_{i+1,i}(a_{i+1}) = a_{i} ∀ i ≥ 0.

Let us introduce a topology on the set |M| of points of M. Let U be an open subset of some M^{p}. Denote by U_{i}, i ≥ p, the preimage of U in M^{p+i} under bundles (5). The subset of points (6) such that a_{i} ∈ U_{i} for all i ≥ p is called the elementary open subset of |M| corresponding to U and is denoted by S(U). By definition, elementary open subsets form a base of the topology on |M|.

Let us define the structure sheaf of functions on |M|. Each (complex-analytic) function f: U → ℂ determines the following function on S(U)

(a_{0}, a_{1}, …, a_{i}, …) ↦ f(a_{p}).

Such functions on S(U) are said to be elementary. Now let Z be an open subset of |M|. A function g: Z → ℂ belongs to the structure sheaf if and only if for each point a ∈ Z there is an elementary open subset S(U) such that a ∈ S(U) ⊂ Z and the restriction of g to S(U) is an elementary function.

• If

M_{1} = {M_{1}^{i}, φ_{i+1,i}^{1}}, M_{2} = {M_{2}^{i}, φ_{i+1,i}^{2}}

are two elementary objects of INF then a morphism ψ: M_{1} → M_{2} is given by α, k ∈ ℤ and a system of maps

ψ_{i}: M_{1}^{i+α} → M_{2}^{i}, i ≥ k,

satisfying

∀ i ≥ k φ_{i+1,i}^{2} ∘ ψ_{i+1} = ψ_{i} ∘ φ_{i+α+1,i+α}^{1}.

• Now an object of INF is a topological space with a sheaf of complex-valued functions that is locally isomorphic to an elementary object of INF. A mapping of objects of INF is a morphism if locally it is a morphism of elementary objects.

Remark 1. Although this definition is rather sketchy, it is sufficient for us, because all objects of INF considered in this paper are open subobjects of elementary objects.

Example 1. With each finite-dimensional manifold M we associate the following elementary object of INF

→ M → M → ⋯ → M,

where all arrows are the identity mappings. This construction identifies the category of finite-dimensional manifolds with a subcategory of INF.

Let M be an object of INF. The sheaf of vector fields on M is defined in the standard way as the sheaf of derivations of the structure sheaf. It is a sheaf of modules over the structure sheaf of algebras.

In particular, if M is elementary object (5) then a tangent vector at a point (6) of M is a sequence

(v_{0}, v_{1}, …, v_{i}, …), v_{i} ∈ T_{a_{i}}M^{i}, (φ_{i+1,i})_{∗}(v_{i+1}) = v_{i} ∀ i ≥ 0.

The vector space of all tangent vectors at a point a is denoted by T_{a}M.

A distribution on M is a locally free subsheaf of submodules of the vector fields sheaf. In other words, a distribution D of rank k distinguishes for each point a of M a subspace

D_{a} ⊂ T_{a}M, dim D_{a} = k,

such that locally there are vector fields X_{1}, …, X_{k} that span the subspaces D_{a}.

For a finite-dimensional manifold W and an object M of INF, one defines the object M × W of INF as follows. It is sufficient to consider the case when M is elementary object (5). Then M × W is the elementary object

→ M^{i+1} × W ⟶^{φ_{i+1,i}×id} M^{i} × W → ⋯ → M^{0} × W.

Now one easily extends Definition 1 of bundles to the case when M_{1}, M_{2} are objects of INF. However, we always assume the fibres W to be finite-dimensional manifolds.

In what follows, when we speak of functions on an object of INF, we always assume that the functions belong to the structure sheaf.

For the sake of simplicity, below objects of INF are also called manifolds, and morphisms of INF are called mappings.

1.3. PDEs as manifolds with distributions. Let π: E → M be a bundle of finitedimensional manifolds and

θ ∈ E, π(θ) = x ∈ M.

Consider a local section f of π whose graph passes through the point θ. Denote by [f]_{x}^{k} the class of all local sections whose graphs are tangent to the graph of f at θ with order ≥ k. The set

J^{k}(π) = { [f]_{x}^{k} | f is a local section of π, x ∈ M }

carries a natural structure of a manifold and is called the manifold of k-jets of the bundle π. Moreover, the natural projections

π_{k}: J^{k}(π) → M, [f]_{x}^{k} ↦ x,
π_{k,k−1} : J^{k}(π) → J^{k−1}(π), [f]_{x}^{k} ↦ [f]_{x}^{k−1},

are bundles. The infinite sequence of bundles

(7)  ⋯ → J^{k}(π) ⟶^{π_{k,k−1}} J^{k−1}(π) → ⋯ → J^{1}(π) ⟶^{π_{1,0}} J^{0}(π) = E

determines an object of INF that is called the manifold of infinite jets of π and is denoted by J^{∞}(π).

For each local section f of π we have the local sections

j_{k}(f): M → J^{k}(π), x ↦ [f]_{x}^{k},

of the bundles π_{k}, k = 0, 1, . . .. These sections determine the local section

j_{∞}(f): M → J^{∞}(π)

of the natural projection π_{∞}: J^{∞}(π) → M.

There is a unique distribution C on J^{∞}(π) such that for any point x ∈ M and any local section f of π over a neighborhood of x we have

(8)  C_{j_{∞}(f)(x)} = j_{∞}(f)_{∗}(T_{x}M).

This distribution is of rank dim M and is called the Cartan distribution of J^{∞}(π).

Consider a system of PDEs of order k imposed on sections of the bundle π. We assume that it determines a submanifold E^{0} ⊂ J^{k}(π) of the manifold J^{k}(π) such that the mapping π_{k}|_{E^{0}} : E^{0} → M is a bundle. Then a local section f of π is a solution of the system of PDEs if and only if the graph of j_{k}(f) is contained in E^{0}.

For each l ∈ ℤ_{+} the l-th prolongation of E^{0} is the set

E^{l} = { [f]_{x}^{k+l} ∈ J^{k+l}(π) | the graph of j_{k}(f) is tangent to E^{0}

with order ≥ l at [f]_{x}^{k} ∈ E^{0} },

l = 0, 1, . . .. Restricting the maps π_{k+l,k+l−1} to E^{l} and preserving the same notation for these restrictions, we obtain the sequence of maps

(9)  ⋯ → E^{l} ⟶^{π_{k+l,k+l−1}} E^{l−1} → ⋯ → E^{0}.

Imposing natural conditions of regularity, we assume that all E^{l} are submanifolds of J^{k+l}(π), while mappings (9) are bundles. The obtained object E of INF is called the infinite prolongation of the initial system of PDEs.

In what follows all considered systems of PDEs are supposed to satisfy these regularity assumptions and, therefore, possess infinite prolongations. Below such object E of INF is sometimes simply called an equation.

The distribution C is tangent to E. Its restriction to E is denoted by C_{E} and is called the Cartan distribution of E. It satisfies [C_{E}, C_{E}] ⊂ C_{E}. Since E is infinitedimensional, this does not generally imply existence and uniqueness of maximal integral submanifolds.

Definition 3. Let E be an object of INF and D be a distribution on it. A subset E′ ⊂ E is called a subequation of the pair (E, D) if E′ is a submanifold of codimension l < ∞ and D is tangent to E′. More precisely, this means the following. We have E′ ≠ ∅, and for each point a ∈ E′ there are a neighborhood a ∈ U ⊂ E and functions f_{1}, …, f_{l} on U such that

• E′ ∩ U = {q ∈ U | f_{1}(q) = ⋯ = f_{l}(q) = 0},

• for any b ∈ U the differentials d_{b}f_{1}, …, d_{b}f_{l} ∈ T_{b}^{∗}E are linearly independent,

• the ideal of functions on U generated by f_{1}, …, f_{l} is preserved by the action of

vector fields from D. In this case E′ is also an object of INF with the distribution D|_{E′}. The number l is called the codimension of the subequation E′.

A pair (E, D) is said to be irreducible if E is connected as a topological space and there is no subequation E′ ⊂ E of finite nonzero codimension.

Let E be the infinite prolongation of a system of PDEs. Then subequations of E are subequations of the pair (E, C_{E}), and E is called irreducible if the pair (E, C_{E}) is irreducible.

Remark 2. The term ‘subequation’ is motivated by the fact that a pair (E, C_{E}), as we agreed above, is sometimes called an equation.

1.4. Coordinate description. Consider a bundle π: E → M. Let x_{1}, …, x_{n} be local coordinates in M and u^{1}, …, u^{d} be local coordinates in fibres of π. For a symmetric multi-index σ = i_{1} … i_{k} set

(10)  u_{σ}^{j} = \frac{∂^{k}u^{j}}{∂x_{i_{1}} … ∂x_{i_{k}}} .

These functions along with x_{1}, …, x_{n} form a system of local coordinates for the infinitedimensional space J^{∞}(π). The topology on J^{∞}(π) is the following. Choose a finite number u_{σ_{1}}^{j_{1}}, …, u_{σ_{r}}^{j_{r}} of coordinates (10) and consider the mapping

J^{∞}(π) → ℂ^{n+r}, a ↦ (x_{1}(a), …, x_{n}(a), u_{σ_{1}}^{j_{1}}(a), …, u_{σ_{r}}^{j_{r}}(a)).

The preimages of open subsets of ℂ^{n+r}, r ∈ ℤ_{+}, under such mappings are by definition open subsets of J^{∞}(π) and form a base of the topology on J^{∞}(π). Admissible functions on open subsets of J^{∞}(π) may depend on x_{1}, …, x_{n} and a finite number of coordinates (10). Below all functions are supposed to be admissible.

The total derivative operators

(11)  D_{x_{i}} = \frac{∂}{∂x_{i}} + ∑_{σ,j} u_{σi}^{j} \frac{∂}{∂u_{σ}^{j}} , i = 1, …, n,

are commuting vector fields on J^{∞}(π) and span the Cartan distribution.

Consider a system of PDEs

(12)  F_{α}(x_{i}, u^{k}, u_{σ}^{j}, …) = 0, α = 1, …, s,

in the bundle π. The basic idea of the described approach is to treat (12) not as differential equations in u^{k}, but as analytic equations in variables (10) and x_{i}.

The differential consequences of (12) are

(13)  D_{x_{i_{1}}} … D_{x_{ir}} (F_{α}) = 0, i_{k} = 1, …, n, α = 1, …, s, r = 0, 1, ….

The infinite prolongation E ⊂ J^{∞}(π) of system (12) is distinguished by equations (13). The vector fields D_{x_{i}} are tangent to E, and their restrictions to E will be denoted by the same symbol D_{x_{i}}. They span the Cartan distribution C_{E} of E.

Example 2. Consider a scalar evolution equation in two independent variables x, t

(14)  u_{t} = F (x, t, u, u_{1}, u_{2}, …, u_{p}), u_{k} = \frac{∂^{k}u}{∂x^{k}}, u = u_{0}.

Its infinite prolongation has the natural coordinates x, t, u_{k}, k ≥ 0, since using differential consequences of (14) all t-derivatives are expressed in terms of these. The total derivative operators are written in these coordinates as follows

D_{x} = \frac{∂}{∂x} + ∑_{j≥0} u_{j+1} \frac{∂}{∂u_{j}} , D_{t} = \frac{∂}{∂t} + ∑_{j≥0} D_{x}^{j}(F)\frac{∂}{∂u_{j}} .

1.5. Differential coverings.

Definition 4. Let E be an object of INF endowed with a distribution D such that [D, D] ⊂ D. A (differential) covering of (or over) the pair (E, D) is given by a bundle of finite rank

(15)  τ: Ẽ → E

and a distribution D^{τ} on Ẽ such that • [D^{τ}, D^{τ}] ⊂ D^{τ},

• for each a ∈ Ẽ the differential τ_{∗} maps the space (D^{τ})_{a}⊂ T_{a}Ẽ isomorphically

onto the space D_{τ(a)} ⊂ T_{τ(a)}E.

An invertible mapping φ: Ẽ → Ẽ such that τ ∘φ = τ is called a gauge transformation. The covering given by the same bundle τ and the new distribution φ_{∗}(D^{τ}) on Ẽ is said to be (gauge) equivalent to the initial covering.

Similarly, a morphism between two coverings τ_{i}: E_{i} → E, i = 1, 2, over the same pair (E, D) is a mapping φ: E_{1} → E_{2} such that τ_{1} = τ_{2} ∘ φ and φ_{∗}(D^{τ_{1}}) ⊂ D^{τ_{2}}.

A τ-vertical vector field X ∈ D(Ẽ) is called a (gauge) symmetry of τ if [X, D^{τ}] ⊂ D^{τ}. This means that the local flow of X (if it exists) consists of automorphisms of τ. The Lie algebra of symmetries is denoted by Sym τ.

Covering (15) is said to be irreducible if both pairs (E, D) and (Ẽ, D^{τ}) are irreducible.

Example 3. Let us show that usual topological coverings are a particular case of this construction. Let M be a finite-dimensional manifold and D be the whole tangent bundle of M. Coverings of rank 0 over (M, D) are just topological coverings τ: M̃ → M, where dim M̃ = dim M and D^{τ} is the whole tangent bundle of M̃.

If the distribution on E is clearly fixed, we speak of coverings over E (without mentioning the distribution).

Let now E be the infinite prolongation of a system of PDEs (12). In this case we fix D to be the Cartan distribution C_{E}.

Let us describe a covering (15) in local coordinates. Recall that locally C_{E} is spanned by D_{x_{i}}. Therefore, locally there is a unique n-tuple of vector fields

(16)  D̃_{x_{i}} ∈ D^{τ}, i = 1, …, n,

on the manifold Ẽ such that

τ_{∗}(D̃_{x_{i}}) = D_{x_{i}}, (17)

[D̃_{x_{i}}, D̃_{x_{j}}] = 0, ∀ i, j = 1, …, n. (18)

Moreover, vector fields (16) span the distribution D^{τ}.

If X ∈ Sym τ then we have

(19)  [X, D̃_{x_{i}}] = 0, i = 1, …, n.

Below in this section we consider equations in two independent variables x and t, i.e., n = 2. Locally the bundle τ is trivial

(20)  τ: E × W → E, dim W = m < ∞.

Let w^{1}, …, w^{m} be local coordinates in W.

From (17) we have

(21)  D̃_{x} = D_{x} + A, D̃_{t} = D_{t} + B,

where

(22)  A = ∑_{j=1}^{m} a^{j} \frac{∂}{∂w^{j}} , B = ∑_{j=1}^{m} b^{j} \frac{∂}{∂w^{j}}

are τ-vertical vector fields on E × W. Condition (18) is written as

(23)  D_{x}B − D_{t}A + [A, B] = 0,

where

D_{x}B = ∑_{j=1}^{m} D_{x}(b^{j})\frac{∂}{∂w^{j}} , D_{t}A = ∑_{j=1}^{m} D_{t}(a^{j})\frac{∂}{∂w^{j}} .

A covering equivalent to the one given by A = B = 0 is called trivial.

The manifold E × W is itself isomorphic to the infinite prolongation of the system that consists of equations (12) and the following additional equations

(24)  \begin{aligned}\frac{∂w^{j}}{∂x} & = a^{j}(x, t, w^{k}, u_{σ}^{i}, …), \\ \frac{∂w^{j}}{∂t} & = b^{j}(x, t, w^{k}, u_{σ}^{i}, …), \end{aligned} j = 1, …, m.

This overdetermined system is consistent modulo (12) if and only if (23) holds on E. The vector fields D_{x} + A, D_{t} + B are the restrictions of the total derivative operators to E × W. That is, the distribution D^{τ} is the Cartan distribution of this system.

Gauge transformations correspond to invertible changes of variables

(25)  x ↦ x, t ↦ t, u_{σ}^{i} ↦ u_{σ}^{i}, w^{j} ↦ g^{j}(x, t, w^{k}, u_{σ}^{i}, …), j = 1, …, m,

in (24). A covering is trivial if and only if it is obtained by such change of variables from the trivial system

\frac{∂w^{j}}{∂x} = \frac{∂w^{j}}{∂t} = 0, j = 1, …, m.

Therefore, classification of coverings over E up to local isomorphism is equivalent to classification of consistent modulo (12) systems (24) up to locally invertible changes of variables (25).

Example 4. Consider a covering of rank 1

(26)  \frac{∂w}{∂x} = a(x, t, w, u, u_{1}, …, u_{k}), \frac{∂w}{∂t} = b(x, t, w, u, u_{1}, …, u_{k})

over the infinite prolongation of equation (14). After a gauge transformation

w ↦ f(x, t, w, u, u_{1}, …, u_{r}), \frac{∂f}{∂w} ≠ 0,

system (26) changes to the following system

\begin{aligned}\frac{∂w}{∂x} & = \frac{1}{\frac{∂f}{∂w}} (a(x, t, f, u, u_{1}, …, u_{k}) − D_{x}f), \\ \frac{∂w}{∂t} & = \frac{1}{\frac{∂f}{∂w}} (b(x, t, f, u, u_{1}, …, u_{k}) − D_{t}f), \end{aligned}

f = f(x, t, w, u, u_{1}, …, u_{r}),

which represents an equivalent to (26) covering.

Recall that in the case of two independent variables x, t a conserved current of E is a pair of functions (f, g) on E satisfying

(27)  D_{t}f = D_{x}g.

Two conserved currents (f_{1}, g_{1}) and (f_{2}, g_{2}) are called equivalent if there is a function h such that

(28)  f_{2} − f_{1} = D_{x}(h), g_{2} − g_{1} = D_{t}(h).

For a conserved current (27) the pair of vector fields

A = f(x, t, u_{σ}^{i}, …)\frac{∂}{∂w}, B = g(x, t, u_{σ}^{i}, …)\frac{∂}{∂w}

satisfies (23) and determines a covering of rank 1.

Equivalent conserved currents (28) determine equivalent coverings. Indeed, the corresponding gauge transformation is w ↦ w + h.

1.6. Coverings as transformations of PDEs. Consider two systems of PDEs

F_{α}(x, t, u^{1}, …, u^{d_{1}}, \frac{∂^{p+q}u^{j}}{∂x^{p}∂t^{q}} , …)= 0, α = 1, …, s_{1}, (29)

G_{α}(x, t, v^{1}, …, v^{d_{2}}, \frac{∂^{p+q}v^{j}}{∂x^{p}∂t^{q}} , …)= 0, α = 1, …, s_{2}, (30)

and a mapping

(31)  u^{j} = φ^{j}(x, t, v^{1}, …, v^{d_{2}}, \frac{∂^{p+q}v^{l}}{∂x^{p}∂t^{q}} , …), j = 1, …, d_{1},

such that the following conditions hold.

(1) For each local solution v^{1}(x, t), …, v^{d_{2}}(x, t) of system (30) functions (31) constitute a local solution of (29).

(2) For each local solution u^{1}(x, t), …, u^{d_{1}}(x, t) of (29) the system that consists of equations (30) and (31) is consistent and possesses locally a general solution

v^{1}(x, t, c_{1}, …, c_{m}), …, v^{d_{2}}(x, t, c_{1}, …, c_{m})

dependent on a finite number of complex parameters c_{1}, …, c_{m}.

Example 5. Miura transformation (2) satisfies these conditions with m = 1.

Consider the following trivial bundles

π: ℂ^{d_{1}+2} → ℂ^{2}, (x, t, u^{1}, …, u^{d_{1}}) ↦ (x, t),
π̃: ℂ^{d_{2}+2} → ℂ^{2}, (x, t, v^{1}, …, v^{d_{2}}) ↦ (x, t),

and their infinite jet spaces J^{∞}(π) and J^{∞}(π̃).

Denote by D_{x}, D_{t} and D̃_{x}, D̃_{t} the total derivative operators on J^{∞}(π) and J^{∞}(π̃) respectively. One has

\frac{∂^{p+q}u^{j}}{∂x^{p}∂t^{q}} = D_{x}^{p}D_{t}^{q}(u^{j}), \frac{∂^{p+q}v^{j}}{∂x^{p}∂t^{q}} = D̃_{x}^{p}D̃_{t}^{q}(v^{j}).

Formulas (31) suggest to consider the mapping

(32)  τ: J^{∞}(π̃) → J^{∞}(π)

defined as follows

(33)  τ^{∗}(x) = x, τ^{∗}(t) = t, τ^{∗}(u^{j}) = φ^{j}, τ^{∗}(\frac{∂^{p+q}u^{j}}{∂x^{p}∂t^{q}} )= D̃_{x}^{p}D̃_{t}^{q}(φ^{j}).

Then we obtain

(34)  τ_{∗}(D̃_{x}) = D_{x}, τ_{∗}(D̃_{t}) = D_{t}.

Let E ⊂ J^{∞}(π) and Ẽ ⊂ J^{∞}(π̃) be the infinite prolongations of systems (29) and (30) respectively. Conditions 1 and 2 above need rigorous analytical explanation, which we do not consider. Instead, following [1, 8], we say that Conditions 1 and 2 are by definition equivalent to the fact that τ(Ẽ) = E and the mapping

(35)  τ|Ẽ: Ẽ → E

is a bundle of rank m. Then from (34) we obtain that (35) is a covering.

According to construction (24), every covering of a system of PDEs is locally isomorphic to a covering of this form.

1.7. Actions of Lie algebras on manifolds. Let 𝔤 be a Lie algebra over ℂ. Recall that an action of the Lie algebra 𝔤 on a complex manifold W is a homomorphism 𝔤 → D(W). For a ∈ W let ev_{a}: D(W) → T_{a}W be the evaluation mapping. For an action ρ: 𝔤 → D(W) the subalgebra {v ∈ 𝔤 | ev_{a}ρ(v) = 0} is called the isotropy subalgebra of the point a.

An action ρ is said to be transitive if the mapping ev_{a}ρ: 𝔤 → T_{a}W is surjective for each a ∈ W. An action ρ is called free if ker ev_{a}ρ = 0 for any a ∈ W.

A bundle W → W′ is called the quotient map with respect to an action ρ: 𝔤 → D(W) if all vector fields from ρ(𝔤) are tangent to the fibres and the induced action on each fibre is transitive.

A morphism from one action ρ_{1} : 𝔤 → D(W_{1}) to another action ρ_{2} : 𝔤 → D(W_{2}) is a mapping ψ: W_{1} → W_{2} such that

(36)  ∀ a ∈ W_{1} ∀ v ∈ 𝔤 ψ_{∗}(ev_{a}ρ_{1}(v))= ev_{ψ(a)}ρ_{2}(v).

The following statement is obvious.

Lemma 1. Let ψ: W_{1} → W_{2} be a morphism of transitive actions ρ_{i}: 𝔤 → D(W_{i}), i = 1, 2. Then ψ(W_{1}) is open in W_{2}.

Let G be a connected complex Lie group associated with a finite-dimensional Lie algebra 𝔤. For g ∈ G set

L_{g}: G → G, a ↦ ga, R_{g}: G → G, a ↦ ag.

A vector field X ∈ D(G) is said to be right invariant if

(37)  ∀ g ∈ G (R_{g})_{∗}(X) = X,

and X is said to be left invariant if

(38)  ∀ g ∈ G (L_{g})_{∗}(X) = X.

Denote by D_{li}, D_{ri} ⊂ D(G) the subalgebras of left invariant and right invariant vector fields respectively. It is well known that

(39)  D_{li} ≅ D_{ri} ≅ 𝔤.

and the actions of the algebras D_{li}, D_{ri} on G are free and transitive.

By isomorphisms (39), we have the free transitive action σ: 𝔤 → D(G) of 𝔤 on G by right invariant vector fields. Let H ⊂ G be a connected Lie subgroup and 𝔥 ⊂ 𝔤 be the corresponding Lie subalgebra. Consider the quotient space G/H with the canonical projection p: G → G/H.

Due to equation (37), all right invariant vector fields are mapped by p_{∗} to well-defined vector fields on G/H. Consider the arising transitive action

σ_{𝔥} = p_{∗} ∘ σ: 𝔤 → D(G/H)

of 𝔤 on G/H. The following lemma is easy to prove.

Lemma 2. Let U be a connected open subset of G/H. Let X ∈ D(U) commute with all vector fields from σ_{𝔥}(𝔤). Then there is V ∈ D_{li} such that X = p_{∗}(V).

And vice versa, if V ∈ D_{li} is projectable to G/H then p_{∗}(V) commutes with all vector fields from σ_{𝔥}(𝔤). An element V ∈ D_{li} ≅ 𝔤 is projectable to G/H if and only if [V, 𝔥] ⊂ 𝔥.

In particular, if U is a connected open subset of G then the algebra

{V ∈ D(U) | [V, σ(𝔤)] = 0}

coincides with D_{li} ≅ 𝔤.

Lemma 3. Let W be a connected finite-dimensional manifold. Suppose that an action ρ: 𝔤 → D(W) is free and transitive. Then the Lie algebra

{V ∈ D(W) | [V, ρ(𝔤)] = 0}

is isomorphic to 𝔤 and acts on W freely and transitively as well.

Proof. It is well known that in this case the action ρ is locally isomorphic to the action σ: 𝔤 → D(G). By Lemma 2, we obtain that for any a ∈ W there is a neighborhood a ∈ U ⊂ W such that

{V ∈ D(U) | [V, ρ(𝔤)] = 0} ≅ 𝔤,
∀ b ∈ U ∀ v ∈ T_{b}W ∃! V ∈ D(U) : ev_{b}V = v, [V, ρ(𝔤)] = 0.

Since W is connected, this implies the statement of the lemma. □

Lemma 4. Let 𝔤 be a (possibly infinite-dimensional) Lie algebra, W_{1} and W_{2} be connected finite-dimensional manifolds, and ψ: W_{1} → W_{2} be a morphism of transitive actions ρ_{i}: 𝔤 → D(W_{i}), i = 1, 2. Suppose that ψ is a bundle with connected fibres and the algebra

𝔰 = {V ∈ D(W_{1}) | ψ_{∗}(V) = 0, [V, ρ_{1}(𝔤)] = 0}

acts freely and transitively on each fibre of ψ. Let 𝔥 ⊂ 𝔤 be the isotropy subalgebra of a point a ∈ W_{2} with respect to the action ρ_{2}. Then all vector fields from ρ_{1}(𝔥) are tangent to the fibre F = ψ^{−1}(a) ⊂ W_{1} and the image of the algebra ρ_{1}(𝔥) in D(F) is isomorphic to 𝔰.

Proof. The fact that all vector fields from ρ_{1}(𝔥) are tangent to F is obvious. Denote by 𝔣 the image of ρ_{1}(𝔥) in D(F). The algebra {V ∈ D(F) | [V, 𝔰] = 0} includes 𝔣 and is, by Lemma 3, isomorphic to 𝔰. Since dim 𝔣 ≥ dim F = dim 𝔰, we obtain

𝔣 = {V ∈ D(F) | [V, 𝔰] = 0} ≅ 𝔰.
□

1.8. Zero-curvature representations. Let 𝔤 be a Lie algebra over ℂ. Let E be an open subset of the infinite prolongation of a system of PDEs in two independent variables x, t such that D_{x}, D_{t} are well defined on E.

A pair of functions

(40)  M, N: E → 𝔤

is called a 𝔤-valued zero-curvature representation (ZCR in short) if

(41)  D_{x}(N) − D_{t}(M) + [M, N] = 0.

We suppose that all coefficients of the vector-valued functions (40) are admissible (i.e., belong to the structure sheaf).

Then each action ρ: 𝔤 → D(W) induces the covering structure in the bundle τ: E × W → E given by

D̃_{x} = D_{x} + ρ(M), D̃_{t} = D_{x} + ρ(N).

Equation (23) for A = ρ(M) and B = ρ(N) follows from (41).

For a morphism of actions ψ: W_{1} → W_{2} the mapping

id × ψ: E × W_{1} → E × W_{2}

is a morphism of the corresponding coverings.

Example 6. Let 𝔤 be a finite-dimensional Lie algebra. Clearly, a 𝔤-valued ZCR dependent polynomially on a parameter λ can be treated as a ZCR with values in the infinite-dimensional Lie algebra 𝔤 ⊗_{ℂ} ℂ[λ]. Then by the above construction each action of 𝔤 ⊗_{ℂ} ℂ[λ] determines a covering.

1.9. Translation-invariant coverings. In what follows we mainly consider translation-invariant PDEs (12) such that F_{α} do not depend on the independent variables x_{i}. In this case it is convenient to exclude the variables x_{i} from the set of coordinates on J^{∞}(π) and E. That is, admissible functions may depend on (10), but not on x_{i}. Besides, in this case we consider total derivative operators (11) without the term ∂/∂x_{i}.

The obtained manifold and the obtained distribution on it are called the translationinvariant infinite prolongation and the translation-invariant Cartan distribution of the translation-invariant system (12) respectively. Differential coverings of the translationinvariant infinite prolongation are called translation-invariant coverings.

Assume that there are two independent variables x, t. Then a differential covering (24) is translation-invariant if and only if a^{j}, b^{j} do not depend on x, t either. Making this restriction, we in fact do not loose any coverings, since, according to [7], with arbitrary covering (24) of rank m we can associate the following translation-invariant covering of rank m + 2

\frac{∂v^{1}}{∂x} = 1, \frac{∂v^{2}}{∂x} = 0, \frac{∂w^{j}}{∂x} = a^{j}(v^{1}, v^{2}, w^{k}, u_{σ}^{i}, …),

\frac{∂v^{1}}{∂t} = 0, \frac{∂v^{2}}{∂t} = 1, \frac{∂w^{j}}{∂t} = b^{j}(v^{1}, v^{2}, w^{k}, u_{σ}^{i}, …)

(we replaced x, t by v^{1}, v^{2} in the right-hand side of (24)). The fibres of this covering have the coordinates v^{1}, v^{2}, w^{1}, …, w^{m}.

Example 7. Consider a translation-invariant evolution equation

(42)  u_{t} = F (u, u_{1}, u_{2}, …, u_{p}), u_{k} = \frac{∂^{k}u}{∂x^{k}}, u = u_{0}.

Its translation-invariant infinite prolongation has the coordinates u_{k}, k ≥ 0. The total derivative operators are written in these coordinates as follows

(43)  D_{x} = ∑_{j≥0} u_{j+1} \frac{∂}{∂u_{j}} ,

(44)  D_{t} = ∑_{j≥0} D_{x}^{j}(F)\frac{∂}{∂u_{j}}

and span the translation-invariant Cartan distribution.

Let us rewrite the translation-invariance condition in coordinate-free terms.

Recall that a connection in a bundle π: E → M is given by a distribution D on E such that for any a ∈ E the mapping π_{∗}: D_{a} → T_{π(a)}M is an isomorphism of vector spaces. Then for each open subset U ⊂ M we have the natural linear mapping

∇: D(U) → D(π^{−1}(U))

that is uniquely defined by the following condition

∀ V ∈ D(U) ∇(V) ∈ D, π_{∗}(∇(V)) = V.

The connection is said to be flat if

∀ V_{1}, V_{2} ∈ D(U) ∇([V_{1}, V_{2}]) = [∇(V_{1}), ∇(V_{2})].

Consider the natural mapping

π_{∞,0} : J^{∞}(π) → E

arising from (7). Let Z be an open subset of E. Recall [1, 9, 10] that for any vector field X ∈ D(Z) there is a unique vector field S(X) ∈ D(π_{∞,0}^{−1}(Z)) such that

(45)  [S(X), C] ⊂ C, (π_{∞,0})_{∗}(S(X)) = X,

where C is the Cartan distribution on J^{∞}(π).

Fix a flat connection in the bundle π. An equation E ⊂ J^{∞}(π) is said to be translation-invariant (with respect to this flat connection) if for any vector field V on an open subset of M the vector field S(∇(V)) is tangent to E.

Vector fields of the form S(∇(V)) span another distribution D′ of rank dim M on J^{∞}(π). Let a ∈ M. The submanifold E′ = E ∩ π_{∞}^{−1}(a) is the translation-invariant infinite prolongation. To obtain the translation-invariant Cartan distribution C_{E′} on it, one projects the Cartan distribution C_{E} to E′ parallel to the distribution D′. The obtained distribution C_{E′} is involutive, but may be singular at some points of E′ (e.g., the points u_{i} = 0, i ≥ 1, in Example 7), and we exclude these singular points from the translation-invariant infinite prolongation. It is clear from the next example that locally the structure of the pair (E′, C_{E′}) does not depend on a ∈ M.

Example 8. As in Subsection 1.4, let

(46)  π: ℂ^{d+n} → ℂ^{n}, (x_{1}, …, x_{n}, u^{1}, …, u^{d}) ↦ (x_{1}, …, x_{n}).

Consider the flat connection given by ∇(∂/∂x_{i}) = ∂/∂x_{i}. It is well known that locally any flat connection is isomorphic to this one.

Since we have S(∂/∂x_{i}) = ∂/∂x_{i}, an equation E ⊂ J^{∞}(π) is translation-invariant with respect to this flat connection if and only if it can be given by a system (12) such that F_{α} do not depend on x_{i}.

1.10. Wahlquist-Estabrook coverings. Consider a translation-invariant evolution equation (42) satisfying ∂F/∂u_{p} ≠ 0. In order to describe locally all its translationinvariant coverings, one must solve equation (23) for

(47)  \begin{aligned}A & = ∑_{j=1}^{m} a^{j}(w^{1}, …, w^{m}, u, …, u_{k})\frac{∂}{∂w^{j}} , \\ B & = ∑_{j=1}^{m} b^{j}(w^{1}, …, w^{m}, u, …, u_{k})\frac{∂}{∂w^{j}} , \end{aligned}

for arbitrary k, m ∈ ℤ_{+}. If k is less than the order p of (42) then the covering is said to be of Wahlquist-Estabrook type.

Consider the following example.

Proposition 1 ([22, 20, 8]). For the KdV equation

(48)  u_{t} = u_{3} + u_{1}u, u_{i} = \frac{∂^{i}u}{∂x^{i}},

any Wahlquist-Estabrook covering

D_{x}B − D_{t}A + [A, B] = 0,

A = A(w^{1}, …, w^{m}, u, u_{1}, u_{2}), B = B(w^{1}, …, w^{m}, u, u_{1}, u_{2})

is of the form

A = X_{1} + \frac{1}{3}uX_{2} + \frac{1}{6}u^{2}X_{3}, (49)

B = (\frac{1}{3}u_{2} + \frac{1}{6}u^{2})X_{2} + (\frac{1}{9}u^{3} − \frac{1}{6}u_{1}^{2} + \frac{1}{3}uu_{2})X_{3} − X_{4}+ (50)

+\frac{1}{3}u[X_{1}, [X_{1}, X_{2}]] + \frac{1}{18}u^{2}[X_{2}, [X_{1}, X_{2}]] + \frac{1}{3}u_{1}[X_{2}, X_{1}],

where the vector fields X_{i} depend only on w^{1}, …, w^{m} and are subject to the relations

[X_{1}, X_{3}] = [X_{2}, X_{3}] = [X_{1}, X_{4}] = [X_{2}, [X_{2}, [X_{2}, X_{1}]]] = 0, (51) [X_{1}, [X_{1}, [X_{2}, X_{1}]]=[X_{4}, X_{2}], [X_{1}, [X_{2}, [X_{2}, X_{1}]]]=[X_{1}, X_{2}]+[X_{4}, X_{3}]. (52)

Remark 3. The KdV equation (48) differs from the one described in the introduction, but one is obtained from the other by a suitable scaling transformation u ↦ cu for some c ∈ ℂ.

Let 𝔉 be the free Lie algebra generated by the letters X_{1}, X_{2}, X_{3}, X_{4}. Let 𝔏 be the quotient of 𝔉 over relations (51), (52). Then formulas (49), (50) determine a ZCR of (48) with values in 𝔏 such that every Wahlquist-Estabrook covering arises from an action of 𝔏 by the construction of Section 1.8. The algebra 𝔏 is called the Wahlquist-Estabrook prolongation algebra of (48).

A similar description of Wahlquist-Estabrook coverings is known for many equations (42) (see, e.g., [2, 20, 8]).

Let us describe the algebra 𝔏 more explicitly. Below for q ∈ 𝔰𝔩_{2}(ℂ) and f(λ) ∈ ℂ[λ] we write the element

q ⊗ f(λ) ∈ 𝔰𝔩_{2}(ℂ) ⊗_{ℂ} ℂ[λ]

simply as qf(λ).

Proposition 2 ([19, 20]). The Lie algebra 𝔏 is isomorphic to the direct sum of the Lie algebra 𝔰𝔩_{2}(ℂ) ⊗_{ℂ}ℂ[λ] and the 5-dimensional Heisenberg algebra H. The algebra H has a basis

r_{−3}, r_{−1}, r_{0}, r_{1}, r_{3}

with the commutator table [r_{−1}, r_{1}] = [r_{3}, r_{−3}] = r_{0}, the other commutators being zero. The isomorphism is given by

(53)  X_{1} =r_{1}− \frac{1}{2}y+ \frac{1}{2}zλ, X_{2} =r_{−1}+z, X_{3} =r_{−3}, X_{4} =r_{3}− \frac{1}{2}yλ+ \frac{1}{2}zλ^{2},

where h, y, z is a basis of 𝔰𝔩_{2} with the relations

[h, y] = 2y, [h, z] = −2z, [y, z] = h.

Remark 4. One of the main ideas of this paper is to introduce Lie algebras playing similar role for coverings (23), (47) with arbitrary k.

The set of coverings of the form (47) is invariant under gauge transformations of the form

(54)  w^{i} ↦ f^{i}(w^{1}, …, w^{m}, u, …, u_{k−p}).

In order to define these Lie algebras, we find for coverings (23), (47) a canonical form with respect to the action of gauge transformations (54).

Since for Wahlquist-Estabrook coverings transformations (54) do not depend on u_{i}, i ≥ 0, all Wahlquist-Estabrook coverings are automatically in the canonical form.

Coverings (23), (47) with arbitrary k were also studied in [3]. However, gauge transformations were not considered there. Because of this, the authors of [3] had to impose some additional constraints on vector fields (47).

2. Analogs of the fundamental group for differential coverings

2.1. An instructive example. To motivate the next constructions, we present a description of some coverings of the KdV equation

(55)  u_{t} = u_{3} + u_{1}u.

The analogous description of all translation-invariant coverings of (55) will be given in Section 4.

The operators D_{x}, D_{t} below are given by (43), (44) with F = u_{3} + u_{1}u.

Theorem 1. Any translation-invariant covering (23) of the form

(56)  A = A(w^{1}, …, w^{m}, u, u_{1}, u_{2}, u_{3}), B = B(w^{1}, …, w^{m}, u, u_{1}, u_{2}, u_{3})

is locally equivalent to a covering of the form

A = X_{1} + \frac{1}{3}uX_{2} + \frac{1}{6}u^{2}X_{3} + f_{1}C, (57)

B = (\frac{1}{3}u_{2} + \frac{1}{6}u^{2})X_{2} + (\frac{1}{9}u^{3} − \frac{1}{6}u_{1}^{2} + \frac{1}{3}uu_{2})X_{3} − X_{4}+ (58)

+\frac{1}{3}u[X_{1}, [X_{1}, X_{2}]] + \frac{1}{18}u^{2}[X_{2}, [X_{1}, X_{2}]] + \frac{1}{3}u_{1}[X_{2}, X_{1}] + g_{1}C,

where the vector fields X_{i}, C depend only on w^{1}, …, w^{m} and satisfy

(59)  [C, X_{i}] = 0, i = 1, 2, 3, 4,

in addition to relations (51), (52). Here (f_{1}, g_{1}) is a conserved current of (55)

f_{1} = u_{1}^{2} − \frac{1}{3}u^{3}, g_{1} = 2u_{1}u_{3} − u_{2}^{2} − u^{2}u_{2} + 2uu_{1}^{2} − \frac{1}{4}u^{4},

D_{t}f_{1} = D_{x}g_{1}.

Proof. It is easy to obtain that A does not depend on u_{2}, u_{3} and is a polynomial of degree 2 in u_{1}

(60)  A = u_{1}^{2}A_{2}(w^{1}, …, w^{m}, u) + u_{1}A_{1}(w^{1}, …, w^{m}, u) + A_{0}(w^{1}, …, w^{m}, u).

We want to get rid of the term u_{1}A_{1} by switching to a locally gauge equivalent covering. Namely, consider an arbitrary point u_{i} = a_{i} ∈ ℂ, w^{j} = w_{0}^{j} ∈ ℂ where vector fields (56) are defined. We will find a gauge transformation defined on a neighborhood of this point that kills the term u_{1}A_{1}.

To this end, let

A_{1}(w^{1}, …, w^{m}, u) = ∑_{j} c^{j}(w^{1}, …, w^{m}, u)\frac{∂}{∂w^{j}} .

Consider the system of ordinary differential equations

\frac{d}{du}f^{j}(w^{1}, …, w^{m}, u) = c^{j}(f^{1}, …, f^{m}, u), j = 1, …, m,

dependent on the parameters w^{1}, …, w^{m}. Consider its local solution on a neighborhood of the point u = a_{0}, w^{j} = w_{0}^{j} with the initial condition f^{j}(w^{1}, …, w^{m}, a_{0}) = w^{j}. Then the formulas

(61)  u_{k} ↦ u_{k}, w^{j} ↦ f^{j}(w^{1}, …, w^{m}, u), k ≥ 0, j = 1, …, m,

define locally a gauge transformation φ such that

φ_{∗}(D_{x} + A) = D_{x} + A′, φ_{∗}(D_{t} + B) = D_{t} + B′,

where the vector field A′ is of the form (60) without the linear in u_{1} term (compare with Example 4).

Now it is straightforward to show that the vector fields A, B are of the form (57),

(58) with the relations

[X_{2}, X_{3}] = [X_{1}, X_{4}] = [C, X_{i}] = 0, i = 1, 2, 3,

[C, X_{4}] + \frac{1}{6}[X_{1}, X_{3}] = 0,

[C, X_{4}] + \frac{1}{3}[X_{1}, X_{3}] + \frac{1}{6}[X_{3}, [X_{1}, [X_{1}, X_{2}]]] = \frac{1}{18}[X_{2}, [X_{2}, [X_{2}, X_{1}]]],

[X_{3}, [X_{2}, [X_{1}, X_{2}]]] = 0, [X_{1}, [X_{1}, [X_{2}, X_{1}]] = [X_{4}, X_{2}],

[X_{1}, [X_{2}, [X_{2}, X_{1}]]]=[X_{1}, X_{2}]+[X_{4}, X_{3}]. (62)

From these relations it follows that [X_{1}, X_{3}] and (ad^{3}X_{2})(X_{1}) commute with X_{1}, X_{2}. Now applying ad^{2}X_{2} to (62) we obtain (ad^{3}X_{2})(X_{1}) = 0, which implies (59), (51), (52). □

2.2. The definition of the fundamental algebras. Consider a system of PDEs in two independent variables x, t. The results of this section are applicable to the following two situations.

(1) The manifold E is the infinite prolongation defined in Section 1.3, and C_{E} is the Cartan distribution on it.

(2) The system of PDEs is translation-invariant, the manifold E is the translationinvariant infinite prolongation defined in Section 1.9, and C_{E} is the translationinvariant Cartan distribution.

However, all examples of this paper belong to the second situation.

Without loss of generality, we can assume E to be connected. Moreover, we assume that the total derivative operators D_{x}, D_{t} are well defined on E. This is not a big restriction, because most of our results are local and locally this is always the case.

Remark 5. In fact the main Definition 5 below can be readily generalized for PDEs in any number of independent variables. However, since all PDEs considered in this paper are in two independent variables, for the sake of clarity we prefer to give this simplified version.

Remark 6. Below in this section we use the following notation. For an open subset E′ of E and a finite-dimensional manifold W, the mapping

(63)  E′ × W → E′

is always the projection to the first factor. For a function f on E, its restriction to E′ is denoted by the same symbol f.

According to Section 1.5, a covering structure in the trivial bundle (63) is uniquely determined by a pair of vector fields A, B ∈ D(E′ × W) that are vertical with respect to projection (63) and satisfy relation (23).

We have the natural embedding D(W) ⊂ D(E′ × W). A vector field X ∈ D(E′ × W) belongs to D(W) if and only if it is vertical with respect to (63) and its coefficients do not depend on coordinates of E.

Inspired by Theorem 1, let us give the following definition.

Definition 5. We say that E possesses fundamental algebras if there are finite sets A_{k}, B_{k}, k ∈ ℤ_{+}, of functions on E satisfying the relations

(64)  A_{k} ⊂ A_{k+1}, B_{k} ⊂ B_{k+1} ∀ k

such that for any connected open subset E_{1} of E the following conditions hold.

(1) Let τ: Ẽ → E_{1} be a covering of E_{1}. Then for any point a ∈ Ẽ there are a neighborhood a ∈ Ẽ_{1} ⊂ Ẽ and k ∈ ℕ such that for E_{2} = τ(Ẽ_{1}) ⊂ E the covering τ|Ẽ_{1} : Ẽ_{1} → E_{2} is isomorphic to a covering E_{2} ×W → E_{2} of the following canonical form

[D_{x} + A, D_{t} + B] = 0, (65)

A = ∑_{f∈A_{k}} fM_{f}, B = ∑_{g∈B_{k}} gN_{g}, (66)

M_{f}, N_{g} ∈ D(W). (67)

(2) Any morphism φ: E_{1} × W_{1} → E_{1} × W_{2} between two coverings of the form

E_{1} × W_{i} → E_{1}, i = 1, 2,

A^{i} = ∑_{f∈A_{k}} fM_{f}^{i}, B^{i} = ∑_{g∈B_{k}} gN_{g}^{i},

[D_{x} + A^{i}, D_{t} + B^{i}] = 0, M_{f}^{i}, N_{g}^{i} ∈ D(W_{i}), i = 1, 2,

is of the form φ = id × ψ, where

ψ: W_{1} → W_{2}, ψ_{∗}(M_{f}^{1}) = M_{f}^{2}, ψ_{∗}(N_{g}^{1}) = N_{g}^{2}.

(3) Let X ∈ D(E_{1} × W) be a symmetry of a covering E_{1} × W → E_{1} given by vector

fields

D_{x} + A, D_{t} + B ∈ D(E_{1} × W)

satisfying (65), (66), (67). Then X ∈ D(W) and

[X, M_{f}] = [X, N_{g}] = 0 ∀ f ∈ A_{k}, ∀ g ∈ B_{k}.

(4) Consider the manifold E_{1} × W with the distribution spanned by D_{x} + A, D_{t} + B of the form (65), (66), (67) and let E′ be a subequation of it. Then locally E′ is of the form E_{2} × W′, where E_{2} is an open subset of E_{1} and W′ is a submanifold of W such that vector fields (67) are tangent to W′.

In particular, E_{1} × W is irreducible if and only if W is connected and the Lie algebra generated by vector fields (67) acts on W transitively.

In this case fundamental algebras 𝔣_{k} are defined as follows. Let 𝔮_{k} be the free Lie algebra generated by the letters M_{f}, N_{g} for f ∈ A_{k}, g ∈ B_{k}. Let us treat (66) as functions on E with values in 𝔮_{k}. Consider the ideal I_{k} of 𝔮_{k} generated by the elements

∑_{g∈B_{k}} D_{x}(g)(a)N_{g} − ∑_{f∈A_{k}} D_{t}(f)(a)M_{f} + ∑_{f∈A_{k}, g∈B_{k}} f(a)g(a)[M_{f}, N_{g}], a ∈ E,

and set 𝔣_{k} = 𝔮_{k}/I_{k}.

Then (66) becomes an 𝔣_{k}-valued ZCR of E. For an action

(68)  ρ: 𝔣_{k} → D(W)

denote by τ(ρ) the covering E × W → E corresponding to (68) by the construction of Section 1.8.

From (64) we have the natural epimorphism

(69)  p_{k}: 𝔮_{k} → 𝔮_{k−1}

that maps the generators

(70)  M_{f}, N_{g}, f ∈ A_{k} \ A_{k−1}, g ∈ B_{k} \ B_{k−1},

to zero. It is easily seen that p_{k}(I_{k}) ⊂ I_{k−1}. Therefore, epimorphisms (69) determine the epimorphisms

(71)  ⋯ → 𝔣_{k} → 𝔣_{k−1} → ⋯ → 𝔣_{1} → 𝔣_{0}.

Example 9. From Theorem 1 for the KdV equation (55) we can take

A_{1} = {1, u, u^{2}, u^{3}, u_{1}^{2}},
B_{1} = {u^{i_{0}}u_{1}^{i_{1}}u_{2}^{i_{2}}u_{3}^{i_{3}} | i_{n} ∈ ℤ_{+}, 0 ≤ 2i_{0} + 3i_{1} + 4i_{2} + 5i_{3} ≤ 8}.

For example, in this case we have

M_{u_{1}^{2}} = −3M_{u^{3}} = \frac{1}{2}N_{u_{1}u_{3}} = −N_{u_{2}^{2}} = C,

M_{u} = N_{u_{2}} = \frac{1}{3}X_{2}, N_{uu_{3}} = N_{u_{1}u_{2}} = 0.

The algebra 𝔣_{1} is isomorphic to the quotient of the free Lie algebra generated by the letters X_{1}, X_{2}, X_{3}, X_{4}, C over relations (51), (52), (59). Formulas (57), (58) determine a ZCR of (55) with values in 𝔣_{1} such that each covering of the form (56) is equivalent to a covering determined by an action of 𝔣_{1}. The algebra 𝔣_{0} is isomorphic to the algebra 𝔏 from Section 1.10.

For k > 3, coverings of (55) of the form

A = A(w^{1}, …, w^{m}, u, u_{1}, …, u_{k}), B = B(w^{1}, …, w^{m}, u, u_{1}, …, u_{k})

are determined in a similar way by actions of higher algebras 𝔣_{k−2}, which will be studied in Section 4.2.

Remark 7. Consider the identity covering E → E. It has canonical form (65), (66),

(67) with M_{f} = N_{g} = 0 and W equal to a point. From Condition 4 we see that any connected open subset of the equation E itself must be irreducible.

Remark 8. Consider an action (68) and let l ≥ k. Consider the epimorphism φ: 𝔣_{l} → 𝔣_{k} from (71) and the action ρφ: 𝔣_{l} → D(W). By the construction of epimorphisms (71), we have τ(ρφ) = τ(ρ).

Therefore, when we consider a finite number of coverings determined by actions

ρ_{i}: 𝔣_{k_{i}} → D(W_{i}), i = 1, …, s,

we can assume that all the actions are defined on the same algebra 𝔣_{k}, where

k = max{k_{1}, …, k_{s}}.

Below in this section we suppose everywhere that E possesses fundamental algebras (71) and E_{1} is a connected open subset of E.

Theorem 2. For any covering τ: Ẽ → E_{1} each point a ∈ Ẽ lies in a locally unique irreducible subequation Ẽ_{a} ⊂ Ẽ. The image τ(Ẽ_{a}) is open in E_{1}, and τ|Ẽ_{a} is a covering.

Proof. It is sufficient to prove this statement locally. Then we can assume that one has Ẽ = E_{1} × W and τ = τ(ρ) for some action ρ: 𝔣_{k} → D(W). Let

a = (q, z) ∈ E_{1} × W, q ∈ E_{1}, z ∈ W.

By Proposition 3 below, locally there is a unique submanifold W′ ⊂ W such that z ∈ W′, all vector fields from ρ(𝔣_{k}) are tangent to W′, and the induced action on W′ is transitive. By Condition 4 of Definition 5, the submanifold Ẽ_{a} = E × W′ ⊂ Ẽ is the required irreducible subequation.

Proposition 3 ([13]). Let 𝔤 be an arbitrary Lie algebra over ℂ and ρ: 𝔤 → D(W) be an action of 𝔤 on a complex-analytic manifold W. Then for each point z ∈ W there is submanifold z ∈ W′ ⊂ W such that all vector fields from ρ(𝔤) are tangent to W′ and the action of 𝔤 on W′ is transitive. The submanifold W′ is locally unique and dim W′ = dim ev_{z}(ρ(𝔤)).

□

Consider a covering τ: Ẽ → E_{1}, where Ẽ is connected. Condition 1 of Definition 5 determines locally an action of 𝔣_{k} on fibres of τ. Due to Condition 2 these local actions produce a well-defined global action

ρ(τ): 𝔣_{k} → D(Ẽ)

such that τ_{∗} ∘ ρ(τ) = 0.

Theorem 3. The covering τ is irreducible if and only if the action ρ(τ) is transitive on each fibre of τ.

Proof. It is sufficient to prove this locally, which is done similarly to the proof of Theorem 2. □

Theorem 4. Consider two coverings τ_{i}: Ẽ^{i} → E_{1}, i = 1, 2, and a mapping φ: Ẽ^{1} → Ẽ^{2} such that the diagram

Ẽ^{1}  \overline{φ →} Ẽ^{2}
[figure: τ_{1}; τ_{2}]
E_{1}

is commutative.

(1) The mapping φ is a morphism of coverings if and only if it is a morphism of the actions ρ(τ_{1}) and ρ(τ_{2}).

(2) If τ_{1} and τ_{2} are irreducible and φ is a morphism of coverings then φ(Ẽ^{1}) is open in Ẽ^{2}.

Proof. It is sufficient to prove both statements locally.

(1) This follows from Condition 2 of Definition 5.

(2) This follows from the previous statement and Lemma 1. □

Remark 9. Recall that a covering of a connected finite-dimensional manifold M is connected if and only if the corresponding action of π_{1}(M) is transitive. Theorem 3 suggests that in PDE geometry irreducible equations play the role of ‘connected’ objects. Then Theorem 2 is the analog for PDEs of the decomposition into connected components of a topological space.

2.3. Regular coverings and their symmetry algebras. In the present form the analogy of (71) with the topological fundamental group is not sufficiently helpful, because canonical form (66) and the vector fields M_{f}, N_{g} ∈ D(W) have no invariant (coordinate-free) meaning. In order to recover algebras 𝔣_{k} in an invariant way, recall that the topological fundamental group can be expressed in terms of automorphism groups of coverings. Studying differential coverings, it is more convenient to consider infinitesimal automorphisms, i.e., symmetries.

From Condition 3 of Definition 5, for each action ρ: 𝔣_{k} → D(W) we obtain

(72)  Sym τ(ρ) = {v ∈ D(W) | [v, ρ(𝔣_{k})] = 0}.

Recall that a connected topological covering M̃ → M is said to be regular if the action of its automorphism group on M̃ is free and transitive on each fibre. Similarly, we call an irreducible differential covering τ: Ẽ → E regular if the action on Ẽ of the algebra Sym τ is free and transitive on each fibre of τ. In particular, τ is the quotient map with respect to this action, and dim Sym τ = rank τ.

Theorem 5. A covering τ: Ẽ → E_{1} is regular if and only if the action on Ẽ of the subalgebra ρ(τ)(𝔣_{k}) ⊂ D(Ẽ) is free and transitive on each fibre of τ. In this case one has dim ρ(τ)(𝔣_{k}) = rank τ and Sym τ ≅ ρ(τ)(𝔣_{k}).

Proof. It is sufficient to prove this locally, and the local version follows from (72) and Lemma 3. □

Each ideal 𝔦 of 𝔣_{k} with codim 𝔦 < ∞ determines a regular covering as follows. Consider the canonical epimorphism ψ: 𝔣_{k} → 𝔣_{k}/𝔦. Let σ: 𝔣_{k}/𝔦 → D(G) be the natural action by right invariant vector fields on the simply connected Lie group G whose Lie algebra is the finite-dimensional algebra 𝔣_{k}/𝔦. For any open subset U ⊂ G we have the transitive action σψ: 𝔣_{k} → D(U). By Theorem 5, the corresponding covering τ(σψ) is regular, and every regular covering is locally isomorphic to a covering of this form. By Remark 8, if 𝔦_{n} ⊂ 𝔣_{n}, n ≥ k, is the preimage of 𝔦 under epimorphism (71) then 𝔣_{n}/𝔦_{n} ≅ 𝔣_{k}/𝔦 and the corresponding regular coverings are also isomorphic.

Let 𝔦_{1}, 𝔦_{2} ⊂ 𝔣_{k} be two ideals of finite codimension. Consider the simply connected Lie groups G_{1}, G_{2} associated with the Lie algebras 𝔣_{k}/𝔦_{1}, 𝔣_{k}/𝔦_{2}. Let U_{i} ⊂ G_{i}, i = 1, 2, be connected open subsets.

Suppose that the corresponding regular coverings are connected by a morphism φ

E_{1} × U_{1}  \overline{φ →} E_{1} × U_{2}
[figure: τ_{1}; τ_{2}]
E_{1}

From Condition 2 of Definition 5 it follows that 𝔦_{1} ⊂ 𝔦_{2} and φ = id×ψ, where ψ: U_{1} → U_{2} is a morphism of actions of 𝔣_{k}. By Theorem 5, we have Sym τ_{i} ≅ 𝔣_{k}/𝔦_{i}, i = 1, 2. The mapping

φ: E_{1} × U_{1} → φ(E_{1} × U_{1}) = E_{1} × ψ(U_{1})

is the quotient mapping with respect to the action of the subalgebra 𝔦_{2}/𝔦_{1} ⊂ Sym τ_{1} on the manifold E_{1} × U_{1}.

Similarly to Theorem 4, this local description of regular coverings and morphisms connecting them implies the following global result.

Theorem 6. Consider two regular coverings τ_{i}: Ẽ^{i} → E_{1}, i = 1, 2, and let

Ẽ^{1}  \overline{φ →} Ẽ^{2}
[figure: τ_{1}; τ_{2}]
E_{1}

be a morphism of them. Then there is k ∈ ℕ and two ideals 𝔦_{1}, 𝔦_{2} of 𝔣_{k} of finite codimension such that

• one has

(73)  Sym τ_{i} = 𝔣_{k}/𝔦_{i}, i = 1, 2,

• we have 𝔦_{1} ⊂ 𝔦_{2}, the subset φ(Ẽ^{1}) is open in Ẽ^{2}, and the mapping φ: Ẽ^{1} → φ(Ẽ^{1}) is the quotient mapping with respect to the action of the subalgebra 𝔦_{2}/𝔦_{1} ⊂ Sym τ_{1} on the manifold Ẽ^{1},

• the differential φ_{∗} of φ induces an epimorphism of algebras Sym τ_{1} → Sym τ_{2}. In terms of isomorphisms (73) it is the natural epimorphism 𝔣_{k}/𝔦_{1} → 𝔣_{k}/𝔦_{2} corresponding to the inclusion 𝔦_{1} ⊂ 𝔦_{2}.

In contrast to fundamental algebras (71), the system of symmetry algebras of regular coverings is a coordinate-free canonical invariant of a system of PDEs, since symmetry algebras are coordinate-independent objects. Thus we recover in an invariant way not algebras (71) themselves, but all their finite-dimensional quotients.

2.4. Quasi-finite Lie algebras. We present here some results on Lie algebras.

Definition 6. A Lie algebra 𝔤 is said to be quasi-finite if for any subalgebra 𝔥 ⊂ 𝔤 of finite codimension there is an ideal of 𝔤 that is of finite codimension and is contained in 𝔥.

Theorem 7. Let 𝔤 be a quasi-finite algebra. Then for any transitive action ρ: 𝔤 → D(W) on a connected finite-dimensional manifold W the algebra ρ(𝔤) is finite-dimensional.

Proof. Let a ∈ W and 𝔥 ⊂ 𝔤 be the isotropy subalgebra of a. Since 𝔤 is quasi-finite and codim 𝔥 = dim W < ∞, there is an ideal 𝔦 of 𝔤 such that 𝔦 ⊂ 𝔥 and codim 𝔦 < ∞. It is well known that in the complex-analytic situation the image ρ(𝔥) of the isotropy subalgebra cannot contain any nontrivial ideal of ρ(𝔤). Therefore, ρ(𝔦) = 0 and

dim ρ(𝔤) ≤ codim 𝔦 < ∞.
□

Theorem 8. Let 𝔤 be a quasi-finite Lie algebra and ρ: 𝔤̃ → 𝔤 be an epimorphism such that

(74)  [ker ρ, 𝔤̃] = 0

(that is, 𝔤̃ is a central extension of 𝔤). Then 𝔤̃ is also quasi-finite.

Proof. Let 𝔥 ⊂ 𝔤̃ be a subalgebra of finite codimension. Clearly, the subset

(75)  𝔥_{1} = {a ∈ 𝔥 | [𝔤̃, a] ⊂ 𝔥]}

is also a subalgebra of finite codimension. By assumption, there is an ideal 𝔦 of 𝔤 that is of finite codimension and is contained in ρ(𝔥_{1}).

The subspace ρ^{−1}(𝔦) ∩ 𝔥 is of finite codimension and is contained in 𝔥. Let us prove that ρ^{−1}(𝔦) ∩ 𝔥 is an ideal of 𝔤̃.

Let a ∈ ρ^{−1}(𝔦) ∩ 𝔥 and v ∈ 𝔤̃. Then a = h + z, where h ∈ 𝔥_{1} and z ∈ ker ρ. Combining (74) and (75), we obtain [v, a] ∈ 𝔥. Besides, since ρ^{−1}(𝔦) is an ideal of 𝔤̃, we have [v, a] ∈ ρ^{−1}(𝔦). Therefore, [v, a] ∈ ρ^{−1}(𝔦) ∩ 𝔥. □

Let 𝔤 be a Lie algebra over ℂ and A be a commutative associative algebra over ℂ. Then the space 𝔤 ⊗_{ℂ} A has the following natural Lie algebra structure

(76)  [g_{1} ⊗ a_{1}, g_{2} ⊗ a_{2}] = [g_{1}, g_{2}] ⊗ a_{1}a_{2}, g_{1}, g_{2} ∈ 𝔤, a_{1}, a_{2} ∈ A.

Theorem 9. If 𝔤 is finite-dimensional and semisimple then the Lie algebra 𝔤 ⊗_{ℂ} A is quasi-finite.

Proof. Let 𝔥 ⊂ 𝔤 ⊗_{ℂ} A be a subalgebra of finite codimension. Then the subspace

(77)  Z = {f ∈ A | 𝔤 ⊗ f ⊂ 𝔥}

is of finite codimension in A. Since [𝔤, 𝔤] = 𝔤, the subspace Z is a subring of A. The subspace Z′ = {f ∈ Z | fA ⊂ Z} is of finite codimension and is an ideal of the ring A. Therefore, the subspace 𝔤 ⊗ Z′ is an ideal of 𝔤 ⊗ A of finite codimension, and from (77) we have 𝔤 ⊗ Z′ ⊂ 𝔥. □

2.5. Local structure of irreducible coverings. Below we suppose that algebras (71) are quasi-finite.

Consider a subalgebra 𝔥 ⊂ 𝔣_{k} of finite codimension. Let 𝔦(𝔥) be the maximal ideal of 𝔣_{k} that is contained in 𝔥. Since 𝔣_{k} is quasi-finite, we have codim 𝔦(𝔥) < ∞.

Let G be the simply connected Lie group whose Lie algebra is 𝔣_{k}/𝔦(𝔥) and H ⊂ G be the connected Lie subgroup whose Lie subalgebra is 𝔥/𝔦(𝔥). According to Section 1.7, the algebra 𝔣_{k} acts on G by right invariant vector fields, which are projected also to the quotient space G/H. Denote by σ(𝔣_{k}, 𝔥) the arising transitive action of 𝔣_{k} on the manifold W (𝔣_{k}, 𝔥) = G/H. We have ker σ(𝔣_{k}, 𝔥) = 𝔦(𝔥).

Remark 10. Let G be a Lie group associated with a Lie algebra 𝔤. Generally, not for every subalgebra 𝔥 ⊂ 𝔤 there is a Lie subgroup whose Lie subalgebra is 𝔥. However, for us it is sufficient to consider the local Lie subgroup, which always exists. In this case the symbol G/H denotes the quotient space not of the whole group G, but of some neighborhood of the unity element.

As above, let E_{1} be a connected open subset of E. Consider the manifold E_{1}(𝔣_{k}, 𝔥) = E_{1} × W (𝔣_{k}, 𝔥) and the covering

τ(𝔣_{k}, 𝔥): E_{1}(𝔣_{k}, 𝔥) → E_{1}

corresponding to the action σ(𝔣_{k}, 𝔥) of 𝔣_{k}.

Theorem 10. The following statements hold.

(1) Every irreducible covering τ of E_{1} is locally isomorphic to a covering τ(𝔣_{k}, 𝔥) for some k ∈ ℕ and 𝔥 ⊂ 𝔣_{k}.

(2) We have

(78)  Sym τ(𝔣_{k}, 𝔥) ≅ 𝔫(𝔥)/𝔥,

where

(79)  𝔫(𝔥) = {v ∈ 𝔣_{k} | [v, 𝔥] ⊂ 𝔥}.

(3) The covering τ(𝔣_{k}, 𝔥) is regular if and only if 𝔥 is an ideal of 𝔣_{k}.

Proof. (1) By Conditions 1 and 4 of Definition 5, locally we have τ = τ(ρ) for some transitive action ρ: 𝔣_{k} → D(W). Let a ∈ W and consider the isotropy subalgebra 𝔥 ⊂ 𝔣_{k} of a. By Theorem 7, the actions ρ and σ(𝔣_{k}, 𝔥) are locally isomorphic. Then the coverings τ(ρ) and τ(𝔣_{k}, 𝔥) are locally isomorphic as well.

(2) Formulas (78) and (79) follow from formula (72) and Lemma 2.

(3) This follows from Theorem 5. □

Recall that for any connected topological covering τ: M̃ → M there is a commutative diagram of coverings

\begin{aligned}M′ & & \overline{p →} & M̃ \\ & [figure: r] & [figure: τ] \\ & & M \end{aligned} ; (80)

where r is regular and p is the quotient mapping with respect to the action of some automorphism subgroup of r. Let us construct an analog of diagram (80) for differential coverings.

Theorem 11. Any irreducible covering τ: Ẽ → E_{1} is locally included in a commutative diagram of irreducible coverings

E′  \overline{p  →} Ẽ
[figure: r; τ]
E_{1}

such that the following assertions hold.

(1) The covering r is regular.

(2) The covering p is the quotient morphism with respect to the action on E′ of some subalgebra 𝔥 of Sym r.

(3) The algebra Sym τ coincides with the quotient 𝔫/𝔥, where

𝔫 = {v ∈ Sym r | [v, 𝔥] ⊂ 𝔥},

and the action of 𝔫/𝔥 on Ẽ is induced by the action of 𝔫 on E′.

Proof. By Theorem 10 (1), it is sufficient to prove the statements for τ = τ(𝔣_{k}, 𝔥_{1}), where 𝔥_{1} is a subalgebra of 𝔣_{k} of finite codimension.

Recall that 𝔦(𝔥_{1}) is the maximal ideal of 𝔣_{k} that is contained in 𝔥_{1}. By Theorem 10 (3), the covering r = τ(𝔣_{k}, 𝔦(𝔥_{1})) is regular. The inclusion of Lie algebras

𝔦(𝔥_{1}) ⊂ 𝔥_{1} ⊂ 𝔣_{k}

determines a surjective morphism

W (𝔣_{k}, 𝔦(𝔥_{1})) → W (𝔣_{k}, 𝔥_{1})

of actions of 𝔣_{k}, which determines the surjective morphism

p: E_{1}(𝔣_{k}, 𝔦(𝔥_{1})) → E_{1}(𝔣_{k}, 𝔥_{1}), τ(𝔣_{k}, 𝔥_{1}) ∘ p = τ(𝔣_{k}, 𝔦(𝔥_{1})),

of the corresponding coverings.

By formulas (78) and (79), one has Sym r = 𝔣_{k}/𝔦(𝔥_{1}). By construction, the morphism p is the quotient map with respect to the action of 𝔥 = 𝔥_{1}/𝔦(𝔥_{1}) ⊂ Sym r on the manifold E_{1}(𝔣_{k}, 𝔦(𝔥_{1})). Finally, the last statement of the theorem follows from formulas (78), (79).

□

For a subalgebra 𝔥 of 𝔣_{k} of finite codimension, denote by 𝔥_{l} the preimage of 𝔥 in 𝔣_{l}, l ≥ k, under epimorphisms (71). By Remark 8, one obtains

E_{1}(𝔣_{l}, 𝔥_{l}) ≅ E_{1}(𝔣_{k}, 𝔥), τ(𝔣_{l}, 𝔥_{l}) ≅ τ(𝔣_{k}, 𝔥) ∀ l ≥ k.

If 𝔥̃ ⊂ 𝔥_{l} is a subalgebra of finite codimension then we have the natural surjective morphism W (𝔣_{l}, 𝔥̃) → W (𝔣_{l}, 𝔥_{l}) of actions of 𝔣_{l}, which determines a covering

τ(𝔥, 𝔥̃): E_{1}(𝔣_{l}, 𝔥̃) → E_{1}(𝔣_{l}, 𝔥_{l}) ≅ E_{1}(𝔣_{k}, 𝔥).

Let 𝔦 be an ideal of 𝔥_{l} with codim 𝔦 < ∞ (but not necessarily an ideal of 𝔣_{l}). By formulas (78), (79), the covering

(81)  τ(𝔥, 𝔦): E_{1}(𝔣_{l}, 𝔦) → E_{1}(𝔣_{k}, 𝔥)

is regular and Sym τ(𝔥, 𝔦) ≅ 𝔥_{l}/𝔦. The following theorem shows that locally every regular covering of E_{1}(𝔣_{k}, 𝔥) is of this form.

Theorem 12. Consider an irreducible covering τ: Ẽ → E_{1} and the corresponding action ρ(τ): 𝔣_{k} → D(Ẽ). Let a ∈ Ẽ and 𝔥 ⊂ 𝔣_{k} be the isotropy subalgebra of a.

Then for any connected neighborhood Ẽ′ ⊂ Ẽ of a the symmetry algebra of any regular covering over Ẽ′ is isomorphic to a finite-dimensional quotient of 𝔥_{l} for some l ≥ k.

And vice versa, for any l ≥ k and any ideal 𝔦 of 𝔥_{l} of finite codimension there is a regular covering τ′ over a neighborhood of a such that Sym τ′ = 𝔥_{l}/𝔦.

Proof. Let τ′: E′′ → Ẽ′ be a regular covering. Consider the connected open subset E_{2} = τ(Ẽ′) of E_{1} and the commutative diagram of coverings

E′′  \overline{τ′ →} Ẽ′
[figure: ττ′; τ]
E_{2}

Since the question is essentially local, we can assume that the above diagram is of the form

E_{2} × W_{1} \overline{\underrightarrow{id × ψ}} E_{2} × W_{2}
[figure: τ(ρ_{1}); τ(ρ_{2})]
E_{2}

where ρ_{i}: 𝔣_{l} → D(W_{i}), i = 1, 2, are transitive actions for some l ≥ k and ψ: W_{1} → W_{2} is a morphism of actions. The point a is of the form a = (q, z), q ∈ E_{2}, z ∈ W_{2}. The algebra h_{l} is the isotropy algebra of z with respect to the action ρ_{2}. Then the first statement of the theorem follows from Lemma 4 for 𝔤 = 𝔣_{l}.

The second statement of the theorem follows from construction (81). □

This theorem is the analog of the fact that for a connected topological covering M̃ → M one has π_{1}(M̃) ⊂ π_{1}(M).

Since 𝔤 in Lemma 4 is allowed to be infinite-dimensional, the first statement of Theorem 12 holds even if algebras (71) are not quasi-finite.

Theorem 13. In the notation of Theorem 12, the symmetry algebra of any regular covering over Ẽ′ is isomorphic to a finite-dimensional quotient of 𝔥_{l} for some l ≥ k even if the fundamental algebras are not quasi-finite.

2.6. Necessary conditions for existence of Bäcklund transformations. Consider two systems of PDEs

(82)  E_{i} = \{F_{α}^{i}(x, t, u^{i1}, …, u^{id_{i}}, \frac{∂^{p+q}u^{ij}}{∂x^{p}∂t^{q}} , …)= 0, α = 1, …, s_{i}\}, i = 1, 2,

A Bäcklund transformation between E_{1} and E_{2} is given by another system

(83)  F_{α}(x, t, v^{1}, …, v^{d}, \frac{∂^{p+q}v^{j}}{∂x^{p}∂t^{q}} , …)= 0, α = 1, …, s.

and two mappings

(84)  u^{ij} = φ^{ij}(x, t, v^{1}, …, v^{d}, \frac{∂^{p+q}v^{l}}{∂x^{p}∂t^{q}} , …), j = 1, …, d_{i}, i = 1, 2,

such that for each i = 1, 2 one has

• for each local solution v^{1}(x, t), …, v^{d}(x, t) of (83) functions (84) form a local solution of (82),

• for each local solution u^{i1}(x, t), …, u^{id_{i}}(x, t) of (82) the system that consists of equations (83) and (84) is consistent and possesses locally a general solution

v^{1}(x, t, c_{1}, …, c_{r_{i}}), …, v^{d}(x, t, c_{1}, …, c_{r_{i}})

dependent on a finite number of complex parameters c_{1}, …, c_{r_{i}}. Similarly to Section 1.6, these conditions mean by definition that the infinite prolongation Ẽ of (83) covers both E_{1} and E_{2}

Ẽ ; [figure: τ_{1}; τ_{2}] ; E_{1} E_{2} ; (85)

where the covering τ_{i} is of rank r_{i}. We allow E_{i} to be not the whole infinite prolongation, but some nonempty open subset of it.

If systems (82) are translation-invariant then we can make (83) translation-invariant as well using the trick from Section 1.9 : replace x, t in F_{α} and φ^{ij} by the new dependent variables w^{1}, w^{2} respectively and add to (83) the following equations

\frac{∂w^{1}}{∂x} = \frac{∂w^{2}}{∂t} = 1, \frac{∂w^{1}}{∂t} = \frac{∂w^{2}}{∂x} = 0.

After this substitution coverings (85) become translation-invariant.

Example 10. Consider two different coverings from the modified KdV equation to the KdV equation

v_{t} =v_{xxx}−6v^{2}v_{x}

[figure: u=v_{x}−v^{2}; u=−v_{x}−v^{2}]

u_{t} =u_{xxx}+6uu_{x}  u_{t} =u_{xxx}+6uu_{x}

This diagram presents a Bäcklund auto-transformation of the KdV equation. See, e.g, [15, 16] for more examples of Bäcklund transformations.

Theorem 14. Suppose that two systems E_{i}, i = 1, 2, possess fundamental algebras

(86)  ⋯ → 𝔣_{k+1}^{i} → 𝔣_{k}^{i} → ⋯ → 𝔣_{1}^{i} → 𝔣_{0}^{i}, i = 1, 2,

and the algebras 𝔣_{k}^{1} are quasi-finite. Let 𝔤 be a finite-dimensional Lie algebra. Suppose that for any k_{1}, k_{2} ∈ ℤ_{+} and any subalgebras

(87)  𝔥^{i} ⊂ 𝔣_{k_{i}}^{i}, codim 𝔥^{i} < ∞, i = 1, 2,

there is an epimorphism 𝔥^{1} → 𝔤, but there is no epimorphism 𝔥^{2} → 𝔤. Then there is no Bäcklund transformation between E_{1} and E_{2}.

Proof. Suppose that there is a Bäcklund transformation. By the above construction, it determines a diagram (85) of coverings. Let a ∈ Ẽ. By Theorem 2, locally there is a unique irreducible subequation Ẽ_{a} ⊂ Ẽ that contains a. The subset τ_{i}(Ẽ_{a}) is open in E_{i}, and the coverings

τ_{i}|Ẽ_{a}: Ẽ_{a} → τ_{i}(Ẽ_{a}), i = 1, 2,

are irreducible. Consider the action

ρ(τ_{1}|Ẽ_{a}): 𝔣_{k}^{1} → D(Ẽ_{a})

and let 𝔥^{1} ⊂ 𝔣_{k}^{1} be the isotropy subalgebra of a. By Theorem 12, an epimorphism 𝔥^{1} → 𝔤 implies that over a connected neighborhood of a ∈ Ẽ_{a} there is a regular covering with symmetry algebra equal to 𝔤. Applying Theorem 13 to this regular covering and the covering τ_{2}|Ẽ_{a}, we obtain that 𝔤 is isomorphic to a quotient of some subalgebra 𝔥^{2} of 𝔣_{l}^{2} of finite codimension. Thus we get a contradiction. □

3. Coverings of scalar evolution equations

In this section we prove some technical results, which will be needed in Sections 4 and 5. Consider a translation-invariant evolution equation

u_{t} = F (u, u_{1}, u_{2}, …, u_{p}), \frac{∂F}{∂u_{p}} ≠ 0, (88)

u_{i} = \frac{∂^{i}u}{∂x^{i}}, u = u_{0}.

Let E be a connected open subset of the translation-invariant infinite prolongation of this equation described in Example 7.

Let W be a connected open subset of ℂ^{m} with coordinates w^{1}, …, w^{m} and

(89)  u_{i} = a_{i} ∈ ℂ, i = 0, 1, …,

be a point of E. Consider a covering E × W → E given by vector fields

\begin{aligned}A & = ∑_{j=1}^{m} a^{j}(w^{1}, …, w^{m}, u, …, u_{k})\frac{∂}{∂w^{j}} , \\ B & = ∑_{j=1}^{m} b^{j}(w^{1}, …, w^{m}, u, …, u_{k})\frac{∂}{∂w^{j}} , \end{aligned}

D_{x}B − D_{t}A + [A, B] = 0. (90)

Below we sometimes omit the dependence on the coordinates w^{i} in vector fields on E × W.

Remark 11. Below in this section we say that locally there is a gauge transformation with certain properties if for any w ∈ W a gauge transformation with these properties exists on a neighborhood of the point (a, w) ∈ E × W, where a is the fixed point (89) of E.

Lemma 5. We have

(91)  \frac{∂A}{∂u_{s}} = 0 ∀ s > k − p + 1.

Moreover, locally there is a gauge transformation

w^{i} ↦ f^{i}(w^{1}, …, w^{m}, u, …, u_{k−p}), i = 1, …, m,

such that the transformed vector field D_{x} + A satisfies for all s ≥ 1

(92)  \frac{∂A}{∂u_{s}} (u, …, u_{s−1}, a_{s}, a_{s+1}, …, a_{k}) = 0 ∀ u, …, u_{s−1}.

Proof. Differentiating equation (90) with respect to u_{s} for s > k and using the form (43), (44) of D_{x} and D_{t}, one immediately obtains (91).

Now suppose that (92) holds for all s > n, where 0 < n ≤ k − p + 1. It easily seen that this property is preserved by any gauge transformation of the form

(93)  w^{i} ↦ f^{i}(w^{1}, …, w^{m}, u, …, u_{n−1})

By induction on k − n, it remains to find a gauge transformation (93) such that the transformed vector field D_{x} + A satisfies (92) for s = n. Let

\frac{∂A}{∂u_{n}} (u, …, u_{n−1}, a_{n}, a_{n+1}, …, a_{k}) = ∑_{j} c^{j}(w^{1}, …, w^{m}, u, …, u_{n−1})\frac{∂}{∂w^{j}} .

Similarly to the proof of Theorem 1, consider the system of ordinary differential equations

\frac{d}{du_{n−1}} f^{j}(w^{1}, …, w^{m}, u, …, u_{n−1}) = c^{j}(f^{1}, …, f^{m}, u, …, u_{n−1}),

j = 1, …, m,

dependent on the parameters w^{1}, …, w^{m} and u, …, u_{n−2}. Its local solution with the initial condition

f^{j}(w^{1}, …, w^{m}, u, …, u_{n−2}, a_{n−1}) = w^{j}

determines the required transformation (93). □

Lemma 6. Consider two coverings

D_{x}B_{i} − D_{t}A_{i} + [A_{i}, B_{i}] = 0,

A_{i} = ∑_{j=1}^{m_{i}} a_{i}^{j}(w_{i}^{1}, …, w_{i}^{m_{i}}, u, …, u_{k_{i}})\frac{∂}{∂w_{i}^{j}} ,

B_{i} = ∑_{j=1}^{m_{i}} b_{i}^{j}(w_{i}^{1}, …, w_{i}^{m_{i}}, u, …, u_{k_{i}})\frac{∂}{∂w_{i}^{j}} , i = 1, 2,

such that both A_{1} and A_{2} satisfy (92) for all s ≥ 1 and some point (89). Let

(94)  w_{2}^{j} = φ^{j}(w_{1}^{1}, …, w_{1}^{m_{1}}, u, u_{1}, …), j = 1, …, m_{2},

determine a morphism of these coverings, i.e.,

(D_{x} + A_{1})φ^{j} = a_{2}^{j}(φ^{1}, …, φ^{m_{2}}, u, …, u_{k_{2}}), (95)

(D_{t} + B_{1})φ^{j} = b_{2}^{j}(φ^{1}, …, φ^{m_{2}}, u, …, u_{k_{2}})

for all j = 1, …, m. Then functions (94) do not actually depend on any u_{i}, i ≥ 0.

Proof. Let r ≥ 0 be the maximal integer such that at least one of functions (94) depends nontrivially on u_{r}. Differentiate (95) with respect to u_{r+1} and substitute u_{i} = a_{i} for i ≥ r + 1. Taking into account (92) for s = r + 1, we obtain that the right-hand side is zero, while on the left-hand side we get ∂φ^{j}/∂u_{r}. Therefore, ∂φ^{j}/∂u_{r} = 0 for all j, which contradicts to our assumption. □

Lemma 7. Consider covering (90) satisfying (92) for all s ≥ 1 and let

S = ∑_{j=1}^{m} s^{j}(w^{1}, …, w^{m}, u, u_{1}, …)\frac{∂}{∂w^{j}}

be a symmetry of it. Then S does not actually depend on any u_{i}, i ≥ 0.

Proof. Analyzing the equation [D_{x} + A, S] = 0 from (19), this is proved similarly to the previous lemma. □

Lemma 8. Consider covering (90) satisfying (92) for all s ≥ 1. Let E′ be a subequation of E × W. Then locally E′ is of the form E_{1} × W′, where E_{1} is an open subset of E and W′ is a submanifold of W such that all vector fields

{A(u, …, u_{k}), B(u, …, u_{k}) ∈ D(W) | u, …, u_{k} ∈ ℂ}

are tangent to W′.

Proof. According to Definition 3, a subequation of codimension l is given by functions

f_{i}(w^{1}, …, w^{m}, u, u_{1}, …), i = 1, …, l,

defined on an open subset U ⊂ E × W such that • f_{1}(c) = ⋯ = f_{l}(c) = 0 for some c ∈ U,

• the differentials

d_{b}f_{i} ∈ T_{b}^{∗}(E × W), i = 1, …, l,

are linearly independent for each b ∈ U,

• the ideal I of functions on U generated by f_{1}, …, f_{l} is preserved by the action

of the vector fields D_{x} + A, D_{t} + B. Let z ∈ W be the image of c under the projection E × W → W. To prove the lemma, it is sufficient to find a set of functions

(96)  g_{α}(w^{1}, …, w^{m}), α ∈ Λ,

defined on a neighborhood of z such that the ideal of functions on a neighborhood U′ ⊂ U of c generated by functions (96) coincides with I|_{U′}.

Let r be the maximal integer such that at least one of the functions f_{1}, …, f_{l} depends nontrivially on u_{r}. Note that f_{i} are defined on an open subset V of ℂ^{r+1} × W with the coordinates u_{0}, …, u_{r}, w^{1}, …, w^{m}, the subset

M = {q ∈ V | f_{1}(q) = ⋯ = f_{l}(q) = 0}

is a submanifold of codimension l in V, and I|_{V} coincides with the ideal of functions on V that vanish on M. Thus we essentially have a question of finite-dimensional complex analysis.

Since

\frac{∂}{∂u_{r+1}} (I) ⊂ I, (D_{x} + A)(I) ⊂ I,

we have

(97)  \frac{∂}{∂u_{r+1}} ((D_{x} + A)(f_{i}))= \frac{∂f_{i}}{∂u_{r}} + \frac{∂A}{∂u_{r+1}} (f_{i}) ∈ I.

Substituting u_{i} = a_{i}, i ≥ r + 1, to (97), from (92) for s = r + 1 we obtain ∂f_{i}/∂u_{r} ∈ I. Therefore, the vector field ∂/∂u_{r} is tangent to M, which allows to generate I on some neighborhood of c by functions that do not depend on u_{i} for i ≥ r. Proceeding by induction on r, one completes the proof. □

Applying this lemma to the identity covering E → E, we obtain the following.

Theorem 15. Any connected open subset of the translation-invariant infinite prolongation of any evolution equation (88) is irreducible.

Let us introduce some auxiliary notions.

Definition 7. For each i ∈ ℤ_{+}, let V_{i} be a connected open subset of ℂ such that for all but a finite number of i we have V_{i} = ℂ. Set

D = {(u_{0}, u_{1}, …, u_{i}, …) | u_{i} ∈ V_{i}}.

Let P be an algebra of functions on D such that each f ∈ P is a complex-analytic function dependent on a finite number of the variables u_{i}, i ≥ 0. The algebra P is said to be perfect if for each function f(u_{0}, …, u_{r}) ∈ P and any i ∈ ℤ_{+} the following conditions hold.

(1) One has ∂f/∂u_{i} ∈ P.

(2) There is g(u_{0}, …, u_{r}) ∈ P such that ∂g/∂u_{i} = f.

(3) For any s < r and any fixed numbers a_{i} ∈ V_{i}, i ≥ s, we have

f(u_{0}, …, u_{s−1}, a_{s}, a_{s+1}, …, a_{r}) ∈ P.

(4) For all j ≥ 1 we have u_{j} ∈ P. Then each function f ∈ P is also called perfect.

Example 11. Let V_{i} = ℂ and P be the algebra of polynomials in u_{i}, i ≥ 0. Evidently, the algebra P is perfect.

Fix open subsets V_{i} ⊂ ℂ satisfying the assumptions of Definition 7 and a perfect algebra P.

Definition 8. Consider a vector field

(98)  A = ∑_{j=1}^{m} a^{j}(w^{1}, …, w^{m}, u, u_{1}, …, u_{k})\frac{∂}{∂w^{j}}

defined on an open subset of W × V_{0} × ⋯ × V_{k}.

A vector field

(99)  S = ∑_{j=1}^{m} s^{j}(w^{1}, …, w^{m})\frac{∂}{∂w^{j}}

is said to be 1-primitive (with respect to A) if [S, A] = 0. Now by induction on q ∈ ℕ a vector field (99) is called q-primitive (with respect to A and P) if the commutator [S, A] can be presented as a sum ∑_{j=1}^{N} f_{j}S_{j}, where S_{j} are (q − 1)-primitive fields and f_{j} are perfect functions. In particular, one has (ad^{q}A)(S) = 0.

A vector field

(100)  S = ∑_{j=1}^{m} s^{j}(w^{1}, …, w^{m}, u, u_{1}, …)\frac{∂}{∂w^{j}}

is said to be primitive (without any prefix) if one has S = ∑_{j=1}^{N} f_{j}S_{j}, where f_{j} are perfect functions and S_{j} are q-primitive vector fields for some q.

Remark 12. Below all primitive vector fields are primitive with respect to A and P, where P is a fixed perfect algebra and A arises from a covering (90).

Evidently, primitive vector fields form a module over the algebra P.

Lemma 9. Consider an arbitrary vector field (100) defined on a neighborhood of the point u_{i} = a_{i} ∈ V_{i}, i ≥ 0. Consider a covering (90) satisfying (92) for all s ≥ 1.

(1) If

(101)  \frac{∂}{∂u_{i}} (D_{x}(S) + [A, S])= 0 ∀ i > 0

then ∂S/∂u_{i} = 0 for all i ≥ 0.

(2) If D_{x}(S) + [A, S] is primitive then S is primitive.

(3) If S is primitive and the function F in (88) is perfect then [B, S] is primitive.

Proof. (1) Let r be the maximal integer such that ∂S/∂u_{r} ≠ 0. From (92) for s = r + 1 we have

(102)  \frac{∂}{∂u_{r+1}} (D_{x}(S) + [A, S])(u, …, u_{r}, a_{r+1}, …, a_{k}) = \frac{∂S}{∂u_{r}} .

Combining this with (101) for i = r + 1, we obtain ∂S/∂u_{r} = 0.

(2) Again let r be the maximal integer such that ∂S/∂u_{r} ≠ 0. Then (102) holds. Since D_{x}(S) + [A, S] is primitive, vector field (102) is also primitive, by the properties of perfect functions. Therefore, by Condition 2 of Definition 7, there is a primitive field S′ such that S̃ = S −S′ does not depend on u_{i}, i ≥ r. Then D_{x}(S̃)+[A, S̃] is primitive, and by induction on r one completes the proof.

(3) Applying ad S to (90), we obtain

(103)  D_{x}([S, B])+[A, [S, B]] = [S, D_{t}A] − [[S, A], B].

By assumption, for some q one has

(104)  S = ∑_{j} f_{j}S_{j}, f_{j} are perfect, S_{j} are q-primitive.

Let us prove that [S, B] is primitive by induction on q. For q = 1 the right-hand side of (103) is zero. Applying Part 1 of this lemma to the vector field [S, B], we obtain that [S, B] is 1-primitive.

Now assume that the statement holds for q − 1. Consider an arbitrary vector field S satisfying (104). Let us prove that [S, B] is primitive.

By formula (44), we have

(105)  [S, D_{t}A] = ∑_{j=0}^{k} D_{x}^{j}(F)[S, \frac{∂}{∂u_{j}} A].

Since F is perfect, the functions D_{x}^{j}(F) are also perfect. Besides, for any primitive X the vector fields [X, ∂A/∂u_{j}] are also primitive for all j. Therefore, (105) is primitive.

Since [S, A] is a linear combination of (q−1)-primitive fields, the vector field [[S, A], B] is also primitive by the induction assumption. Thus the right-hand side of (103) is primitive and we can apply Part 2 of this lemma to [S, B]. □

4. Coverings of the KdV equation

In this section we return to the KdV equation

(106)  u_{t} = u_{3} + u_{1}u.

Our final goal here is Theorem 17.

4.1. The canonical form of coverings.

Theorem 16. For any covering of equation (106)

D_{x}B − D_{t}A + [A, B] = 0, (107)

A = A(u, u_{1}, …, u_{k}), B = B(u, u_{1}, …, u_{k})

(we omit the dependence on fibre coordinates w^{j}) locally there is an equivalent covering such that A, B are polynomial in u_{i} and A satisfies (92) for all s ≥ 1 and a_{i} = 0, i ≥ 1.

Proof. Consider an arbitrary point u_{i} = a_{i} ∈ ℂ, w^{j} = w_{0}^{j} ∈ ℂ where the vector fields A and B are defined. All local gauge transformations in this proof will be defined on a neighborhood of this point. By Lemma 5, we can assume that (92) holds for all s ≥ 1.

Remark 13. It would be most convenient to take a_{i} = 0 from the beginning. However, since we consider coverings over arbitrary open subsets of the translation-invariant infinite prolongation of (106), we do not know in advance whether A, B are defined around this point. We will show by induction that after a suitable gauge transformation the vector fields A, B become polynomial in u_{i} and, therefore, are uniquely extended to the whole space of variables u, …, u_{k}.

To clarify further arguments, let us first determine the form of A, B with respect to the highest derivatives u_{i}, i ≥ k −3. A straightforward analysis of equation (107) shows that A does not depend on u_{k}, u_{k−1} and is a polynomial of degree 2 in u_{k−2}, while B is polynomial in u_{k}, u_{k−1}, u_{k−2}. Therefore, following the strategy of Remark 13, we can find a gauge transformation such that the transformed A satisfies (92) with a_{i} = 0 for i ≥ k − 2.

Then (92) for s = k − 2 implies

(108)  A = u_{k−2}^{2}A_{2}(u, …, u_{k−3}) + A_{0}(u, …, u_{k−3}).

Further analysis shows that A_{2} does not depend on u_{k−3} and B is of the form

(109) B = 2u_{k−2}u_{k}A_{2} − u_{k−1}^{2}A_{2} + B_{11}(u, …, u_{k−3})u_{k−2}u_{k−1}+ + B_{10}(u, …, u_{k−3})u_{k−1} + B_{0}(u, …, u_{k−2}).

Differentiating (107) with respect to u_{k}, u_{k−2}, we obtain

2D_{x}(A_{2}) + B_{11} + 2[A_{0}, A_{2}] = 0,

while differentiation with respect to u_{k−1}, u_{k−1} implies

−D_{x}(A_{2}) + B_{11} − [A_{0}, A_{2}] = 0.

Therefore,

(110)  D_{x}(A_{2}) + [A, A_{2}] = 0,

which by Lemma 9 (1) says that A_{2} does not depend on u_{i}, i ≥ 0, and [A, A_{2}] = 0. That is, A_{2} is 1-primitive with respect to A.

Definition 9. Let r ∈ ℤ_{+} and r < k. A vector field

A = ∑_{j=1}^{m} a^{j}(w^{1}, …, w^{m}, u, u_{1}, …, u_{k})\frac{∂}{∂w^{j}}

is said to be r-simple if it satisfies (92) for all s ≥ 1 with a_{i} = 0, i ≥ k − r, and some a_{1}, …, a_{k−r−1} ∈ ℂ.

Lemma 10. (1) For each r < k and any covering (107) there is a locally gauge

equivalent covering with r-simple A.

(2) If a covering (107) has r-simple A then the vector fields

(111)  \begin{aligned}A′ & = A(u, …, u_{k}) − A(u, …, u_{k−r−1}, 0, …, 0), \\ B′ & = B(u, …, u_{k}) − B(u, …, u_{k−r+1}, 0, …, 0) \end{aligned}

are primitive with respect to A and P, where P is the perfect algebra constructed in Example 11.

Proof. For r = 2 we proved these statements above. Suppose that the statements of hold for some r = l ≤ k − 2 and let us prove them for r = l + 1.

By assumption, each covering is locally equivalent to a covering (107) with l-simple A. Then by Part 2 of the lemma we have

(112)  \begin{aligned}A & = A′(u, …, u_{k−2}) + Ã(u, …, u_{k−l−1}), \\ B & = B′(u, …, u_{k}) + B̃(u, …, u_{k−l+1}), \end{aligned}

where

(113)  Ã = A(u, …, u_{k−l−1}, 0, …, 0), B̃ = B(u, …, u_{k−l+1}, 0, …, 0),

and the primitive vector fields A′, B′ are given by (111) for r = l.

We can rewrite (107) as follows

(114)  D_{x}B̃ − D_{t}Ã + [Ã, B̃] + P = 0,

where

(115)  P = D_{x}B′ − D_{t}A′ + [A, B′] + [A′, B]

is primitive. Indeed, the fact that D_{x}B′, D_{t}A′, [A, B′] are primitive follows immediately from the fact that A′, B′ are primitive, while [A′, B] is primitive by Lemma 9 (3). In particular, P is polynomial in u_{i}, i ≥ 0.

From equation (114) it follows easily that Ã, B̃ are polynomial in u_{k−l−1}, u_{k−l}, u_{k−l+1}. Therefore, A(u, …, u_{k}) and B(u, …, u_{k}) are defined for u_{i} = a_{i}, i ≤ k − l − 2, and arbitrary values of u_{j}, j ≥ k − l − 1. By Lemma 5, after some gauge transformation

w^{i} ↦ g^{i}(w^{1}, …, w^{m}, u, …, u_{k−l−2})

A becomes (l + 1)-simple, which proves Part 1 of the lemma for r = l + 1.

To prove Part 2, consider an arbitrary covering (107) with (l + 1)-simple A, where l ≤ k − 2. Since (l + 1)-simple A is also l-simple, we again have representation (112) and equation (114), where (115) is primitive.

Similarly to formulas (108) and (109), from (114) we obtain

Ã = P_{1} + u_{k−l−1}^{2}A′_{2}(u, …, u_{k−l−3}) + A′_{0}(u, …, u_{k−l−2}), (116)

B̃ = P_{2} + 2u_{k−l−1}u_{k−l+1}A′_{2} − u_{k−l}^{2}A′_{2} + B′_{11}(u, …, u_{k−l−2})u_{k−l−1}u_{k−l}+ (117)

+B′_{10}(u, …, u_{k−l−2})u_{k−l} + B′_{0}(u, …, u_{k−l−1}),

where P_{1}, P_{2} are primitive. Similarly to (110), this implies that D_{x}A′_{2} + [A, A′_{2}] is also primitive. By Lemma 9 (2), the vector field A′_{2} is primitive.

Then the vector fields

A(u, …, u_{k}) − A(u, …, u_{k−l−2}, 0, …, 0),
B(u, …, u_{k}) − B(u, …, u_{k−l}, 0, …, 0)

are also primitive, which proves Part 2 of the lemma for r = l + 1. □

By the above lemma for r = k−1, we obtain that after a suitable gauge transformation one has

A = A′′(u, …, u_{k−2}) + A′′_{0}(u),
B = B′′(u, …, u_{k}) + B′′_{0}(u, u_{1}, u_{2}),

where A′′, B′′ are primitive and A is (k − 1)-simple. Now it is straightforward to prove that A′′_{0}, B′′_{0} are polynomial in u, u_{1}, u_{2}. □

4.2. The fundamental algebras. From the above proof it follows that for each k ≥ 3 there are finite subsets

M_{k} ⊂ ℤ_{+}^{k−1}, N_{k} ∈ ℤ_{+}^{k+1}

such that the following statement holds. If a covering (107) of equation (106) satisfies (92) for all s ≥ 1 with a_{i} = 0, i ≥ 1, then it is of the form

(118)  \begin{aligned}A & = ∑_{(i_{0},...,i_{k−2})∈M_{k}} u_{0}^{i_{0}} … u_{k−2}^{i_{k−2}}A_{i_{0}...i_{k−2}}, \\ B & = ∑_{(i_{0},...,i_{k})∈N_{k}} u_{0}^{i_{0}} … u_{k}^{i_{k}}B_{i_{0}...i_{k}}, \end{aligned}

where the vector fields

(119)  A_{i_{0}...i_{k−2}}, B_{i_{0}...i_{k}}

do not depend on u_{i}, i ≥ 0.

Let us show that this canonical form of coverings satisfies Definition 5 if we take

A_{k} = {u_{0}^{i_{0}} … u_{k−2}^{i_{k−2}} | (i_{0}, …, i_{k−2}) ∈ M_{k}},
B_{k} = {u_{0}^{i_{0}} … u_{k}^{i_{k}} | (i_{0}, …, i_{k}) ∈ N_{k}}.

Relation (64) is obvious. Condition 1 of Definition 5 follows from Theorem 16. Let

S = ∑_{j=1}^{m} s^{j}(w^{1}, …, w^{m}, u, u_{1}, …)\frac{∂}{∂w^{j}}

be a symmetry of the covering given by vector fields (118), i.e.,

(120)  [D_{x} + A, S] = [D_{t} + B, S] = 0.

By Lemma 7, S does not depend on u_{i}, i ≥ 0. Then (120) implies that S commutes with all vector fields (119), which proves Condition 3 of Definition 5. Conditions 2 and 4 follow analogously from Lemmas 6 and 8 respectively.

Vector fields (118) satisfy (107) if and only if certain Lie algebra relations hold for (119). Denote by 𝔣_{k−2} the quotient of the free Lie algebra generated by letters (119) over these relations. We obtain the system of fundamental algebras

(121)  ⋯ → 𝔣_{k+1} → 𝔣_{k} → ⋯ → 𝔣_{1} → 𝔣_{0}

for equation (106). In particular, the algebras 𝔣_{1} and 𝔣_{0} are described in Example 9. Denote by 𝔞_{k−2} the subalgebra of 𝔣_{k−2} generated by A_{i_{0},...,i_{k−2}}.

Lemma 11. We have

(122)  B_{i_{0}...i_{k}} ∈ 𝔞_{k−2} for i_{0} + ⋯ + i_{k} > 0.

Proof. For (i_{0}, …, i_{k}) ∈ N_{k} set

r(i_{0}, …, i_{k}) = max{s | i_{s} > 0}.

Let us prove (122) by induction on k − r(i_{0}, …, i_{k}).

For (i_{0}, …, i_{k}) ∈ N_{k} with r(i_{0}, …, i_{k}) = k it follows from (109).

Suppose that (122) holds for all (i_{0}, …, i_{k}) ∈ N_{k} with r(i_{0}, …, i_{k}) ≥ l + 1. Differentiate (107) with respect to u_{l+1} and substitute u_{i} = 0 for i ≥ l +1. Since A satisfies (92) for s = l + 1 and a_{i} = 0, we obtain (122) for (i_{0}, …, i_{k}) ∈ N_{k} with r(i_{0}, …, i_{k}) = l. □

Combining (122) and (107), one gets

(123)  [B_{0...0}, 𝔞_{k−2}] ⊂ 𝔞_{k−2}.

Substituting u_{i} = 0 to (107), we obtain also

(124)  [A_{0...0}, B_{0...0}] = 0.

Let us specify the structure of (118). For k = 3 it was described in Theorem 1. Similarly to the proof of Theorem 16, one obtains that for k ≥ 4 vector fields (118) have the form

A = A_{k−2}(u_{k−2}^{2} − \frac{2k − 3}{3}uu_{k−2}^{2})+A′_{k−2}u_{k−3}^{2} + A_{0}(u, …, u_{k−4}), (125)

B = 2u_{k−2}u_{k}A_{k−2} + B_{0}(u, …, u_{k−1}), (126)

where

A_{k−2} = A_{0...02}, A′_{k−2} = A_{0...020},

[A_{k−2}, A] = 0, (127) [A_{k−2}, B] = 3[A_{0}, A′_{k−2}]. (128)

and A_{0}, B_{0} are polynomial in u_{i}.

Equation (127) implies

(129)  [A_{k−2}, 𝔞_{k−2}] = 0.

Combining this with (122) and (128), we obtain

(130)  [A_{k−2}, B_{0...0}] = 3[A_{0...0}, A′_{k−2}].

Moreover, taking into account (124) and applying ad^{s}B_{0...0} to (130), we obtain

(131)  −(ad^{s+1}B_{0...0})(A_{k−2}) = 3[A_{0...0}, (ad^{s}B_{0...0})(A′_{k−2})] ∀ s ≥ 0.

By the definition of 𝔣_{n} and formulas (125), (126), for each n ≥ 2 the algebra 𝔣_{n−1} is isomorphic to the quotient of 𝔣_{n} over the ideal 𝔦_{n} generated by A_{n}. From (122), (123), and (131) we obtain that 𝔦_{n} ⊂ 𝔞_{n}. Moreover, (129) implies

(132)  [𝔦_{n}, 𝔞_{n}] = 0.

Lemma 12. For each n ≥ 1 we have the relation

(133)  −(ad^{n}B_{0...0})(A_{n}) = 0

in the algebra 𝔣_{n}.

Proof. For n = 1 this statement follows from (59). By induction on n, suppose that (133) holds for n−1. By formula (125), the generator A′_{n} ∈ 𝔣_{n} is mapped to A_{n−1} ∈ 𝔣_{n−1} by the natural epimorphism 𝔣_{n} → 𝔣_{n}/𝔦_{n} ≅ 𝔣_{n−1}. Therefore, (ad^{n−1}B_{0...0})(A′_{n}) ∈ 𝔦_{n}. Combining this with (131) and (132), we obtain (133). □

From the above results it follows that the elements

c_{i} = (ad^{i}B_{0...0})(A_{n}), i = 0, …, n − 1,

span the ideal 𝔦_{n}. The element c_{n−1} belongs to the center of 𝔣_{n}. Moreover, for each i = 0, …, n − 1 the image of c_{i} belongs to the center of the quotient

𝔣_{n}/⟨c_{i+1}, …, c_{n−1}⟩.

Thus we have the following statement.

Lemma 13. For each n ≥ 2 the algebra 𝔣_{n} is obtained from 𝔣_{n−1} applying the operation of one-dimensional central extension no more than n times.

Let us now prove the main result of this section.

Theorem 17. The KdV equation (106) possesses fundamental algebras (121). Each algebra 𝔣_{k} is quasi-finite and is obtained from the algebra 𝔰𝔩_{2}(ℂ)⊗_{ℂ}ℂ[λ] applying several times the operation of one-dimensional central extension.

Proof. It was shown above that (121) are fundamental algebras of (106). Let us prove that algebras (121) are quasi-finite.

By Theorem 9, the algebra

𝔤 = 𝔰𝔩_{2}(ℂ) ⊗_{ℂ} ℂ[λ]

is quasi-finite. From (59) it follows that 𝔣_{1} is the trivial central extension of the algebra 𝔏 from Proposition 2.

Since the Heisenberg algebra H is nilpotent, the algebra 𝔣_{1} is obtained from 𝔤 applying 6 times the operation of one-dimensional central extension. Therefore, by Theorem 8, the algebra 𝔣_{1} is also quasi-finite. Finally, combining Lemma 13 and Theorem 8, we obtain that all fundamental algebras (121) are quasi-finite. □

It is well known that 𝔰𝔩_{2}(ℂ) ⊗_{ℂ} ℂ[λ] has no nontrivial central extensions. Combining this with Theorem 17, we obtain the following specification of the structure of 𝔣_{k}.

Theorem 18. Each algebra 𝔣_{k} is isomorphic to the direct sum of 𝔰𝔩_{2}(ℂ) ⊗ ℂ[λ] and a finite-dimensional nilpotent algebra.

5. Coverings of the Krichever-Novikov equation

Consider the Krichever-Novikov (KN) equation [11, 17, 18]

(134)  u_{t} = u_{3} − \frac{3}{2} \frac{u_{2}^{2}}{u_{1}} + \frac{h(u)}{u_{1}}, u_{k} = \frac{∂^{k}u}{∂x^{k}},

where h(u) is a polynomial of degree 3 with coefficients in ℂ. If the roots of the polynomial h(u) are distinct then equation (134) is said to be nonsingular.

The main goal of this section is Theorem 22.

5.1. The canonical form of coverings. We want to have an analog of Theorem 16 for equation (134). The straightforward repetition of the proof of Theorem 16 is not possible, because (134) is not polynomial in u_{1}.

To overcome this, we need to introduce a perfect algebra that contains the function 1/u_{1}. By Condition 2 of Definition 7, this algebra must contain also ∫ 1/u_{1}du_{1}.

To this end, choose a half-line L ⊂ ℂ from 0 to ∞ such that V_{1} = ℂ \ L is simply connected. Let ln u_{1} be a single-valued branch of the logarithm defined on V_{1}. Set V_{i} = ℂ, i ≠ 1, and let P be the algebra of polynomials in

(135)  u_{i}, i ≥ 0, \frac{1}{u_{1}} , ln u_{1}.

Then P is a perfect algebra. Indeed, all conditions of Definition 7 are obvious except of Condition 2. The latter follows from the fact that for any a ∈ ℤ, b ∈ ℤ_{+} there is a polynomial g in u_{1}, 1/u_{1}, ln u_{1} such that ∂g/∂u_{1} = u_{1}^{a} ln^{b} u_{1}.

Remark 14. Thus for equation (134) we study not the whole translation-invariant infinite prolongation, but the open dense subset

{(u_{0}, u_{1}, …) | u_{1} ∈ ℂ \ L, u_{i} ∈ ℂ ∀ i ≠ 1}

of it.

In Theorem 16, we proved that every covering of the KdV equation is locally equivalent to a covering in the canonical form satisfying (92) for all s ≥ 1 and a_{i} = 0, i ≥ 1. For equation (134) the point u_{i} = 0 is also crucial. However, one cannot prove the same statement for coverings of (134), because 1/u_{1} and ln u_{1} are not defined at u_{1} = 0. Let us make necessary modifications.

Definition 10. A vector field

(136)  S = ∑_{j=1}^{m} s^{j}(w^{1}, …, w^{m}, u, …, u_{k})\frac{∂}{∂w^{j}}

is said to be u_{1}-free if each function s^{j}(w^{1}, …, w^{m}, u, …, u_{k}) is polynomial in

(137)  u_{i}, i ≥ 1, \frac{1}{u_{1}} , ln u_{1}

with coefficients dependent on u, w^{1}, …, w^{m} and the coefficient at the monomial u_{1} is zero. (This coefficient is well defined because the functions

u_{1}^{a} ln^{b} u_{1}, a ∈ ℤ, b ∈ ℤ_{+},

are linearly independent.)

Definition 11. Let r ∈ ℤ_{+} and r ≤ k − 2. A vector field

A = ∑_{j=1}^{m} a^{j}(w^{1}, …, w^{m}, u, u_{1}, …, u_{k})\frac{∂}{∂w^{j}}

is said to be weakly r-simple if it satisfies (92) for all s ≥ 2 with a_{i} = 0, i ≥ k − r, and some a_{2}, …, a_{k−r−1} ∈ ℂ.

In contrast to r-simple vector fields, a weakly r-simple vector field does not necessarily satisfy (92) for s = 1.

Remark 15. In this section perfect functions are elements of the perfect algebra P defined above.

Lemma 14. If in Lemmas 6, 7, 8, 9 one replaces the condition that A satisfies (92) for all s ≥ 1 by the condition that A is u_{1}-free and weakly (k − 2)-simple then the conclusions of these lemmas remain valid.

Proof. Let us prove that Lemma 9 (1) remains valid, since the other statements are proved analogously.

So assume that A(u, …, u_{k}) is u_{1}-free and weakly (k − 2)-simple and that equation (101) holds. By Definition 11, A satisfies (92) for all s ≥ 2. Therefore, the equations

(138)  \frac{∂S}{∂u_{i}} = 0 ∀ i ≥ 1

are proved in the same way as in Lemma 9 (1).

Let us prove that ∂S/∂u is also equal to zero. From (101) for i = 1 we have

(139)  \frac{∂S(u)}{∂u} + [\frac{∂A}{∂u_{1}} , S(u)] = 0.

Since A is u_{1}-free and (138) holds, the vector field [∂A/∂u_{1}, S(u)] is either zero or depends nontrivially on some u_{i}, i ≥ 1. Combining this with (138) and (139), we obtain

\frac{∂S}{∂u} = [\frac{∂A}{∂u_{1}} , S(u)] = 0.

□

Theorem 19. For any covering of equation (134)

D_{x}B − D_{t}A + [A, B] = 0, (140) A = A(u, u_{1}, …, u_{k}), B = B(u, u_{1}, …, u_{k}) (141)

(we omit the dependence on fibre coordinates w^{j}) locally there is an equivalent covering such that

(1) A, B are polynomial in (135),

(2) A is (k − 2)-simple and u_{1}-free.

Proof. Let (141) be defined on a neighborhood of a point u_{i} = a_{i}.

Lemma 15. (1) For each r ≤ k − 2 and any covering (140) there is a locally gauge

equivalent covering with r-simple A.

(2) If a covering (140) has r-simple A then the vector fields

\begin{aligned}A′ & = A(u, …, u_{k}) − A(u, …, u_{k−r−1}, 0, …, 0), \\ B′ & = B(u, …, u_{k}) − B(u, …, u_{k−r+1}, 0, …, 0) \end{aligned}

are primitive with respect to A and P.

Proof. This is proved similarly to Lemma 10. Formulas (116) and (117) for l ≤ k − 3 are replaced by

Ã = P_{1} + \frac{u_{k−l−1}^{2}}{u_{1}^{2}} A′_{2} + A′_{0}(u, …, u_{k−l−2}),

B̃ = P_{2} + 2\frac{u_{k−l−1}u_{k−l+1}}{u_{1}^{2}} A′_{2} − \frac{u_{k−l}^{2}}{u_{1}^{2}} A′_{2} + B′_{11}(u, …, u_{k−l−2})u_{k−l−1}u_{k−l}+

+B′_{10}(u, …, u_{k−l−2})u_{k−l} + B′_{0}(u, …, u_{k−l−1}),

where P_{1}, P_{2}, A′_{2} are primitive. □

By the above lemma for r = k − 2, after a suitable gauge transformation we have

(142)  \begin{aligned}A & = A′ + A_{0}(u, u_{1}), \\ B & = B′ + B_{0}(u, u_{1}, u_{2}, u_{3}), \end{aligned}

where the vector fields

(143)  \begin{aligned}A′ & = A(u, …, u_{k}) − A(u, u_{1}, 0, …, 0), \\ B′ & = B(u, …, u_{k}) − B(u, u_{2}, u_{3}, 0, …, 0) \end{aligned}

are primitive and A is (k − 2)-simple.

Substituting (142) to (140), it is straightforward to obtain that

(144)  A_{0} = C + \frac{1}{u_{1}} A_{1}(u) + u_{1}A_{2}(u) + A_{3}(u),

where C is primitive.

The vector field A remains weakly (k − 2)-simple and polynomial in (137) after any gauge transformation of the form

(145)  w^{i} ↦ f^{i}(w^{1}, …, w^{m}, u).

Let us find a gauge transformation (145) such that A becomes u_{1}-free. To this end, let

∑_{j=1}^{m} c^{j}(w^{1}, …, w^{m}, u)\frac{∂}{∂w^{j}}

be the coefficient of A at the monomial u_{1} and consider the system of ordinary differential equations

\frac{d}{du}f^{j}(w^{1}, …, w^{m}, u) = c^{j}(f^{1}, …, f^{m}, u), j = 1, …, m,

dependent on the parameters w^{1}, …, w^{m}. Its local solution with the initial condition

f^{j}(w^{1}, …, w^{m}, a_{0}) = w^{j}, j = 1, …, m,

determines gauge transformation (145) that makes A u_{1}-free.

By Lemma 14, in Lemma 15(2) for r = k − 2 the condition that A is (k − 2)- simple can be replaced by the condition that A is weakly (k − 2)-simple and u_{1}-free. Therefore, after this gauge transformation vector fields (143) remain primitive and we have formula (144) with primitive u_{1}-free C and A_{2}(u) = 0.

Now it is straightforward to show that

(146)  A(u, u_{1}, 0, …, 0), B(u, u_{2}, u_{3}, 0, …, 0)

are polynomial in (135). Therefore, A and B satisfy the conditions of the theorem. □

5.2. The fundamental algebras. Consider the following set of perfect functions

Z = {(ln^{a} u_{1})u_{0}^{i_{0}}u_{1}^{i_{1}}u_{2}^{i_{2}} … u_{k}^{i_{k}} | i_{1} ∈ ℤ, a, i_{0}, i_{2}, …, i_{k} ∈ ℤ_{+}, |i_{1}| + a + i_{0} + i_{2} + ⋯ + i_{k} > 0}.

Similarly to the case of the KdV equation, from the proof of Theorem 19 it follows that for each k ≥ 3 there are finite subsets

A′_{k}, B′_{k} ⊂ Z, A′_{k} ⊂ A′_{k+1}, B′_{k} ⊂ B′_{k+1}

such that the following statement holds. If a covering (140) of equation (134) has (k − 2)-simple u_{1}-free A then it is of the form

(147)  A = ∑_{f∈A′_{k}} fA_{f} + A_{1}, B = ∑_{g∈B′_{k}} gB_{g} + B_{1},

where the vector fields

(148)  A_{f}, B_{g}, A_{1}, B_{1}

do not depend on u_{i}, i ≥ 0.

Let us show that the conditions of Definition 5 hold, if we set

A_{k} = A′_{k} ∪ {1}, B_{k} = B′_{k} ∪ {1}.

Indeed, Condition 1 follows from Theorem 19. Conditions 2, 3, 4 hold because, by Lemma 14, Lemmas 6, 7, 8 are applicable to the canonical form of coverings described in Theorem 19.

Vector fields (147) satisfy (140) if and only if certain Lie algebra relations hold for (119). Denote by 𝔣_{k−2}^{KN} the quotient of the free Lie algebra generated by letters (119) over these relations. We obtain the system of fundamental algebras

(149)  ⋯ → 𝔣_{n+1}^{KN} → 𝔣_{n}^{KN} → ⋯ → 𝔣_{1}^{KN} → 𝔣_{0}^{KN}

for equation (134).

Proposition 4 ([17]). For each integer n ≥ 2 there is a conserved current D_{t}f_{n} = D_{x}g_{n} of the form

f_{n} = \frac{u_{n}^{2}}{u_{1}^{2}} + f̃_{n}(u, …, u_{n−1}), g_{n} = 2\frac{u_{n}u_{n+2}}{u_{1}^{2}} + g̃_{n}(u, …, u_{n+1}),

where f̃_{n}, g̃_{n} are polynomials in 1/u_{1}, u_{i}, i ≥ 0.

Similarly to Lemma 5, we can find equivalent conserved currents

f′_{n} = f_{n} + D_{x}(h_{n}(u, …, u_{n−2})), g′_{n} = g_{n} + D_{t}(h_{n}(u, …, u_{n−2}))

such that • the functions f′_{n}, g′_{n} are perfect,

• we have

\frac{∂f′_{n}}{∂u_{s}} (u, …, u_{s−1}, 0, …, 0) = 0 ∀ s ≥ 2,

• f′_{n} is polynomial in (137) with zero coefficient at the monomial u_{1}.

Example 12. We have

f′_{2} = \frac{u_{2}^{2}}{u_{1}^{2}} + \frac{2}{3} \frac{h(u)}{u_{1}^{2}},

g′_{2} = 2\frac{u_{2}u_{4}}{u_{1}^{2}} − \frac{u_{3}^{2}}{u_{1}^{2}} − \frac{4}{3} \frac{h(u)u_{3}}{u_{1}^{3}} − 4\frac{u_{2}^{2}u_{3}}{u_{1}^{3}} +

+\frac{9}{4} \frac{u_{2}^{4}}{u_{1}^{4}} − h(u)\frac{u_{2}^{2}}{u_{1}^{4}} + 2\frac{dh(u)}{du} \frac{u_{2}}{u_{1}^{2}} − \frac{1}{3} \frac{h(u)^{2}}{u_{1}^{4}} .

Return to algebras (149). Let 𝔞_{k} ⊂ 𝔣_{k−2}^{KN} be the subalgebra generated by A_{f}, f ∈ A′_{k}, and 𝔞̃_{k} ⊂ 𝔣_{k−2}^{KN} be the subalgebra generated by 𝔞_{k−2} and A_{1}. Similarly to Lemma 11, we obtain

B_{g} ∈ 𝔞_{k−2} ∀ g ∈ B′_{k}, (150) [B_{1}, 𝔞̃_{k−2}] ⊂ 𝔞_{k−2}. (151)

For k ≥ 5 vector fields (147) can be rewritten as follows

A = f′_{k−2}A^{k−2} + \frac{u_{k−3}^{2}}{u_{1}^{2}} Ã^{k−2} + A_{0}(u, …, u_{k−4}), (152)

B = g′_{k−2}A^{k−2} + B_{0}(u, …, u_{k−1}), (153)

where

A^{k−2} = A_{u_{k−2}^{2}}_{u_{1}^{−2}}, Ã^{k−2} = A_{u_{k−3}^{2}}_{u_{1}^{−2}},

[A^{k−2}, A] = 0, (154)

[A^{k−2}, B] = 3[A_{0}, Ã^{k−2}]. (155)

Equation (154) implies

[A^{k−2}, 𝔞̃_{k−2}] = 0.

Combining this with (150) and (155), we obtain

[A^{k−2}, B_{1}] = 3[A_{1}, Ã^{k−2}], (156)

[Ã^{k−2}, 𝔞_{k−2}] = 0. (157)

Taking into account (157), (151) and applying ad^{s}B_{1} to (156), we obtain

(158)  −(ad^{s+1}B_{1})(A^{k−2}) = 3[A_{1}, (ad^{s}B_{1})(Ã^{k−2})] ∀ s ≥ 0.

Similarly to Section 4.2, the obtained identities imply that for each k ≥ 5 the algebra 𝔣_{k−2}^{KN} is obtained from 𝔣_{k−3}^{KN} applying several times the operation of one-dimensional central extension.

Let us describe the algebras 𝔣_{i}^{KN} for i = 0, 1, 2.

Theorem 20. Any covering of equation (134) of the form

D_{x}B − D_{t}A + [A, B] = 0,
A = A(u, u_{1}, u_{2}, u_{3}, u_{4}), B = B(u, u_{1}, u_{2}, u_{3}, u_{4})

is locally equivalent to a covering of the form

A = f′_{2}C + \frac{1}{u_{1}} A_{1}(u) + V_{1},

B = g′_{2}C − \frac{u_{3}}{u_{1}^{2}} A_{1} + \frac{u_{2}^{2}}{2u_{1}^{3}} A_{1} + \frac{2u_{2}}{u_{1}} \frac{∂A_{1}}{∂u} − \frac{h(u)}{3u_{1}^{3}}A_{1}+

+\frac{2}{u_{1}} [A_{1}, \frac{∂A_{1}}{∂u}] − 2u_{1} \frac{∂^{2}A_{1}}{∂u^{2}} + V_{2},

where A_{1} = A_{10} + uA_{11} + u^{2}A_{12}, the vector fields C, V_{i}, A_{1k} do not depend on u_{i}, i ≥ 0, and are subject to the following relations

[C, V_{i}] = [C, A_{1k}] = [V_{1}, V_{2}] = [V_{i}, A_{1k}] = 0 i = 1, 2, k = 0, 1, 2, (159)

2h(u)\frac{∂A_{1}}{∂u} − \frac{dh(u)}{du}A_{1} − 3[A_{1}, [A_{1}, \frac{∂A_{1}}{∂u}]] = 0. (160)

Proof. This is proved by a straightforward computation following the scheme of the proof of Theorem 19. Relation (160) was obtained in [6]. □

Equation (160) determines some relations between the vector fields A_{1k}, k = 0, 1, 2. Let us describe the quotient of the free Lie algebra generated by A_{1k} over these relations.

Consider the ideal I ⊂ ℂ[v_{1}, v_{2}, v_{3}] generated by the polynomials

(161)  v_{i}^{2} − v_{j}^{2} + \frac{8}{3}(e_{j} − e_{i}), i, j = 1, 2, 3,

where e_{1}, e_{2}, e_{3} are the roots of the polynomial h(u). Set

E = ℂ[v_{1}, v_{2}, v_{3}]/I.

That is, E is the ring of regular functions on the affine elliptic curve in ℂ^{3} defined by polynomials (161). The image of v_{j} ∈ ℂ[v_{1}, v_{2}, v_{3}] in E is denoted by v̄_{j}. Consider also a basis x_{1}, x_{2}, x_{3} of the Lie algebra 𝔰𝔩_{2}(ℂ) ≅ 𝔰𝔬_{3}(ℂ) with the relations

(162)  [x_{1}, x_{2}] = x_{3}, [x_{2}, x_{3}] = x_{1}, [x_{3}, x_{1}] = x_{2}

and endow the space L = 𝔰𝔩_{2} ⊗_{ℂ} E with the Lie algebra structure described in (76).

Proposition 5 ([6]). Suppose that the roots e_{1}, e_{2}, e_{3} of h(u) are distinct. The quotient of the free Lie algebra generated by A_{1k}, k = 0, 1, 2, over relations (160) is isomorphic to the subalgebra ℜ ⊂ L generated by the elements

x_{1} ⊗ v̄_{1}, x_{2} ⊗ v̄_{2}, x_{3} ⊗ v̄_{3} ∈ L.

From (159) we obtain

𝔣_{0}^{KN} = 0, 𝔣_{1}^{KN} ≅ ℜ ⊕ ℂ^{2}, 𝔣_{2}^{KN} ≅ ℜ ⊕ ℂ^{3}.

Theorem 21. The algebra ℜ is quasi-finite.

Proof. Below we assume everywhere that {j, k, l} = {1, 2, 3}. For each j = 1, 2, 3 consider the subspace V_{j} ⊂ ℂ[v_{1}, v_{2}, v_{3}] spanned by the monomials v_{j}^{d_{j}}v_{k}^{d_{k}}v_{l}^{d_{l}} satisfying

(163)  d_{j} ≡ d_{k} + 1 ≡ d_{l} + 1 mod 2.

Denote by R_{j} the image of V_{j} in the quotient space E.

The algebra ℜ was also studied in [14] in connection with coverings of the Landau-Lifshitz equation. In the proof of Lemma 3.1 of [14] it is shown that ℜ = ⊕_{j=1}^{3}⟨x_{j}⟩⊗R_{j}.

Let 𝔥 ⊂ ℜ be a subalgebra of finite codimension. Then the subspace H_{j} = {f ∈ R_{j} | x_{j} ⊗ f ∈ 𝔥} is of finite codimension in R_{j} for each j = 1, 2, 3. In addition, from the definition of R_{j} and relations (162) we have

(164)  R_{j}R_{k} ⊂ R_{l}, H_{j}H_{k} ⊂ H_{l}.

This implies that for all j = 1, 2, 3 the subspace

(165)  H′_{j} = {a ∈ H_{j} | aR_{k} ⊂ H_{l}, aR_{l} ⊂ H_{k}}

is also of finite codimension in R_{j}. From (164) and (165) one gets

(166)  H′_{j}H′_{k} ⊂ H′_{l}, H′_{j}R_{j}R_{k}R_{l} ⊂ H′_{j}.

It is easy to see that R_{j} = ⟨v̄_{j}⟩ + R_{k}R_{l}. Therefore,

(167)  R_{j}^{2} = ⟨v̄_{j}^{2}⟩ + R_{j}R_{k}R_{l}.

For each j = 1, 2, 3 the subspace

(168)  H′′_{j} = {a ∈ H′_{j} | av̄_{k}^{2} ⊂ H′_{j}, av̄_{l}^{2} ⊂ H′_{j}, aR_{k} ⊂ H′_{l}, aR_{l} ⊂ H′_{k}}

is of finite codimension in H′_{j} and, therefore, in R_{j}. By definitions (168), (165) and properties (164), (166), (167), one gets

R_{k}H′′_{j} ⊂ H′′_{l}, R_{l}H′′_{j} ⊂ H′′_{k},

which implies that ⊕_{j=1}^{3}⟨x_{j}⟩⊗H′′_{j} ⊂ 𝔥 is an ideal of ℜ. Since H′′_{j} is of finite codimension in R_{j}, this ideal is of finite codimension in ℜ. □

Collecting the results of this subsection and taking into account Theorems 21 and 8, one obtains the following.

Theorem 22. The nonsingular Krichever-Novikov equation (134) possesses fundamental algebras (149), where 𝔣_{0}^{KN} = 0. Each 𝔣_{n}^{KN} for n > 0 is quasi-finite and is obtained from ℜ applying several times the operation of one-dimensional central extension.

6. Coverings of the equation u_{t} = u_{xxx}

In this section we study the linear equation

(169)  u_{t} = u_{xxx}.

The following theorem is proved by a straightforward computation.

Theorem 23. Any Wahlquist-Estabrook covering

D_{x}B − D_{t}A + [A, B] = 0,
A = A(u, u_{1}, u_{2}), B = B(u, u_{1}, u_{2})

of equation (169) is of the form

A = u^{2}A_{2} + uA_{1} + A_{0},

B = u_{2}(2uA_{2} + A_{1}) − u_{1}^{2}A_{2} + u_{1}[A_{1}, A_{0}] − \frac{1}{2}u^{2}[A_{1}, [A_{1}, A_{0}]]+

+u[A_{0}, [A_{0}, A_{1}]] + B_{0},

where the vector fields A_{i}, B_{0} depend only on w^{1}, …, w^{m} and are subject to the relations

[A_{0}, A_{2}] = [A_{1}, A_{2}] = 0, (170) [A_{0}, B_{0}] = 0, (171) [A_{1}, [A_{1}, [A_{1}, A_{0}]]] = 0, (172)

[A_{2}, B_{0}] = \frac{3}{2}[A_{0}, [A_{1}, [A_{1}, A_{0}]]], (173)

[B_{0}, A_{1}] = [A_{0}, [A_{0}, [A_{0}, A_{1}]]]. (174)

Denote by 𝔑 the quotient of the free Lie algebra generated by A_{i}, B_{0} over relations (170), (171), (172), (173), (174). Similarly to Section 4.1 one proves the following.

Theorem 24. Equation (169) possesses a system of fundamental algebras, which are obtained from 𝔑 applying several times the operation of one-dimensional central extension.

Let us present some information on the structure of 𝔑.

Theorem 25. There are ideals 𝔑_{i}, i ∈ ℤ_{+}, of 𝔑 such that • 𝔑_{0} = 0, 𝔑_{i} ⊂ 𝔑_{i+1} ∀ i ∈ ℤ_{+},

• the quotient 𝔑_{i+1}/𝔑_{i} is commutative for all i ∈ ℤ_{+},
• the quotient 𝔑/ ∪_{i} 𝔑_{i} is solvable.

Proof. For a subset S of a Lie algebra we denote by ⟨S⟩ the ideal generated by this subset. For simplicity, below the images of A_{i}, B_{0} ∈ 𝔑 in quotients of 𝔑 are denoted by the same symbols A_{i}, B_{0}. From the relations that define the algebra 𝔑 one easily obtains the following.

Lemma 16. Let Q be a quotient algebra of 𝔑 and C be an element of the subalgebra of Q generated by A_{i}. If [A_{0}, C] = [A_{1}, C] = 0 then the ideal ⟨C⟩ ⊂ Q is spanned by the elements (ad^{k}B_{0})(C), k ∈ ℤ_{+}, and is commutative.

Let us construct the required ideals 𝔑_{i}. Set 𝔑_{0} = 0 and 𝔑_{1} = ⟨A_{2}⟩. Combining relation (170) with the above lemma, we see that 𝔑_{1}/𝔑_{0} = 𝔑_{1} is commutative. By induction on i ∈ ℕ, set

(175)  𝔑_{i+1} = ⟨𝔑_{i}, [A_{1}, (ad^{2i−1}A_{0})(A_{1})]⟩.

Lemma 17. For all i ≥ 1 in the quotient algebra 𝔑/𝔑_{i} we have

[(ad^{k}A_{0})(A_{1}), (ad^{l}A_{0})(A_{1})] = 0 ∀ k, l ∈ ℤ_{+} k + l ≤ 2i − 2, (176)

[A_{0}, [A_{1}, (ad^{2i−1}A_{0})(A_{1})]] = [A_{1}, [A_{1}, (ad^{2i−1}A_{0})(A_{1})]] = 0. (177)

Proof. Let us prove this by induction on i. For i = 1 relation (176) is trivial, and relation (177) follows from (172) and (173). Suppose that the statement holds for i = n ≥ 1 and let us prove it for i = n+1. By the induction assumption, relations (176) for i = n hold in 𝔑/𝔑_{n+1}. By definition (175), we have also

(178)  [A_{1}, (ad^{2n−1}A_{0})(A_{1})] = 0.

Applying the Jacobi identity to (178) and taking into account (176) for i = n, we obtain

(179)  [(ad^{k}A_{0})(A_{1}), (ad^{l}A_{0})(A_{1})] = 0 ∀ k, l ∈ ℤ_{+} k + l ≤ 2n − 1.

By the same argument, we have

(180) [(ad^{k}A_{0})(A_{1}), (ad^{l}A_{0})(A_{1})] =

= −[(ad^{k+1}A_{0})(A_{1}), (ad^{l−1}A_{0})(A_{1})] ∀ k + l = 2n.

Using this, we obtain

(181) [(ad^{k}A_{0})(A_{1}), (ad^{l}A_{0})(A_{1})] =

= [(ad^{l}A_{0})(A_{1}), (ad^{k}A_{0})(A_{1})] = 0 ∀ k + l = 2n.

Relations (179) and (181) imply (176) for i = n + 1.

It remains to prove (177) for i = n + 1, that is,

[A_{0}, [A_{1}, (ad^{2n+1}A_{0})(A_{1})]] = 0, (182)

[A_{1}, [A_{1}, (ad^{2n+1}A_{0})(A_{1})]] = 0. (183)

Relation (183) follows easily from the Jacobi identity combined with (176) for i = n+1. Similarly to (180) we have

(184) [(ad^{k}A_{0})(A_{1}), (ad^{l}A_{0})(A_{1})] =

= −[(ad^{k+1}A_{0})(A_{1}), (ad^{l−1}A_{0})(A_{1})] ∀ k + l = 2n + 1.

Set I = [A_{1}, (ad^{2n+1}A_{0})(A_{1})]]. Using (184), one gets

I = (−1)^{n}[(ad^{n}A_{0})(A_{1}), (ad^{n+1}A_{0})(A_{1})] =

= (−1)^{n+1}[(ad^{n−1}A_{0})(A_{1}), (ad^{n+2}A_{0})(A_{1})].

Applying ad A_{0} to this equality, we obtain

(185) [A_{0}, I] = (−1)^{n}[(ad^{n}A_{0})(A_{1}), (ad^{n+2}A_{0})(A_{1})] =

= (−1)^{n+1}([(ad^{n}A_{0})(A_{1}), (ad^{n+2}A_{0})(A_{1})]+

+ [(ad^{n−1}A_{0})(A_{1}), (ad^{n+3}A_{0})(A_{1})]).

On the other hand, applying ad B_{0} to [(ad^{n−1}A_{0})(A_{1}), (ad^{n}A_{0})(A_{1})] = 0 and taking into account (174) and (171), one gets

[(ad^{n+2}A_{0})(A_{1}), (ad^{n}A_{0})(A_{1})] + [(ad^{n−1}A_{0})(A_{1}), (ad^{n+3}A_{0})(A_{1})] = 0.

Combining this with (185), we obtain [A_{0}, I] = 0, which proves relation (183). □

By Lemma 16, relation (177) implies that 𝔑_{i+1}/𝔑_{i} is commutative. Relation (176) says that in the quotient algebra 𝔑/ ∪_{i} 𝔑_{i} we have

(186)  [(ad^{k}A_{0})(A_{1}), (ad^{l}A_{0})(A_{1})] = 0 ∀ k, l ∈ ℤ_{+},

which implies that this quotient of 𝔑 is solvable. □

Theorem 26. The algebra 𝔑 is not quasi-finite.

Proof. In the quotient algebra 𝔑/ ∪_{i} 𝔑_{i} denote c_{k} = (ad^{k}A_{0})(A_{1}). Consider the subalgebra 𝔤 of 𝔑/ ∪_{i} 𝔑_{i} generated by B_{0} and c_{k}. Obviously, for a quasi-finite algebra any quotient algebra and any subalgebra of finite codimension are also quasi-finite. Therefore, it is sufficient to prove that the algebra 𝔤 is not quasi-finite.

Relations (186) say that [c_{k}, c_{l}] = 0, while relations (174) and (171) imply [B_{0}, c_{k}] = c_{k+3}. Let m_{k}, k ∈ ℤ_{+}, be a sequence of nonzero complex numbers satisfying m_{k+3} = −(k+1)m_{k}. Consider the following transitive action of 𝔤 on the manifold M = {(x, y) ∈ ℂ^{2} | x ≠ 0, y ≠ 0}

c_{k} ↦ \frac{m_{k}}{x^{k+1}} \frac{∂}{∂y} , B_{0} ↦ \frac{1}{x^{2}} \frac{∂}{∂x}.

By Theorem 7, since the image of 𝔤 in D(M) is infinite-dimensional, the algebra 𝔤 is not quasi-finite. □

7. Non-existence results for Bäcklund transformations

Theorem 27. Equation (169) is not connected by any Bäcklund transformation neither with the KdV equation nor with the nonsingular Krichever-Novikov equation.

Proof. Below a Lie subalgebra denoted by 𝔥, 𝔥^{1}, or 𝔥^{2} is always supposed to be of finite codimension. The following lemma is obvious.

Lemma 18. Let 𝔤 be a finite-dimensional semisimple Lie algebra. Suppose that a Lie algebra 𝔤_{1} is obtained from a Lie algebra 𝔤_{2} applying several times the operation of onedimensional central extension. Then each of the following properties holds for i = 1 if and only if it holds for i = 2.

• There are a subalgebra 𝔥 ⊂ 𝔤_{i} and an epimorphism 𝔥 → 𝔤.

• For any subalgebra 𝔥 ⊂ 𝔤_{i} there is an epimorphism 𝔥 → 𝔤.

Set 𝔤 = 𝔰𝔩_{2}(ℂ). Let us prove first that there is no Bäcklund transformation between equation (169) and the nonsingular Krichever-Novikov equation. Combining Lemma 18 with Theorems 22, 24, and 14, we see that it is sufficient to prove that for any subalgebras 𝔥^{1} ⊂ ℜ, 𝔥^{2} ⊂ 𝔑 there is an epimorphism 𝔥^{1} → 𝔤, but there is no epimorphism 𝔥^{2} → 𝔤.

There is a natural family of epimorphisms ℜ → 𝔤 parameterized by the points of the affine curve in ℂ^{3} given by polynomials (161). Namely, for a point (a_{1}, a_{2}, a_{3}) of the curve the generator x_{i} ⊗ v̄_{i} of ℜ is mapped to a_{i}x_{i} ∈ 𝔤. Since 𝔥^{1} is of finite codimension in ℜ, there are polynomials f_{i}(v̄_{1}, v̄_{2}, v̄_{3}) and a point (a_{1}, a_{2}, a_{3}) of the curve such that x_{i} ⊗ f_{i} belongs to 𝔥^{1} and f_{i}(a_{1}, a_{2}, a_{3}) ≠ 0 for all i = 1, 2, 3. Then the restriction to 𝔥^{1} of the corresponding homomorphism ρ: ℜ → 𝔤 is surjective, since the elements

ρ(x_{i} ⊗ f_{i}) = f_{i}(a_{1}, a_{2}, a_{3})x_{i}, i = 1, 2, 3,

span 𝔤.

Non-existence of an epimorphism 𝔥^{2} → 𝔤 follows from Theorem 25. Indeed, suppose that there is an epimorphism ρ: 𝔥^{2} → 𝔤. Since 𝔥^{2}∩𝔑_{i} is solvable, we have ρ(𝔥^{2}∩𝔑_{i}) = 0 for all i. Therefore, there is an epimorphism

𝔥^{2}/(𝔥^{2} ∩ (∪_{i}𝔑_{i}))→ 𝔤,

which is impossible, since 𝔑/ ∪_{i} 𝔑_{i} is solvable.

Let us now prove that there is no Bäcklund transformation between equation (169) and the KdV equation. Since, according to Theorem 17, each fundamental algebra of the KdV equation is obtained from 𝔤 ⊗ ℂ[λ] applying several times the operation of one-dimensional central extension, it is sufficient to prove that for any subalgebra 𝔥^{1} ⊂ 𝔤 ⊗ ℂ[λ] there is an epimorphism 𝔥^{1} → 𝔤. Consider the natural family of epimorphisms

ρ_{a}: 𝔤 ⊗ ℂ[λ] → 𝔤, g ⊗ f(λ) ↦ f(a)g, a ∈ ℂ.

Since 𝔥^{1} is of finite codimension, for some of these epimorphisms its restriction to 𝔥^{1} is surjective. □

Acknowledgements

The author is deeply grateful to I. S. Krasilshchik for careful reading of the paper and many valuable comments. The author thanks also R. Martini and V. V. Sokolov for useful discussions.

References

[1] A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor′kova, I. S. Krasil′shchik, A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky, and A. M. Vinogradov. Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Amer. Math. Soc., Providence, RI, 1999.

[2] R. Dodd and A. Fordy. The prolongation structures of quasipolynomial flows. Proc. Roy. Soc. London Ser. A 385 (1983), 389–429.

[3] J. D. Finley and J. K. McIver. Prolongations to higher jets of Estabrook-Wahlquist coverings for PDEs. Acta Appl. Math. 32 (1993), 197–225.

[4] V. V. Gorbatsevich, A. L. Onishchik, and E. B. Vinberg. Foundations of Lie theory and Lie transformation groups. Springer-Verlag, Berlin, 1997.

[5] S. Igonin. Horizontal cohomology with coefficients and nonlinear zero-curvature representations. Russian Math. Surveys 58 (2003), 180–182

[6] S. Igonin and R. Martini. Prolongation structure of the Krichever-Novikov equation. J. Phys. A: Math. Gen. 35 (2002), 9801-9810; arxiv.org/nlin.SI/0208006

[7] E. G. Kirnasov. On Wahlquist-Estabrook type coverings over the heat equation. Math. Notes 42 (1987), 732–739.

[8] I. S. Krasilshchik and A. M. Vinogradov. Nonlocal trends in the geometry of differential equations. Acta Appl. Math. 15 (1989), 161-209.

[9] I. S. Krasil′shchik, V. V. Lychagin, A. M. Vinogradov. Geometry of jet spaces and nonlinear partial differential equations. Gordon & Breach, New York, 1986.

[10] J. Krasil′shchik and A. Verbovetsky. Homological Methods in Equations of Mathematical Physics. arxiv.org/math.DG/9808130

[11] I. M. Krichever and S. P. Novikov. Holomorphic bundles over algebraic curves, and nonlinear equations. Russian Math. Surveys 35 (1980), 53–80.

[12] M. Marvan. On zero-curvature representations of partial differential equations. Differential geometry and its applications (Opava, 1992), 103-122. Math. Publ., 1, Silesian Univ. Opava, 1993.

[13] T. Nagano. Linear differential systems with singularities and an application to transitive Lie algebras. J. Math. Soc. Japan 18 (1966), 398–404.

[14] G. H. M. Roelofs and R. Martini. Prolongation structure of the Landau-Lifshitz equation. J. Math. Phys. 34 (1993), 2394–2399.

[15] C. Rogers and W. K. Schief. Bäcklund and Darboux transformations. Cambridge Univ. Press, Cambridge, 2002.

[16] C. Rogers and W. F. Shadwick. Bäcklund transformations and their applications. Academic Press, New York, 1982.

[17] V. V. Sokolov. On the hamiltonian property of the Krichever-Novikov equation. Soviet. Math. Dokl. 30 (1984), 44-46.

[18] S. I. Svinolupov, V. V. Sokolov, and R. I. Yamilov. Bäcklund transformations for integrable evolution equations. Soviet Math. Dokl. 28 (1983), 165–168.

[19] H. N. van Eck. The explicit form of the Lie algebra of Wahlquist and Estabrook. A presentation problem. Nederl. Akad. Wetensch. Indag. Math. 45 (1983), 149–164.

[20] H. N. van Eck. A non-Archimedean approach to prolongation theory. Lett. Math. Phys. 12 (1986), 231–239.

[21] A. Verbovetsky. Notes on the horizontal cohomology. Secondary calculus and cohomological physics (Moscow, 1997), 211-231. Contemp. Math., 219, Amer. Math. Soc., Providence, RI, 1998.

[22] H. D. Wahlquist and F. B. Estabrook. Prolongation structures of nonlinear evolution equations. J. Math. Phys. 16 (1975), 1-7.

Department of Mathematics, Utrecht University, P.O. Box 80010, 3508 TA Utrecht, the Netherlands

  E-mail address: igonin@mccme.ru