The Diffiety Institute Preprint Series

Preprint DIPS–1/97 February 20, 1997

Available via INTERNET: http://ecfor.rssi.ru/˜ diffiety/ http://www.botik.ru/˜ diffiety/ anonymous FTP: ftp://ecfor.rssi.ru/general/pub/diffiety ftp://www.botik.ru/˜ diffiety/preprints

The Diffiety Institute

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

Some Local Properties of Bäcklund Transformations

by

Michal Marvan Some Local Properties of Bäcklund Transformations

Michal Marvan

Abstract

For Bäcklund transformations, treated as relations in the category of diffieties, local conditions of effectivity and normality are introduced, having implications for the solution generating properties. We check them for the pKdV, the sine-Gordon, and the Tzitzéica equation. MS classification numbers: 58G37, 35A30. Key words: Diffiety, Bäcklund transformation, permutability, nonlinear superposition.

Introduction

Bäcklund transformations (BT) represent an efficient and practical tool to generate families of exact solutions of nonlinear partial differential equations of mathematical physics [11, 13]. A typical, although not self-evident property of a given BT is the following: Suppose that every single solution u is transformed to a kparametric family of solutions v(p_{1}, …, p_{k}). Then every l-parametric family of solutions u(q_{1}, …, q_{l}) is transformed to a k + l-parametric family of solutions v(q_{1}, …, q_{l}, p_{1}, …, p_{k}), where the parameters q_{1}, …, q_{l}, p_{1}, …, p_{k} are independent. We shall call it an additivity principle and aim at its geometric explanation.

Bianchi [2] was the first to observe permutability, another phenomenon usually accompanying BT’s and believed to be responsible for nonlinear superposition— an algebraic (non-differential) relation among four solutions of a given PDE, which are pairwise related by BT’s constituting a Bianchi diagram

\begin{aligned}& & v′ \\ & ^{Ψ}↗ & & ↘^{Φ} \\ u & & & & w. \\ & _{Φ}↘ & & ↗_{Ψ} \\ & & v \end{aligned}  (1)

Here arrows indicate two, usually different BT’s (although often belonging to a family of BT’s and differing in the values of what is called a Bäcklund parameter). The permutability property then reads: For any solution u and any solutions v and v′ such that uΦv and uΨv′ there is a unique solution w such that vΨw and v′Φw. A superposition formula w = w(u, v, v′) results (see also [8, 4]).

We demonstrate in Sect. 3 that both principles relate to normality, a local geometric property of BT’s which we check for several well-known equations in Sect. 4. As a pre-condition to normality, effectivity is introduced in Sect. 2.

Throughout the paper we use the very clear geometric picture given by Krasil’shchik and Vinogradov [6] within the category of diffieties—geometric objects representing systems of nonlinear partial differential equations. In [5], I.S. Krasil’shchik presented interesting ideas towards explanation of the permutability in the framework of this category. The present paper appears as an attempt to work through some of the problems opened there.

It is convenient to treat BT’s as correspondences or relations—Bäcklund relations (BR)—studied earlier by Pommaret [12] within the formal theory of differential equations. Other relevant references are [20, 1]. Such notions as the composition and the inversion of relations pass to BR’s very naturally. We shall see below that the permutability includes plain commutativity of BR’s.

1 Preliminaries

1.1. By ∞-dimensional manifolds we mean R^{∞}-modelled smooth manifolds, as introduced in [3] (see also [20, 14]). Here R^{∞} denotes the topological product of countably many copies of the real line R. Recall that a function f : R^{∞} → R is smooth if locally f is a C^{∞}-function of a finite number of arguments only. The notions of a smooth structure, k-dimensional submanifold, k ≤ ∞, the tangent bundle and the fibered manifold have similar properties as their finite-dimensional analogues. Although (see [14]) the ∞-dimensional manifolds we use below (jet spaces) are not Banach manifolds, the immersion theorem still can be proved when the source manifold is finite-dimensional: if dim X < ∞ then immersions X → Y are, locally, diffeomorphisms onto submanifolds. (A map f : X → Y is an immersion if the tangent map T_{x}f : T_{x}X → T_{f(x)}Y is injective for every x ∈ X.)

Consider a submanifold N in a manifold E. By a distribution along N we mean an assignment to every x ∈ N of a linear subspace Ξ_{x} ⊆ T_{x}E. Denote rk_{x} Ξ := dim Ξ_{x}. If rk_{x} Ξ ≤ k for some constant k then Ξ is said to be finitedimensional. Let rk Ξ denote min_{x∈N} rk_{x} Ξ. A finite-dimensional distribution Ξ along N is said to be smooth if, locally, smooth vector fields ξ^{1}, …, ξ^{k} along N exist such that the linear spaces Ξ_{x} are spanned by the vectors ξ_{x}^{1}, …, ξ_{x}^{k}. (A smooth vector field along N is a smooth mapping ξ : N → T E such that ξ_{x} ∈ T_{x}E for every x ∈ N.) We write Ξ = ⟨ξ^{1}, …, ξ^{k}⟩. When Ξ is smooth then for any natural number r the set U_{r} = {x ∈ N | rk_{x} Ξ ≤ r} is open in N.

For any two distributions Ξ, H along N their sum Ξ + H is defined by (Ξ + H)_{x} = Ξ_{x} + H_{x}. Obviously, Ξ + H is smooth whenever Ξ, H are smooth. An intersection of two smooth distributions need not be smooth, since its rank may rise even in isolated points.

LEMMA 1.1. Let Ξ, H be two smooth distributions along a finite-dimensional submanifold N. Let the distributions Ξ, H, Ξ + H be of constant rank within an open domain U ⊂ N. Then the intersection Ξ∩H is of constant rank and smooth in U.

PROOF. Within a coordinate chart U, a smooth n-dimensional distribution may be identified with a smooth mapping from U to the Grassmann manifold G_{n}R^{m} of n-planes in R^{m}, m = dim E. Consider the submanifold P_{r,s,n} := {(X, Y, Z) ∈ G_{r}R^{m}×G_{s}R^{m}×G_{n}R^{m} | X +Y = Z}, n ≤ r+s. Our statement then follows from the fact that the mapping P_{r,s,n} → G_{r+s−n}R^{m} which sends (X, Y, Z) to X ∩ Y is smooth.

By a distribution on N we mean a distribution Ξ along N, tangent to N, i.e., Ξ ⊆ T N. Then Lie brackets [ξ_{i}, ξ_{j}] are well defined for arbitrary smooth fields ξ_{i}, ξ_{j} ∈ Ξ. A smooth distribution Ξ on N is said to be involutive if [ξ_{i}, ξ_{j}] ∈ Ξ for any ξ_{i}, ξ_{j} ∈ Ξ. This is briefly written as [Ξ, Ξ] ⊆ Ξ. For a smooth distribution Ξ = ⟨ξ^{1}, …, ξ^{k}⟩ on N we introduce its involutive or Lie hull ℒ(Ξ) = ℒ⟨ξ^{1}, …, ξ^{k}⟩ as the distribution on N spanned by all the Lie brackets [ξ^{i_{1}}, … [ξ^{i_{r−1}}, ξ^{i_{r}}] …], r = 1, 2, . . .. Clearly ℒ(Ξ) represents the minimal set of vector fields on N containing ξ^{1}, …, ξ^{k} and closed with respect to addition, multiplication by functions f ∈ C^{∞}N, and the Lie bracket. When ℒ(Ξ) is finite-dimensional then it is obviously smooth.

1.2. We refer the reader to [17, 7] for the diffiety theory and its relation to the geometry of partial differential equations. Diffieties are infinite-dimensional smooth manifolds equipped with an involutive distribution, called Cartan distribution and denoted by 𝒞. Morphisms are smooth maps preserving the distribution. In this paper we use a restriction of this category to fibred manifolds over a fixed finite-dimensional manifold M. In this case, a diffiety is identified with a fibred manifold endowed with a flat connection, whose horizontal distribution plays the role of 𝒞. Unlike the full category, the latter category has products (Whitney products) at the manifold level (cf. [9]).

Recall that for a formally integrable system of PDE we have its infinite prolongation E = E^{(∞)}, a subspace in the jet space j^{∞}Y of a fibred manifold Y → M, dim Y < ∞. The link to traditional terms is via special coordinate systems. Local coordinates, say x^{i}, in the base M can be identified with independent variables and local coordinates, say u^{k}, along the fibres of the projection Y → M can be identified with dependent variables. Then the “derivatives” u_{i_{1}...i_{k}}^{k} represent natural local coordinates along the fibres of the projection j^{∞}Y → Y. Vector fields (total derivatives)

D_{i} = \frac{∂}{∂x^{i}} + ∑_{i_{1}...i_{r}} u_{i_{1}...i_{r}i}^{k} \frac{∂}{∂u_{i_{1}...i_{r}}^{k}} ,  (2)

(summation over symmetric multiindices i_{1} … i_{r}) on j^{∞}Y admit a restriction to the submanifold E and the restricted fields then span the Cartan distribution 𝒞 here. Its dimension is equal to dim M. The pair (E, 𝒞) is the corresponding diffiety and carries all the information needed to determine the BT’s.

Projections of diffieties, p : Ẽ → E, which are fibred manifolds of fibre dimension k, and map Cartan distribution onto Cartan distribution are called kdimensional coverings [6]. A covering is called finitary if k < ∞. For a finitary covering, local coordinates w^{κ} along the fibres correspond to Wahlquist–Estabrook pseudopotentials [19].

1.3. Let f : E → E′ be a finitary covering. Vectors tangent to the fibres (fvertical vectors) form a finite-dimensional involutive distribution on E. We shall denote this distribution by Ξ^{f} and call it an f-vertical distribution. Thus, Ξ_{z}^{f} = Ker T_{z}f at any point z ∈ E (T_{z}f : T_{z}E → T_{fz}E′ is the tangent mapping to f at z). It is easily verified that the f-vertical distribution Ξ^{f} on E satisfies the relation [Ξ^{f}, D] ⊆ Ξ^{f} for all f-projectable vector fields D, in particular, for the fields D_{i} spanning the Cartan distribution on E.

Following Sokolov [15], a distribution Ξ on E satisfying the relation

[Ξ, 𝒞] ⊆ Ξ  (3)

is called a pseudosymmetry. Thus, Ξ^{f} is a pseudosymmetry on E.

In [15] the notion of a pseudosymmetry was introduced for evolution equations. The main purpose was factorization, i.e., reconstruction of the projection f : E → E′ out of its determining pseudosymmetry, much in the same way as diffieties are factorized with respect to ordinary symmetries (see [7, pp. 332–335]). Another useful feature known already from [15, Proposition 3] is that pseudosymmetries admit an effective description. We put this result in the context of general diffieties following the guidelines of [7] to profit from the analogue with symmetries.

Consider diffieties E, E′ in “outer” geometry, i.e., as submanifolds of j^{∞}Y, j^{∞}Y′, and assume the map f to be a restriction of a projection g : j^{∞}Y → j^{∞}Y′. To be an outer morphism g must be a prolongation of some nonlinear differential operator j^{∞}Y → Y′ and map E ⊂ j^{∞}Y into E′ ⊂ j^{∞}Y′. As such, g must preserve the Cartan distribution, and Ξ must coincide with Ker T g ∩T E. Let us start with Ker T g, i.e., with pseudosymmetries on j^{∞}Y.

Locally Ξ = ⟨ξ^{(1)}, …, ξ^{(n)}⟩ for smooth vector fields ξ^{(1)}, …, ξ^{(n)}, n = dim Ξ on j^{∞}Y. Introduce a column n-vector ξ = (ξ^{(1)}, …, ξ^{(n)}). Operators of differential geometry are applied to ξ componentwise. Then eq. (3) reads:

[D_{i}, ξ] = A_{(i)}ξ  (4)

for some n × n-matrices A_{(i)} of C^{∞} functions, i = 1, …, dim M.

Since the fields ξ^{(ι)} are necessarily vertical with respect to the projection to M, in coordinates we have ξ^{(ι)} = ξ^{(ι)}_{I}^{k} ∂/∂u_{I}^{k}. Adopt the same notation for the n-tuple ξ so that ξ = ξ_{I}^{k} ∂/∂u_{I}^{k}. Upon inserting D_{i} from (2), eq. (4) becomes (D_{i}ξ_{I}^{k} − ξ_{Ii}^{k}) ∂/∂u_{I}^{k} = A_{(i)}ξ_{I}^{k} ∂/∂u_{I}^{k}, yielding a recurrence formula

ξ_{Ii}^{k} = ∇_{i}^{A}ξ_{I}^{k} := D_{i}ξ_{I}^{k} − A_{(i)}ξ_{I}^{k},  (5)

where ∇_{i}^{A}(ξ) := D_{i}ξ − A_{(i)}ξ. Put ∇_{i_{1}...i_{r}}^{A} = ∇_{i_{1}}^{A} ∘ ⋯ ∘ ∇_{i_{r}}^{A}. Then (5) is resolved as ξ_{I}^{k} = ∇_{I}^{A}ξ^{k} and in analogy with [7] we may put

� _{ξ}^{A} := ξ_{I}^{k} \frac{∂}{∂u_{I}^{k}} = ∇_{I}^{A}ξ^{k} \frac{∂}{∂u_{I}^{k}}  (6)

and conclude that every pseudosymmetry on j^{∞}Y is of the form � _{ξ}^{A} for an appropriate set of matrices A = {A_{(i)} | i = 1, …, dim M} and vector fields ξ = {ξ^{k} ∂/∂u^{k} | k = 1, …, dim_{M} Y }. Note that formula (6) gives [15, eq. (16)] when applied to evolution equations, and reduces to formula [7, eqs. (0.47)] for symmetries when A = 0.

From the Jacobi identity [D_{i}, [D_{j}, ξ]] = [[D_{i}, D_{j}], ξ] + [D_{j}, [D_{i}, ξ]] we get, since [D_{i}, D_{j}] = 0 and the components of ξ are independent, that

D_{i}A_{(j)} − D_{j}A_{(i)} + [A_{(i)}, A_{(j)}] = 0 on E  (7)

for any i ≠ j (cf. [15, eq. (17)]). We conclude that the gl_{n}-valued form

A = A_{(i)} dx^{i}  (8)

is a zero-curvature representation (ZCR) for E. Also, equation (7) is equivalent to [∇_{i}^{A}, ∇_{j}^{A}] being zero, so that ∇_{i_{1}...i_{r}}^{A} is symmetric in lower indices.

2 Bäcklund relations, regularity and effectivity

In [6] a Bäcklund transformation (BT) Φ from a diffiety E_{1} to a diffiety E_{2} was defined as a pair of coverings p over E_{1} and q over E_{2} with a common source Ẽ. We shall additionally assume a commutativity of the diagram

Ẽ ; ^{p}↙  ↘^{q} ; E_{1}  E_{2} ; ↘  ↙ ; M  (9)

which means that the transformation does not touch independent variables.

The coverings p, q will be considered finitary. This condition is crucial for dimension analysis, and practically non-restrictive when dim M = 2.

Say that points e_{1} ∈ E_{1} and e_{2} ∈ E_{2} are Φ-related whenever there is a point e ∈ Ẽ such that pe = e_{1} and qe = e_{2}. For every e_{1} there is typically a whole family of Φ-related points e_{2} (the projection along q of the fibre p^{−1}e_{1} over e_{1}). If this projection is a submanifold (possibly with singularities) then its dimension will be called a generating power of the BT Φ at the point e_{1}. One can identify the generating power with the dimension of the covering p whenever q|_{p^{−1}e_{1}} is an immersion. A simple criterion is derived below.

Pairs of Φ-related points form a subset in the Whitney product E_{1} ×_{M} E_{2}. It is natural to call this subset a Bäcklund relation (BR) (cf. [20, Sect. 7.5]) corresponding to the given BT. We shall denote it by E_{Φ}. Also, we write e_{1}Φe_{2} to express that e_{1}, e_{2} are Φ-related. Following [5] two BT’s are said to be equivalent whenever they determine the same BR.

Compositions and inverses of BR’s are as usual. If E_{Ψ} ⊂ E_{1} ×_{M} E_{2} and E_{Φ} ⊂ E_{2} ×_{M} E_{3} are two BR’s, then a pair (v, u) ∈ E_{2} ×_{M} E_{1} is Φ^{−1}-related iff uΦv, and a pair (u, w) ∈ E_{1} ×_{M} E_{3} is Φ ∘ Ψ-related iff there exists v ∈ E_{2} such that uΨv and vΦw. In terms of BT’s the inversion of a BR amounts to interchanging the two coverings, while composition of BR’s corresponds to the pullback of coverings [5] (see diagram (11) below).

DEFINITION 2.1. A BT Φ is said to be regular if E_{Φ} is a submanifold in E_{1} ×_{M} E_{2}, and the maps p_{Φ} = pr_{1}|_{E_{Φ}} : E_{Φ} → E_{1}, q_{Φ} = pr_{2}|_{E_{Φ}} : E_{Φ} → E_{2} are fibred manifolds. We also say that the BR E_{Φ} is regular.

BR’s resulting from the Wahlquist–Estabrook construction are, as a rule, regular. A simple example of a BR which is not regular is given by Φ ∘ Φ^{−1} for any BT Φ. In general, this question is a matter of the formal integrability theory and as such has been studied in [12].

Let Φ be a regular BT. Then it is easily seen that the submanifold E_{Φ} is a subdiffiety in E_{1} ×_{M} E_{2} and the projections p_{Φ} : E_{Φ} → E_{1}, q_{Φ} : E_{Φ} → E_{2} are coverings (cf. [20]). Otherwise said, when Φ is a regular BT then E_{Φ}, p_{Φ}, q_{Φ} form a BT equivalent to Φ. We call it a minimal BT. A regular BR will be often identified with its minimal BT.

PROPOSITION 2.2. Let Φ be a regular BT (9). Then for any u ∈ E_{1} the set K := qp^{−1}u of all its Φ-related points is a submanifold in E_{2}.

PROOF. Obviously, we may also write K = q_{Φ}p_{Φ}^{−1}u. By regularity, the fibre N := p_{Φ}^{−1}u is a finite-dimensional submanifold in E_{Φ}, and hence in E_{1} ×_{M} E_{2}. At the same time, pr_{1}^{−1} u is another finite-dimensional submanifold in E_{1} ×_{M} E_{2}, containing N as a subset (in fact, N = E_{Φ} ∩pr_{1}^{−1} u) so that N is a submanifold in pr_{1}^{−1} u as well. By general properties of Whitney products the map q_{Φ}|_{pr_{1}^{−1} u} is a diffeomorphism of the fibre pr_{1}^{−1} u onto a fibre in E_{2} over M, and, therefore, the image K = q_{Φ}N of the submanifold N is a submanifold in this fibre, and hence in E_{2}.

Let f_{Φ} : Ẽ → E_{1} ×_{M} E_{2} denote the morphism determined by the pair p, q, so that p_{Φ} ∘ f_{Φ} = p and q_{Φ} ∘ f_{Φ} = q. Consider the diagram

Ẽ ; ↓ f_{Φ} ; E_{Φ} ; ^{p_{Φ}} ↙ ↘^{q_{Φ}} ; E_{1}  E_{2},  (10)

where p_{Φ} ∘ f_{Φ} = p, q_{Φ} ∘ f_{Φ} = q (cf. [12, p. 121]). Obviously, the map f_{Φ} : Ẽ → E_{Φ} is surjective. Observe that the map f_{Φ} does not contribute to the generating power but may increase the fibre dimension. Such a redundant covering f_{Φ} may be revealed as follows. Consider the involutive distribution Ξ^{Φ} := Ξ^{f_{Φ}} of f_{Φ}- vertical vectors (see Sect. 1.3): Ξ_{z}^{Φ} = Ker T_{z}f_{Φ}, z ∈ Ẽ. For a general BT (9), let us call vectors vertical with respect to both projections p and q bivertical. Thus, bivertical vectors lie in the intersection Ξ^{p} ∩ Ξ^{q}. However, within our scheme Ξ^{p} ∩ Ξ^{q} = Ξ^{p_{Φ}∘f_{Φ}} ∩ Ξ^{q_{Φ}∘f_{Φ}} = Ξ^{Φ} and, therefore, f_{Φ}-vertical and bivertical vectors coincide. Hence, if there is a nontrivial covering f_{Φ} then there is, at least locally, a nonzero bivertical vector field. Conversely, when a nonzero bivertical vector field is found, then a factorization process with respect to a pseudosymmetry ([15]) may be applied to reconstruct f_{Φ}. We present an example where a BT is factorized in Sect. 4.3.1.

DEFINITION 2.3. A BT (9) is said to be effective when Ξ_{z}^{p} ∩ Ξ_{z}^{q} = 0 at every point z ∈ Ẽ, i.e., when there is no nonzero bivertical vector.

Obviously, the minimal BT of a regular BR is effective.

LEMMA 2.4. A BT Φ is effective iff for every N = p^{−1}u ⊂ Ẽ, u ∈ E_{1}, the map q|_{N} : N → E_{2} is an immersion.

PROOF. This immediately follows from the equality Ker(q|_{N}) = T N ∩ Ξ^{q}|_{N} = Ξ^{p}|_{N} ∩ Ξ^{q}|_{N}.

PROPOSITION 2.5. Let Φ be a regular effective BT (9). Then the generating power of Φ coincides with the fibre dimension of the covering p.

PROOF. Using the notation of Proposition 2.2, the generating power is (for a given u ∈ E_{1}) equal to dim K = dim qp^{−1}u = dim p^{−1}u, since q|_{p^{−1}u} is an immersion by Lemma 2.4.

3 Normality, additivity and superposition principles

In this section we use the notation introduced in Sect. 1. Below ⟨Ξ, η⟩ stands for Ξ + ⟨η⟩.

DEFINITION 3.1. Consider a BT (9), let Ξ, H denote the involutive distributions Ξ_{x} = Ker T_{x}p, H_{x} = Ker T_{x}q on Ẽ, respectively. The BT is said to be r-normal, where r is a natural number, if

r = inf_{η∈H\{0}} rk ℒ⟨Ξ, η⟩.

A BT for which r = ∞ is said to be normal.

If there is a bivertical field η ≠ 0 then rk ℒ⟨Ξ, η⟩ = rk Ξ = dim p so that r ≤ dim p and therefore Φ cannot be normal. Assuming that Φ is effective one avoids this bound.

PROPOSITION 3.2. Let Φ be a BT (9), K be a finite-dimensional submanifold in E_{1}, Φ be r-normal for some r > dim p + dim K. Set N := p^{−1}K ⊂ Ẽ. Then there exists an open dense subset N^{0} in N such that q|_{N^{0}} : N^{0} → E_{2} is an immersion.

PROOF. Consider involutive distributions Ξ_{z} = Ker T_{z}p, H_{z} = Ker T_{z}q along the finite-dimensional submanifold N ⊆ Ẽ. Then Ξ|_{N} = T_{z}N. Consider the distribution Z_{z} = H_{z} ∩ T_{z}N (possibly non-smooth) on N. Let N^{0} := {z ∈ N | Z_{z} = 0}. Then N^{0} is an open subset in N, and Ker T_{z}(q|_{N^{0}}) = Z_{z} = 0 for any z ∈ N^{0}, so that q|_{N^{0}} : N^{0} → E is an immersion.

Prove that N^{0} is dense in N. Consider the closed subset B = N \ N^{0}. Let us show that B has an empty interior in N. Suppose not, then there is a subset U ⊆ B open in N. By virtue of Lemma 1.1 there exists an open subset V ⊆ U and a non-zero smooth vector field η ∈ Z|_{V}, i.e., both lying in H|_{V} and tangent to N. By r-normality, the C^{∞}V -closed Lie algebra ℒ⟨Ξ|_{V} , η⟩ is of rank ≥ r, but ℒ⟨Ξ|_{V} , η⟩ is tangent to V and dim V = dim N = dim p + dim K < r, which is a contradiction.

This proposition (case r = ∞) can be thought of as a geometric form of the additivity principle (see the Introduction). Let us discuss some of the consequences.

Let E_{1} ←^{p_{Ψ}} E_{Ψ} →^{q_{Ψ}} E_{2} and E_{2} ←^{p_{Φ}} E_{Φ} →^{q_{Φ}} E_{3} be two BR’s (minimal BT’s). The composition Φ∘Ψ is determined by coverings p_{Ψ} ∘p′, q_{Φ} ∘q′ from the commutative diagram

\begin{aligned}& & & & E′ \\ & & & ^{p′}↙ & & ↘^{q′} \\ & & E_{Ψ} & & & & E_{Φ} \\ & ^{p_{Ψ}} ↙ & & ↘^{q_{Ψ}} & & ^{p_{Φ}} ↙ & & ↘^{q_{Φ}} \\ E_{1} & & & & E_{2} & & & & E_{3} \end{aligned}  (11)

where the square is a pullback (the Whitney product) E′ ≅ E_{Ψ} ×_{E_{2}} E_{Φ} (see [5]). Recall that E_{Ψ}×_{E_{2}} E_{Φ} is a submanifold in E_{1}×_{M} E_{2}×_{M} E_{3} consisting of all triples (u, v, w) such that uΨv, vΦw. Here p′ : (u, v, w) ↦ (u, v) and q′ : (u, v, w) ↦ (v, w).

PROPOSITION 3.3. Let BR’s Ψ, Φ be regular, let Φ be r-normal with r > dim p_{Ψ} + dim p_{Φ}. Fix u ∈ E_{1} arbitrary, let P_{u} ⊂ E_{Ψ} ×_{E_{2}} E_{Φ} denote the set of all triples (u, v, w) with uΨv, vΦw. Then there exists an open dense subset P_{u}^{0} ⊆ P_{u} such that the mapping P_{u}^{0} → E_{3}, (u, v, w) ↦ w, is an immersion.

PROOF. First observe that P_{u} = p^{′−1}p_{Ψ}^{−1}u is the fibre of E′ over u. Put L_{u} := p_{Ψ}^{−1}u so that P_{u} = p^{′−1}L_{u}. Since Ψ is regular, K_{u} := q_{Ψ}L_{u} is a submanifold in E_{2} and q_{Ψ}|_{L_{u}} is an embedding (cf. Proposition 2.2 and its proof). It follows by properties of the Whitney product that N_{u} := q′P_{u} is a submanifold in E_{Φ} and q′|_{P_{u}} is an embedding. By Proposition 3.2 (observe that dim K_{u} = dim L_{u} = dim p_{Φ}) there is an open dense subset N_{u}^{0} ⊆ N_{u} such that q_{Φ}|_{N_{u}^{0}} is an immersion. Put P_{u}^{0} := (q′|_{P_{u}})^{−1}N_{u}^{0}. Then q_{Φ} ∘ q′|_{P_{u}^{0}} = q_{Φ}|_{N_{u}^{0}} ∘ q′|_{P_{u}^{0}} is an immersion, acting as (u, v, w) ↦ w.

COROLLARY 3.4. Under the assumptions of Proposition 3.3, the generating power of the BT Φ ∘ Ψ is equal to the sum of the generating powers of Ψ and Φ.

Proposition 3.3 is a geometric way to say that under regularity and normality conditions, for any fixed u ∈ E_{1} and any pair (v, w) in general position among those satisfying uΨv and vΨw it holds that v can be computed by “algebraic means” (avoiding integration of differential equations) out of the corresponding point w ∈ E_{3}. Otherwise said, normality means that there is a differential operator Δ(u, v, w) of order zero with respect to v and such that v is computable by the implicit function theorem from u, w and their derivatives, up to a closed sparse set (for any u) of exceptional pairs (v, w). Exceptional pairs exist: Consider the case when Φ = Ψ^{−1}, fix u. Then any pair (v, u) is exceptional, since on v we then have just one condition, uΨv, and v depends on as many parameters as is the generating power of Ψ.

Note that normality of only one of the BT’s, Φ, is needed in Proposition 3.3. By induction (under enough regularity conditions) it follows that when there is a chain of relations “v_{i+1} results from v_{i} by Φ_{i},” i = 0, …, n − 1, and BT’s Φ_{i}, i = 1, …, n−1 are normal then any of the intermediate solutions v_{i} is determined by “algebraic means” from the pair v_{0}, v_{n}.

Now let us discuss consequences of Proposition 3.3 for nonlinear superposition principles. Rewrite diagram (1) equivalently as

\begin{aligned}& v′ \\ ^{Ψ^{−1}} ↙ & & ↘ ^{Φ} \\ u & & w \\ _{Φ} ↘ & & ↙ _{Ψ^{−1}} \\ & v. \end{aligned}  (12)

THEOREM 3.5 (Weak Permutability Theorem). Let regular BR’s Ψ^{−1} and Φ commute (i.e., let Ψ^{−1} ∘ Φ = Φ ∘ Ψ^{−1}) and let Ψ^{−1} be r-normal for some r > dim q_{Ψ} + dim p_{Φ}.

1^{∘} Let u, v, v′ be points such that uΦv and uΨv′. Then there exists a point w such that vΨw and v′Φw;

2^{∘} For any fixed v′, in the space P_{v′} of all triples (v′, w, v) with v′Φw, vΨw there is an open dense subset P_{v′}^{0} such that the projection (v′, w, v) ↦ v, when restricted to P_{v′}^{0}, is an immersion.

PROOF. 1^{∘} By assumption, v′ is Φ ∘ Ψ^{−1}-related to v, via u. Then v′ is also Ψ^{−1} ∘ Φ-related to v since Φ and Ψ^{−1} commute, and the existence of w such that v′Φw and wΨ^{−1}v, i.e., vΨw follows.

2^{∘} Follows directly from Proposition 3.3.

Some remarks are due: (1) When Ψ is a member of a family of BT’s then often Ψ^{−1} is another member of the same family, but exceptions occur ([10]).

(2) According to our comment after Proposition 3.3 the superposition formula comes out in implicit form Δ(v, v′, w) = 0, where Δ is a differential operator of order zero in w, satisfying the assumptions of the implicit function theorem with respect to w.

(3) Let us turn our attention to the fact that traditional superposition formulas are of rather different form F (u, v, v′, w) = 0, where F is a function. Nevertheless, superposition formulas of the form Δ(v, v′, w) = 0 are known for most equations as well. They usually arise as an intermediate result in the course of derivation of the full superposition formula, and are abandoned afterwords. Let us call the superposition formulas Δ(v, v′, w) = 0 ternary, in contrast with usual, or quaternary superposition formulas F (u, v, v′, w) = 0. Ternary superposition formulas may become important in situations when explicit quaternary superposition formula is not known (cf. Sect. 4.3.4).

(4) It is clear that one cannot guarantee uniqueness of v in local theory. Also, the exceptional set P_{u} \ P_{u}^{0} cannot be generally avoided (consider the case when Φ = Ψ and v = v′).

4 Examples

4.1 The pKdV equation

Consider the potential KdV equation (pKdV), which we write as

u_{t} = −u_{xxx} − 3u_{x}^{2}.  (13)

A 1-parameter family Φ = Φ_{a} of BT’s is due to Wahlquist and Estabrook [18]

v_{x} = −u_{x} − \frac{1}{2}(u − v)^{2} − 2a,  (14)
v_{t} = −v_{xxx} − 3v_{x}^{2}.  (15)

This BT is already in the form of a BR, for the Whitney product E ×_{M} E is (13) together with (15), and equation (14) determines its subdiffiety.

One obvious property is that the latter system is invariant with respect to the u ↔ v exchange. Therefore, the relation Φ_{a} is symmetric: Φ_{a}^{−1} = Φ_{a}. This is, however, not typical for BR’s.

4.1.1 Effectivity

Consider the pKdV diffiety E over M, with coordinates t, x on M, and u, u_{x}, u_{xx}, u_{xxx}, etc. along the fibres. Other jet coordinates, such as u_{t}, u_{tx}, etc. can be expressed from (13) and its differential consequences. On the second copy of E we have likewise natural coordinates t, x, v, v_{x}, v_{xx}, v_{xxx}, etc. Two convenient choices of coordinates on Ẽ are:

(a) t, x, v, u, u_{x}, u_{xx}, u_{xxx}, …,

(b) t, x, u, v, v_{x}, v_{xx}, v_{xxx}, . . ..

The projection p is (t, x, v, u, u_{x}, u_{xx}, …) ↦ (t, x, u, u_{x}, u_{xx}, …) in the coordinate system (a) while the projection q is (t, x, u, v, v_{x}, v_{xx}, …) ↦ (t, x, v, v_{x}, v_{xx}, …) in the coordinate system (b).

On Ẽ, the p-vertical distribution Ξ^{p} is generated by the C^{∞} vector field ξ = ∂/∂v in the coordinate system (a), and the q-vertical distribution Ξ^{q} is generated by the C^{∞} vector field η = ∂/∂u in the coordinate system (b). A coordinate transformation from (b) to (a) induced by the relation (14) gives

η = \frac{∂}{∂u} + (v − u)\frac{∂}{∂u_{x}} + (\frac{1}{2}(v − u)^{2} − 2(u_{x} + a))\frac{∂}{∂u_{xx}} + ⋯  (16)

in the coordinate system (a). The effectivity of the BT Φ follows immediately, since ξ, η are manifestly independent.

4.1.2 Normality

Our proof of normality will use the fact that the fields ξ, η are pseudosymmetries, so that we can apply the formulas of Sect. 1.3. To simplify notation, we put ∂_{0} = ∂/∂u, ∂_{1} = ∂/∂u_{x}, ∂_{2} = ∂/∂u_{xx}, etc. The proof takes several steps.

(A) Let us write η = ∑_{i≥0} η_{i} ∂_{i}. Then η_{0} = 1 and

η_{k+1} = D_{x}η_{k} + (v − u)η_{k}.  (17)

Indeed, with the aid of the truncated expansion (16) we easily check that η_{0} = 1 and establish relations (4) of Sect. 1.3 to be

[D_{x}, η] = (v − u)η, [D_{t}, η] = (u_{xx} − (u_{x} − 2a)(v − u))η.  (18)

Then (17) is the recurrence relation (5). Also, from (18) we get the form (8) as (v − u) dx + 2(u_{xx} − (u_{x} − 2a)(v − u)) dt (a nonlocal conservation law).

(B) There exist fields α, β, γ, local on E and satisfying

η = α + (v − u)β + \frac{1}{2}(v − u)^{2}γ  (19)

(fields α, β, γ are lifted to Ẽ by the triviality of the fibering Ẽ → E).

Assuming the existence let us prove uniqueness. To this end, insert (19) into the first of the formulas (18) to get eq. (4) in the form

[D_{x}, ( \begin{matrix}α \\ β \\ γ \end{matrix} )] = ( \begin{matrix}0 & −2(u_{x} + a) & 0 \\ 1 & 0 & −2(u_{x} + a) \\ 0 & 1 & 0 \end{matrix} ) ( \begin{matrix}α \\ β \\ γ \end{matrix} ) ,  (20)

(we will not need the t-part). Upon introducing the notation α = ∑_{i≥0} α_{i} ∂_{i}, β = ∑_{i≥0} β_{i} ∂_{i}, γ = ∑_{i≥0} γ_{i} ∂_{i}, we pass from (20) to the recurrence formula (5) for the fields α, β, γ in the form

α_{k+1} = D_{x}α_{k} − 2(u_{x} + a)β_{k},
β_{k+1} = D_{x}β_{k} + α_{k} − 2(u_{x} + a)γ_{k},  (21)
γ_{k+1} = D_{x}γ_{k} + β_{k},

along with α_{0} = 1, β_{0} = γ_{0} = 0 coming from η_{0} = 1. The uniqueness of the fields α, β, γ follows.

To prove existence, we now define fields α, β, γ by the formulas (21) and show that (19) holds. It is sufficient to check by induction that η_{k} = α_{k} + (v − u)β_{k} + \frac{1}{2}(v − u)^{2}γ_{k}. But this is easy, hence omitted.

Let us also mention that the ZCR from eq. (8) is sl_{3}-valued and induced from the standard sl_{2}-valued ZCR by a linear representation of sl_{2} in sl_{3}.

(C) It holds ℒ⟨α, β, γ⟩ ⊆ ℒ⟨ξ, η⟩.

It follows from (19) that [ξ, [ξ, η]] = γ, [ξ, η] = β + (v − u)γ, since ξ commutes with α, β, γ (coefficients α^{k}, β^{k}, γ^{k} do not depend on v). This together with (19) implies that α, β, γ ∈ ℒ⟨ξ, η⟩, and the inclusion follows.

In the rest of the proof we verify that rk ℒ⟨α, β⟩ = ∞, which implies the normality due to the inclusion ℒ⟨α, β, γ⟩ ⊆ ℒ⟨ξ, η⟩. First we define special fields on pKdV as follows. Introduce the notation u_{1} = u_{x} + a, u_{2} = u_{xx}, u_{3} = u_{xxx}, etc. A monomial is a product of the form a = cu_{i_{1}} ⋯ u_{i_{k}}, where c is a real constant and 1 ≤ i_{1} ≤ … ≤ i_{k}, 1 ≤ k. For the given monomial we introduce its degree as deg a := k, and its order as ord a := i_{1} + ⋯ + i_{k}. A homogeneous polynomial of degree k and order s is a sum of any number of monomials of the degree k and the order s. The linear space of all homogeneous polynomials of degree k and order s will be denoted by P_{k,s}, 1 ≤ k ≤ s. We also put P_{0,0} = R, and P_{0,s} = 0 for s ≠ 0.

A special field of order r ≥ 0 is any vector field ϕ on E of the form

ϕ = ∑_{0≤i≤j} f_{i,j} ∂_{r+i+j},  (22)

f_{i,j} ∈ P_{i,j}. We also write ord ϕ = r. Observe that f_{0,0} is a constant.

(D) The fields α, β, γ on E are special. Their order r and coefficients f_{0,0} and f_{1,j} are given in the following table:

r  f_{0,0}  f_{1,j}, j ≥ 1
α  0  1  −2u_{j}
β  1  1  −2(j + 1)u_{j}
γ  2  1  −j(j + 3)u_{j}

This follows by induction from (21) and obvious rules D_{x}P_{k,s} ⊆ P_{k,s+1}, (u_{x} + a)P_{k,s} ⊆ P_{k+1,s+1}. The straightforward computations are omitted.

(E) Let ϕ = ∑_{0≤i≤j} f_{i,j} ∂_{r+i+j}, ψ = ∑_{0≤i≤j} g_{k,l} ∂_{s+k+l} be two special fields of orders r, s, respectively. Then their Lie bracket χ = [ϕ, ψ] is a special field of order r + s + 1: χ = ∑_{0≤p≤q} h_{p,q} ∂_{r+s+1+p+q}. Moreover,

h_{0,0} = f_{0,0} ∂_{r}g_{1,r} − g_{0,0} ∂_{s}f_{1,s}.  (23)

Indeed, h_{p,q} = ∑_{i,j} f_{i,j} ∂_{r+i+j}g_{p−i+1,q+r+i} − ∑_{k,l} g_{k,l} ∂_{s+k+l}f_{p−k+1,q+s+k}, by straightforward computation and simple rules ∂_{0}f_{i,j} = 0 for all i, j, ∂_{r}f_{i,j} = 0 for i+j ≤ r, ∂_{r}f_{i,j} ∈ P_{i−1,j−r} otherwise, and P_{i,j} · P_{k,l} ⊆ P_{i+k,j+l}.

(F) Define ψ^{(0)} = α, ψ^{(1)} = β, ψ^{(s+1)} = [α, ψ^{(s)}] for s ≥ 1. Then the fields ψ^{(s)} ∈ ℒ⟨α, β⟩ are C^{∞} independent.

All fields ψ^{(s)} are special, ψ^{(s)} = ∑_{0≤i≤j} g_{k,l}^{(s)} ∂_{s+k+l}, and (23) gives g_{0,0}^{(s)} = f_{0,0} ∂_{0}g_{1,0}^{(s−1)} − g_{0,0}^{(s−1)} ∂_{s}f_{1,s} = 2g_{0,0}^{(s−1)} since, obviously, ∂_{0}g_{1,0}^{(s−1)} = 0 and ∂_{s}f_{1,s} = 2. From g_{0,0}^{(1)} = 1 we conclude that the “leading coefficients” g_{0,0}^{(s)} are nonzero constants. Together with the fact that orders ord ϕ^{(s)} = s are strictly increasing this implies the statement.

It immediately follows that rk ℒ⟨α, β⟩ = ∞. The normality of the BT Ψ is thereby established.

4.1.3 A ternary superposition principle

Following the standard procedure we first compute the composition Φ_{b} ∘ Φ_{a} for Φ_{b} given analogously by the equation (15) above and additional equations

w_{x} = −v_{x} − \frac{1}{2}(v − w)^{2} − 2b,  (24)
w_{t} = −w_{xxx} − 3w_{x}^{2}.  (25)

Recall that w is Φ_{b} ∘Φ_{a}-related to u iff there exists v such that uΦ_{a}v and vΦ_{b}w. For fixed u, w this amounts to an overdetermined system of three equations (14), (15), (24) on a unique unknown v.

Case u ≠ w: From (14), (24) we get a linear algebraic equation on v with a solution

v = \frac{u_{x} − w_{x} + \frac{1}{2}u^{2} − \frac{1}{2}w^{2} + 2a − 2b}{u − w}.  (26)

This is the ternary superposition formula for the pKdV equation. Its domain, i.e., E_{Φ_{a}∘Φ_{b}} is found when substituting v from formula (26) into (14). The resulting condition is

\begin{matrix}(u_{xx} − w_{xx})(u − w) + \frac{1}{2}(u_{x} − w_{x})^{2} − \frac{1}{8}(u − w)^{4} \\ = (u_{x} + w_{x} + a + b)(u − w)^{2} + 2(a − b)^{2}. \end{matrix}  (27)

It is easy to check that then (24), (15) hold as well. Summing up, whenever u ≠ w satisfy (27) then the point v exists and is unique, given by (26).

Case u = w: When u = w, equations (14), (24) are compatible iff a = b and then both are the same, which just expresses the already known fact that Φ_{a} is symmetric. We conclude that for a ≠ b the case u = w is void, while for a = b we have an exceptional case in the sense of Sect. 3.

Equation (27) is obviously symmetric in a, b. Therefore, the two BR’s commute: Φ_{a} ∘Φ_{b} = Φ_{b} ∘Φ_{a}. Thus, for a ≠ b we have all the ingredients of permutability. The traditional quaternary superposition formula for pKdV is derived when writing down a copy v′ = (u_{x} − w_{x} + \frac{1}{2}u^{2} − \frac{1}{2}w^{2} + 2b − 2a)/(u − w) of equation (26) with a, b interchanged and subtracting it from (26) to get (v − v′)(u − w) = 4(a − b).

4.2 The sine-Gordon equation

We consider the sine-Gordon equation u_{tx} = sin u along with a one-parameter family Φ = Φ_{a}, a ≠ 0, of BT’s

v_{x} = −u_{x} + 2a sin \frac{u − v}{2} ,  v_{t} = u_{t} + \frac{2}{a} sin \frac{u + v}{2} .  (28)

A simple computation, which is omitted, shows that Φ_{a}^{−1} = Φ_{−a}.

4.2.1 Effectivity

To establish the effectivity of the BT Φ we consider the corresponding pair of coverings p, q : Ẽ → E. On Ẽ we distinguish two coordinate systems:

(a) t, x, v, u, u_{x}, u_{xx}, u_{xxx}, …, u_{t}, u_{tt}, u_{ttt}, …,

(b) t, x, u, v, v_{x}, v_{xx}, v_{xxx}, …, v_{t}, v_{tt}, v_{ttt}, . . ..

We put ξ = ∂/∂v in the coordinate system (a), η = ∂/∂u in the coordinate system (b). Applying the coordinate transformation from (b) to (a) induced by the relations (28) we get first several terms

η = \frac{∂}{∂u} + a cos \frac{u − v}{2} \frac{∂}{∂u_{x}} + (a^{2} − au_{x} sin \frac{u − v}{2} )\frac{∂}{∂u_{xx}} + ⋯
+ \frac{1}{a} cos \frac{u + v}{2} \frac{∂}{∂u_{t}} + (\frac{1}{a^{2}} − \frac{1}{a}u_{t} sin \frac{u + v}{2} )\frac{∂}{∂u_{tt}} + ⋯

(29)

in the coordinate system (a). We see that ξ, η are manifestly independent, hence the effectivity of the BT follows.

4.2.2 Normality

To simplify notation, we put ∂_{0} = ∂/∂u, ∂_{1} = ∂/∂u_{x}, ∂_{2} = ∂/∂u_{xx}, …, ∂′_{1} = ∂/∂u_{t}, ∂′_{2} = ∂/∂u_{tt}, . . .. The proof takes several steps, mostly analogous to those already exposed for the pKdV equation.

(A) Let us write η = ∑_{i≥0} η_{i} ∂_{i} + ∑_{i>0} η′_{i} ∂′_{i}. Then η_{0} = 1 and (assuming η′_{0} = η_{0})

η_{k+1} = D_{x}η_{k} + a cos \frac{u − v}{2} · η_{k}, η′_{k+1} = D_{t}η′_{k} + \frac{1}{a} cos \frac{u + v}{2} · η′_{k}. (30)

This follows from the relations (4),

[D_{x}, η] = −a cos \frac{u − v}{2} · η, [D_{t}, η] = −\frac{1}{a} cos \frac{u + v}{2} · η,  (31)

which one easily finds with the aid of the expression (29). The one-form (8) is the nonlocal conservation law −a cos \frac{1}{2}(u − v) dx − a^{−1} cos \frac{1}{2}(u + v) dt.

(B) There exist fields α, β, γ, local on E and satisfying

η = α + β cos \frac{1}{2}v + γ sin \frac{1}{2}v  (32)

(supposing that fields α, β, γ are lifted to E_{Φ} by the triviality of the fibering Ẽ → E). The relation (4) with respect to D_{x} is found from (32) and (31):

[D_{x}, ( \begin{matrix}α \\ β \\ γ \end{matrix} )] = ( \begin{matrix}0 & −a cos \frac{1}{2}u & −a sin \frac{1}{2}u \\ −a cos \frac{1}{2}u & 0 & \frac{1}{2}u_{x} \\ −a sin \frac{1}{2}u & −\frac{1}{2}u_{x} & 0 \end{matrix} ) ( \begin{matrix}α \\ β \\ γ \end{matrix} )  (33)

(C) It holds ℒ⟨α, β, γ⟩ ⊆ ℒ⟨ξ, η⟩.

Indeed, using the fact that ξ commutes with α, β, γ, one easily obtains from (32) the formulas β = −2 sin \frac{1}{2}v ·[ξ, η]−4 cos \frac{1}{2}v ·[ξ, [ξ, η]], γ = 2 cos \frac{1}{2}v ·[ξ, η]−4 sin \frac{1}{2}v · [ξ, [ξ, η]], α = η − β cos \frac{1}{2}v − γ sin \frac{1}{2}v.

Continuing with fields α, β, γ would lead us into troubles in step (E) below, due basically to the occurrence of the terms cos \frac{1}{2}u, sin \frac{1}{2}u in (33). However, in the xparts these terms can be removed. Let us introduce a mapping ζ ↦ ζ^{x}, applied to fields ζ = ∑_{i≥0} ζ_{i} ∂_{i} + ∑_{i>0} ζ′_{i} ∂′_{i} on E such that ζ_{i} does not depend on u_{t}, u_{tt}, u_{ttt}, etc. in the coordinate system (a). Namely, we put ζ^{x} = ∑_{i≥0} ζ_{i} ∂_{i}. Obviously, ζ ↦ ζ^{x} is a Lie algebra homomorphism, so that ℒ⟨ξ, η^{x}⟩ = ℒ⟨ξ^{x}, η^{x}⟩ ⊆ ℒ⟨ξ, η⟩^{x}. Hence, to prove the normality it is sufficient to verify that rk ℒ⟨α^{x}, β^{x}, γ^{x}⟩ = ∞.

(C^{∗}) There exist fields α^{∗}, β^{∗}, γ^{∗} on E such that ℒ⟨α^{∗}, β^{∗}, γ^{∗}⟩ = ℒ⟨α^{x}, β^{x}, γ^{x}⟩, and

[D_{x}, ( \begin{matrix}α^{∗} \\ β^{∗} \\ γ^{∗} \end{matrix} )] = ( \begin{matrix}0 & −a & 0 \\ −a & 0 & u_{x} \\ 0 & −u_{x} & 0 \end{matrix} ) ( \begin{matrix}α^{∗} \\ β^{∗} \\ γ^{∗} \end{matrix} ) .  (34)

Indeed, it is sufficient to put α^{∗} = α^{x}, β^{∗} = cos \frac{1}{2}u · β^{x} + sin \frac{1}{2}u · γ^{x}, γ^{∗} = − sin \frac{1}{2}u · β^{x} + cos \frac{1}{2}u · γ^{x}. Recurrence relations for the fields α^{∗}, β^{∗}, γ^{∗}, readily found from eq. (34), are omitted.

Special fields on E are defined as follows. Introduce the notation u_{1} = u_{x}, u_{2} = u_{xx}, u_{3} = u_{xxx}, etc. A monomial, its degree, order and a homogeneous polynomial as well as the linear space P_{k,s} of all homogeneous polynomials of degree k and order s are defined word for word as in the pKdV case. A special field of even type and order r ≥ 0 is any vector field ϕ on E of the form

ϕ = ∑_{0≤2i≤j} a^{r+2i}f_{2i,j} ∂_{r+2i+j},  (35)

where f_{2i,j} ∈ P_{2i,j}. A special field of odd type and order r ≥ 0 is any vector field ϕ on E of the form

ϕ = ∑_{1≤2i+1≤j} a^{r+2i+1}f_{2i+1,j} ∂_{r+2i+j+1},  (36)

where f_{2i+1,j} ∈ P_{2i+1,j}. We also write ord ϕ = r.

(D) Fields α^{∗} − ∂/∂u, β^{∗}, γ^{∗} are special. Their type, order r and coefficients f_{0,0} and f_{1,j} are given in the following table:

type  r  f_{0,0}  f_{1,j}, j ≥ 1
α^{∗} − ∂/∂u  even  2  1  —
β^{∗}  even  1  1  —
γ^{∗}  odd  0  —  u_{j}

(E) Let ϕ = ∑_{1≤2i+1≤j} a^{r+2i+1}f_{2i+1,j} ∂_{r+2i+1+j} be a special field of odd type and order r, let ψ = ∑_{0≤2k≤l} a^{s+2k}g_{2k,l} ∂_{s+2k+l} be a special field of even type and order s. Then their Lie bracket χ = [ϕ, ψ] is a special field of even type and order r + s + 1: χ = ∑_{0≤2p≤q} h_{2p,q} ∂_{r+s+1+2p+q}. Moreover,

h_{0,0} = −g_{0,0} ∂_{s}f_{1,s}.  (37)

Indeed, this is proved by straightforward computation, giving h_{2p,q} = ∑_{i,j} f_{2i+1,j} ∂_{r+2i+1+j}g_{2p−2,q+r+2i+1} − ∑_{k,l} g_{2k,l} ∂_{s+2k+l}f_{2p−2k+1,q+s+2k}.

(F) Define ψ^{(1)} = β^{∗}, ψ^{(s+1)} = [γ^{∗}, ψ^{(s)}] for s ≥ 1. Then the fields ψ^{(s)} ∈ ℒ⟨β^{∗}, γ^{∗}⟩ are C^{∞} independent.

To prove this, first observe that by (E) all fields ψ^{(s)} are special of even type and order s, so that ψ^{(s)} = ∑_{0≤2p≤q} g_{2p,q}^{(s)} ∂_{s+2p+q} and, moreover, g_{0,0}^{(s)} = −g_{0,0} ∂_{s}f_{1,s−1}^{(s−1)} = −(−1)^{s} by (37). Hence, all leading coefficients g_{0,0}^{(s)} are nonzero constants, which together with the fact that orders ord ψ^{(s)} = s are strictly increasing implies the statement.

The normality of the BT Φ is thus established.

4.3 Tzitzéica equation

In 1910 Tzitzéica [16] published a BT for equation q_{tx} = e^{q} − e^{−2q}, which he wrote as

(ln u)_{tx} = u − u^{−2}.  (38)

The transformation was given by a system

p_{xx} = \frac{u_{x}}{u} p_{x} − \frac{a}{u}p_{t}, p_{tx} = up, p_{tt} = \frac{u_{t}}{u} p_{t} − \frac{1}{au}p_{x}.  (39)

along with v = −u + 2p_{x}p_{t}/p^{2}. Here a stands for the Bäcklund parameter. If u solves (38) and p solves (39) then v solves (38) as well. The two coverings over equation (38) constituting this BT are three-dimensional. Natural nonlocal variables are p, p_{x}, p_{t}.

4.3.1 Reduction

The BT (39) is not effective. Indeed, it can be shown that the field ψ = p ∂/∂p + p_{x} ∂/∂p_{x} + p_{t} ∂/∂p_{t} is bivertical. Since non-effectivity prevents normality, let us look for an appropriate reduction. The reduction is achieved when passing to the quotient with respect to the pseudosymmetry ψ. This amounts to writing the equation in terms of invariants of the field ψ. Choosing r = p_{x}/p, s = p_{t}/p as a basis of invariants we transform (39) into

r_{x} = \frac{u_{x}}{u} r − \frac{a}{u}s − r^{2}, r_{t} = s_{x} = u − rs, s_{t} = \frac{u_{x}}{u} s − \frac{1}{au}r − s^{2} (40)

along with v = −u + 2rs. Here r, s are nonlocal variables of a two-dimensional covering. Denote this BT by Φ = Φ_{a}. Again, by routine computation we can verify that Φ_{1}^{−1} = Φ_{−a} (see also remarks in Sect. 4.3.4).

4.3.2 Effectivity

Consider two coordinate systems on Ẽ:

(a) r, s, u, u_{x}, u_{xx}, u_{xxx}, …, u_{t}, u_{tt}, u_{ttt}, …,

(b) r, s, v, v_{x}, v_{xx}, v_{xxx}, …, v_{t}, v_{tt}, v_{ttt}, . . ..

Vector fields written in coordinates (a) as ξ = ∂/∂r, χ = ∂/∂s generate Ξ^{p}. Fields, written in the alternative coordinates (b) as ρ = ∂/∂r, σ = ∂/∂s generate Ξ^{q}. Transformed to the coordinate system (a) they are

\begin{aligned}ρ = \frac{∂}{∂r} + 2s\frac{∂}{∂u} & + & 2(\frac{su_{x}}{u} + u + \frac{2as^{3}}{(u − 2rs)u})\frac{∂}{∂u_{x}} + ⋯ \\ & + & 2(\frac{su_{t}}{u} − \frac{r}{au} − \frac{r}{a(u − 2rs)})\frac{∂}{∂u_{t}} + ⋯, \end{aligned}  (41)

and

\begin{aligned}σ = \frac{∂}{∂s} + 2r\frac{∂}{∂u} & + & 2(\frac{ru_{x}}{u} − \frac{as}{u} − \frac{as}{u − 2rs})\frac{∂}{∂u_{x}} + ⋯ \\ & + & 2(\frac{ru_{t}}{u} + u + \frac{2r^{3}}{a(u − 2rs)u})\frac{∂}{∂u_{t}} + · · · . \end{aligned}  (42)

Obviously, the four fields ξ, χ, ρ, σ are C^{∞} independent. The effectivity of Φ is thus established.

We see that the generating power of this BT’ is two, contrary to the opinion expressed by Tzitzéica in [16]. Tzitzéica assumed the generating power three (four, considering the Bäcklund parameter).

4.3.3 Normality

In many respects the situation is similar to the case of the sine-Gordon equation. An important difference is that now we have two-dimensional coverings. In this case, normality means that ℒ⟨ξ, χ, fρ + gσ⟩ is of infinite rank for any smooth functions f, g on Ẽ. Many details are omitted below. We often do not present explicit results of routine symbolic computations if the conclusions we make are easy.

Denoting ∂_{0} = ∂/∂u, ∂_{1} = ∂/∂u_{x}, ∂_{2} = ∂/∂u_{xx}, …, ∂′_{1} = ∂/∂u_{t}, ∂′_{2} = ∂/∂u_{tt}, …, let us write ρ = ∂/∂r+∑_{i≥0} ρ_{i} ∂_{i}+∑_{i>0} ρ′_{i} ∂′_{i}, σ = ∂/∂s+∑_{i≥0} σ_{i} ∂_{i}+∑_{i>0} σ′_{i} ∂′_{i}.

(A) It holds ρ_{0} = 2s, σ_{0} = 2r, and (assuming ρ′_{0} = ρ_{0}, σ′_{0} = σ_{0})

\begin{aligned}ρ_{k+1} & = & D_{x}ρ_{k} + (\frac{u_{x}}{u} + \frac{2as^{2}}{(u − 2rs)u})ρ_{k} + sσ_{k}, \\ σ_{k+1} & = & D_{x}σ_{k} − \frac{a}{u − 2rs}ρ_{k} + rσ_{k}, \\ ρ′_{k+1} & = & D_{t}ρ′_{k} + sρ′_{k} − \frac{1}{a(u − 2rs)}σ′_{k}, \\ σ′_{k+1} & = & D_{t}σ′_{k} + rρ′_{k} + (\frac{u_{t}}{u} + \frac{2r^{2}}{a(u − 2rs)u})σ′_{k}. \end{aligned}  (43)

Again, commutation relations underlying formulas (43) are easily found by explicit computation with truncated expansions (41), (42).

(B) There exist fields α, β, γ, δ, ϵ, ζ, η, θ, local on E and satisfying

\begin{aligned}ρ = ξ & + \frac{2s}{u}α + 2β + \frac{4as^{3}}{(u − 2rs)u^{2}} δ − 2ar(1 + \frac{u}{u − 2rs})ϵ \\ & + \frac{12as^{2}}{(u − 2rs)u}ζ − \frac{6au}{u − 2rs}η + \frac{18as}{u − 2rs}θ, \end{aligned}  (44)

\begin{aligned}σ = χ & + \frac{2r}{u}α + \frac{2a^{2}}{u} γ + \frac{4ar^{3}}{u − 2rs}ϵ − \frac{2as}{u^{2}}(1 + \frac{u}{u − 2rs})δ \\ & + \frac{12ar^{2}}{u − 2rs}η − \frac{6a}{u − 2rs}ζ − \frac{18ar}{u − 2rs}θ \end{aligned}  (45)

(supposing that fields α, β, γ, δ, ϵ, ζ, η, θ are lifted to E_{Φ} by the triviality of the fibering E_{Φ} → E).

Determining formulas for these fields are derived from (43) to be

\begin{matrix}α_{k+1} = D_{x}α_{k} + a^{2}γ_{k} + 2a^{2}ϵ_{k}, \\ β_{k+1} = D_{x}β_{k} + α_{k} + \frac{u_{x}}{u} β_{k}, \\ γ_{k+1} = D_{x}γ_{k} − \frac{u_{x}}{u} γ_{k} + 3η_{k}, \\ δ_{k+1} = D_{x}δ_{k} + α_{k} − 2\frac{u_{x}}{u} δ_{k}, \\ ϵ_{k+1} = D_{x}ϵ_{k} + 2\frac{u_{x}}{u} ϵ_{k} − 3η_{k}, \\ ζ_{k+1} = D_{x}ζ_{k} + \frac{1}{3}β_{k} + \frac{2}{3}δ_{k} − \frac{u_{x}}{u} ζ_{k}, \\ η_{k+1} = D_{x}η_{k} + \frac{u_{x}}{u} η_{k} − 3θ_{k}, \\ θ_{k+1} = D_{x}θ_{k} + ζ_{k}, \end{matrix} \begin{matrix}α′_{k+1} = D_{t}α′_{k} + uβ′_{k} + \frac{2}{u^{2}} δ′_{k}, \\ β′_{k+1} = D_{t}β′_{k} + \frac{3}{u^{2}} ζ′_{k}, \\ γ′_{k+1} = D_{t}γ′_{k} + \frac{u}{a^{2}} α′_{k}, \\ δ′_{k+1} = D_{t}δ′_{k} + 3uζ′_{k}, \\ ϵ′_{k+1} = D_{t}ϵ′_{k} + \frac{1}{a^{2}u^{2}} α′_{k}, \\ ζ′_{k+1} = D_{t}ζ′_{k} + 3uθ′_{k}, \\ η′_{k+1} = D_{t}η′_{k} + \frac{1}{3u^{2}} γ′_{k} + \frac{2u}{3}ϵ′_{k}, \\ θ′_{k+1} = D_{t}θ′_{k} − uη′_{k}. \end{matrix}

(C) It holds ℒ⟨α, fβ, γ, δ, ϵ, ζ, η, θ⟩ ⊆ ℒ⟨ξ, χ, fρ + gσ⟩.

Indeed, compute, using expressions (44) and (45), Lie brackets [ξ, ω], [χ, ω], [ξ, [ξ, ω]], [ξ, [χ, ω]], [χ, [χ, ω]], [ξ, [ξ, [ξ, ω]]], [ξ, [ξ, [χ, ω]]], [ξ, [χ, [χ, ω]]], [χ, [χ, [χ, ω]]], where ω = fρ + gσ. From the resulting system it is easily seen that

α, fβ + gγ, δ, ϵ, ζ, η, θ ∈ ℒ⟨ξ, χ, fρ + gσ⟩.  (46)

But, γ = 3[δ, η]. Indeed, 0 = [ρ, σ], which is obvious in the coordinate system (b). Substituting from formulas (44) and (45) we get 0 = F (r, s) where F is a rational expression in r, s, whose coefficients involve fields α, …, θ and their brackets. Equating the coefficients to zero we get, among others, 0 = (3[δ, η] − γ)a^{2} + (−3ζ + 3[α, ζ] − [β, δ])a. Then from recursion formulas obtained in step (B) we conclude that expressions for fields α, …, θ may contain only even powers of a. Hence, both expressions −3ζ + 3[α, ζ] − [β, δ] and 3[δ, η] − γ are zero. Our assertion follows from (46) now.

(C^{∗}) Passing to the x-part exactly as in the sine-Gordon case we define fields α^{∗}, β^{∗}, γ^{∗}, δ^{∗}, ϵ^{∗}, ζ^{∗}, η^{∗}, θ^{∗} not containing terms ∂′_{i} with t-derivatives. Thus, it is sufficient to check that rk ℒ⟨α^{∗}, fβ^{∗}, γ^{∗}, …, θ^{∗}⟩ = ∞.

Introduce the notation u_{1} = u_{x}/u, u_{2} = u_{xx}/u, u_{3} = u_{xxx}/u, etc. A monomial, its degree, order and the sets P_{i,j} of homogeneous polynomials are defined word for word as in the pKdV case.

A special field of order r ≥ 0 is any vector field ϕ on E of the form

ϕ = u ∑_{0≤m} ∑_{0≤i≤j} a^{2m}f_{i,j} ∂_{r+j+6m},  (47)

where f_{i,j} ∈ P_{i,j}. We also write ord ϕ = r.

(D) Fields α^{∗}, …, θ^{∗} as well as β^{∗} − δ^{∗}, γ^{∗} − ϵ^{∗} are special. Their order r and coefficients f_{0,0} and f_{1,j} are given in the following table:

r f_{0,0}  f_{1,j}, j ≥ 1
α^{∗}  0  1  u_{j}
β^{∗}  1  1  (j + 2)u_{j}
γ^{∗}  5 −9  −\frac{3}{40}(j^{5} + 10j^{4} + 35j^{3} + 110j^{2} + 324j)u_{j}
δ^{∗}  1  1  ju_{j}
ϵ^{∗}  5 −9 −\frac{3}{40}(j^{5} + 10j^{4} + 35j^{3} + 110j^{2} + 324j + 360)u_{j}
ζ^{∗}  2  1  \frac{1}{2}(j^{2} + j)u_{j}
η^{∗}  4 −3  −\frac{1}{8}(j^{4} + 6j^{3} + 11j^{2} + 30j + 48)u_{j}
θ^{∗}  3  1  \frac{1}{6}(j^{3} + 3j^{2} + 2j + 6)u_{j}
β^{∗} − δ^{∗} 1  0  2u_{j}
γ^{∗} − ϵ^{∗} 5  0  27u_{j}

(E) Consider two special fields, ϕ = u ∑_{0≤m} ∑_{0≤i≤j} a^{2m}f_{i,j} ∂_{r+j+6m} and ψ = u ∑_{0≤n} ∑_{0≤k≤l} a^{2n}f_{k,l} ∂_{s+k+6n}, of orders r and s, respectively. Let r, s > 0. Then the Lie bracket χ = [ϕ, ψ] is a special field of order r + s: χ = u ∑_{0≤n} ∑_{0≤p≤q} a^{2n}h_{p,q} ∂_{r+s+q+6n}, Moreover,

h_{0,0} = f_{0,0} ∂_{r}g_{1,r} − g_{0,0} ∂_{s}f_{1,s}.  (48)

(F) Put ϕ^{(0)} = δ^{∗}, ϕ^{(k+1)} = [γ^{∗} − ϵ^{∗}, ϕ^{(k)}] for k ≥ 0. Then fields ϕ^{(k)} ∈ ℒ⟨γ^{∗}, δ^{∗}, ϵ^{∗}⟩ are C^{∞} independent.

By (E) the bracket [γ^{∗} − ϵ^{∗}, ϕ^{(k)}] is a special field of order 5k + 1 and leading term h_{0,0} = −g_{0,0} ∂_{s}f_{1,s} = (−2)^{k} (since f_{0,0} = 0). Hence, ℒ⟨γ^{∗}, δ^{∗}, ϵ^{∗}⟩ is of infinite rank. The normality of the Tzitzéica equation is thus proved.

4.3.4 Remarks

(1) Tzitzéica claimed that a nonlinear superposition law for his equation exists, but presented no explicit formula. Author’s attempts to find one were unsuccessful, although it was quite easy to write down a weak ternary superposition formula in the form of a large system of polynomial equations determining all the nonlocal variables of the composition Φ_{a} ∘ Φ_{b}. The Tzitzéica equation provides an example where, to author’s best knowledge, the quaternary formula has not been explicitly written yet.

(2) It is still possible to present BR Φ as a subdiffiety in the product E ×_{m} E, determined by a single equation

0 = − 2a^{2}u^{2}v^{2}(vu_{t} − uv_{t})^{3} + a(vu_{t} − uv_{t})^{2}(vu_{x} − uv_{x})^{2}
+ a(u^{2}v^{2} + u + v)(2u^{2}v^{2} − u − v)(vu_{t} − uv_{t})(vu_{x} − uv_{x})
− a(u + v)(2u^{2}v^{2} − u − v)^{3} − u^{2}v^{2}(vu_{x} − uv_{x})^{3}.

It is easily verified that the last equation is invariant under the change u ↔ v, a ↔ −a. The relation Φ_{a}^{−1} = Φ_{−a} then follows.

(3) The above computational results were obtained using procedures developed by the author over the computer algebra system MAPLE.

(4) One is naturally led to a conjecture that all completely integrable equations are normal.

Acknowledgements

The author is indebted to I.S. Krasil’shchik for encouragement, continuous attention to the work, reading the manuscript, advice and critisizm. Thanks are also due to the Supercomputing Centre of the Masaryk University, Brno, for kind permission to use their computational equipment.

The support by grant No. 201/96/0845 of the Czech Grant Agency is gratefully acknowledged.

References

[1] Anderson, R.L. and Fokas, A.S.: Group theoretical nature of Bäcklund transformations, Lett. Math. Phys. 3 (1979) 117–126.

[2] Bianchi, L.: Lezioni di Geometria Differenziale Vol. I, Part II (Zanichelli, Bologna, 1927).

[3] Bernstein, I.N. and Rosenfel’d, B.I.: Homogeneous spaces of infinite dimensional Lie algebras and characteristic classes of foliations, Uspekhi Mat. Nauk 28 (1973) (4) 103–138 (in Russian).

[4] Hirota, R. and Satsuma, J.: A simple structure of superposition formula of the Bäcklund transformation, J. Phys. Soc. Japan 45 (1978) 1741–1750.

[5] Krasil’shchik, I.S.: Notes on coverings and Bäcklund transformations, Preprint ESI 260, Vienna, 1995 (http://www.esi.ac.at).

[6] Krasil’shchik, I.S. and Vinogradov, A.M.: Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations, Acta Applicandae Mathematicae 15 (1989) 161–209.

[7] Krasil’shchik, I.S., Lychagin, V.V. and Vinogradov, A.M.: Geometry of Jet Spaces and Nonlinear Differential Equations (Gordon and Breach, New York, 1986).

[8] Lamb jr., G.L.: Bäcklund transformations at the turn of the century, in: R.M. Miura, ed., Bäcklund Transformations, Proc. Conf. Nashville, Tennessee 1974, Lecture Notes in Mathematics 515 (Springer, Berlin et al., 1975) 69–79.

[9] Marvan, M.: On the 𝒞-spectral sequence with “general” coefficients, in: Differential Geometry and its Applications, Proc. Conf. Brno, 1989 (World Scientific, Singapore, 1990) 361–371.

[10] Marvan, M.: Another look on recursion operators, in: Differential Geometry and its Applications, Proc. Conf. Brno, 1995 (Masaryk University, Brno, 1996) 393–402 (http://www.emis.de/proceedings/).

[11] Miura, R.M.: Introduction, in: R.M. Miura, ed., Bäcklund Transformations, Proc. Conf. Nashville, Tennessee 1974, Lecture Notes in Mathematics 515 (Springer, Berlin et al., 1975).

[12] Pommaret, J.-F.: Problème de Bäcklund géneralisé, C.R. Acad. Sci. Paris 289 (1979) 119–122.

[13] C. Rogers and W.F. Shadwick, Bäcklund Transformations and Their Applications (Academic Press, New York et al., 1982).

[14] Saunders, D.J.: The Geometry of Jet Bundles, London Math. Soc. Lecture Notes Ser. 142 (Cambridge Univ. Press, Cambridge, 1989).

[15] Sokolov, V.V.: Pseudosymmetries and differential substitutions, Funkc. Anal. Prilozh. 22 (1988) 47–56 (in Russian).

[16] Tzitzéica, M.: Sur une nouvelle classe de surface, C.R. Acad. Sci. Paris 150 (1910) 955–956, 1227–1229.

[17] Vinogradov, A.M.: Category of partial differential equations, in: Lecture Notes in Math. 1108 (Springer, Berlin, 1984) 77–102.

[18] Wahlquist, H.D. and Estabrook, F.B.: Bäcklund transformation for solutions of the Korteweg–de Vries equation, Phys. Rev. Lett. 31 (1973) 1386–1390.

[19] Wahlquist, H.D. and Estabrook, F.B.: Prolongation structures and nonlinear evolution equations I, II, J. Math. Phys. 16 (1975) 1–7; 17 (1976) 1293–1297

[20] Zharinov, V.V.: On Bäcklund correspondences, Matem. Sbornik 136 (178) (1988) 274–291 (in Russian). English translation Math. USSR Sbornik 64 (1989) 277–293.