The Diffiety Institute Preprint Series

Preprint DIPS–3/2000 November 5, 2000

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

The Diffiety Institute

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

Complete integrability of the coupled KdV–mKdV system

by

Paul Kersten, Joseph Krasil′shchik

arXiv nlin.SI/0010041

Complete integrability of the coupled KdV–mKdV system

Paul Kersten, Joseph Krasil′shchik

Abstract. The coupled KdV–mKdV system arises as the classical part of one of superextensions of the KdV equation. For this system, we prove its complete integrability, i.e., existence of a recursion operator and of infinite series of symmetries.

Introduction

There are several supersymmetric extensions of the classical Korteweg–de Vries equation (KdV) [5, 8, 9]. One of them is of the form (the so-called N = 2, A = 1 extension [3])

u_{t} = −u_{3} + 6uu_{1} − 3φφ_{2} − 3ψψ_{2} − 3ww_{3} − 3w_{1}w_{2} + 3u_{1}w^{2} + 6uww_{1}

+ 6ψφ_{1}w − 6φψ_{1}w − 6φψw_{1},

φ_{t} = −φ_{3} + 3φu_{1} + 3φ_{1}u − 3ψ_{2}w − 3ψ_{1}w_{1} + 3φ_{1}w^{2} + 6φww_{1},
ψ_{t} = −ψ_{3} + 3ψu_{1} + 3ψ_{1}u + 3φ_{2}w + 3φ_{1}w_{1} + 3ψ_{1}w^{2} + 6ψww_{1},
w_{t} = −w_{3} + 3w^{2}w_{1} + 3uw_{1} + 3u_{1}w,

where u and w are classical (even) independent variables while φ and ψ are odd ones (here and below the numerical subscript at an unknown variable denotes it derivative over x of the corresponding order). Being completely integrable itself, this system gives rise to an interesting system of even equations

u_{t} = −u_{3} + 6uu_{1} − 3ww_{3} − 3w_{1}w_{2} + 3u_{1}w^{2} + 6uww_{1},
w_{t} = −w_{3} + 3w^{2}w_{1} + 3uw_{1} + 3u_{1}w,  (1)

which can be considered as a sort coupling between the KdV (with respect to u) and the modified KdV (with respect to w) equations. In fact, setting w = 0, we obtain

u_{t} = −u_{3} + 6uu_{1},

while for u = 0 we have

w_{t} = −w_{3} + 3w^{2}w_{1}.

1991 Mathematics Subject Classification. 58F07, 58G37, 58H15, 58F37.

Key words and phrases. Coupled KdV–mKdV system, complete integrability, recursion operators, symmetries, conservation laws, coverings, deformations, superdifferential equations.

Partially supported by the INTAS grant 96-0793.

2 Paul Kersten, Joseph Krasil′shchik

In what follows, we prove complete integrability, cf. [2], of system (1) by establishing existence of infinite series of symmetries and/or conservation laws. Toward this end we construct a recursion operator using the techniques of deformation theory introduced in [4] and extensively described and exemplified in [5].

In the first section of the paper the theoretical background is introduced. The second section deals with particular computations and description of basic results.

1. Geometrical and algebraic background

Here we briefly describe the geometrical theory of partial differential equations [1, 6] and algebraic foundations of computational approach to recursion operators [4, 5].

Let π: E → M be a locally trivial vector bundle and π_{k}: J^{k}(π) → M, k = 0, 1, …, ∞, be the bundles of its k-jets. A (nonlinear) partial differential equation (PDE) of order k is a submanifold E ⊂ J^{k}(π), k < ∞. Its lth prolongation is a subset E^{l} ⊂ J^{k+l}(π). There exist natural mappings π_{k+l+1,k+l}: E^{l+1} → E^{l}, and E is said to be formally integrable, if all E^{l} are smooth manifolds while π_{k+l+1,k+l} are smooth fiber bundles. Below, only formally integrable equations are considered.

The inverse limit E^{∞} ⊂ J^{∞}(π) of the system {E^{l}, π_{k+l,k+l−1}} is called the infinite prolongation of E and we consider the bundle π_{E}: E^{∞} → M. This bundle enjoys the following characteristic property:

Proposition 1 (see [6]). Let π_{i}: E_{i} → M, i = 1, 2, be two locally trivial vector bundles and Δ: Γ(π_{1}) → Γ(π_{2}) be a linear differential operator acting from sections of π_{1} to sections of π_{2}. Then there exists a unique differential operator CΔ: Γ(π_{E}^{∗}(π_{1})) → Γ(π_{E}^{∗}(π_{2})) such that

j_{∞}(s)^{∗} ∘ CΔ = Δ ∘ j_{∞}(s)^{∗}  (2)

for any formal solution s of the equation E. The correspondence Δ ↦ CΔ is C^{∞}(E^{∞})-linear and complies with the composition of differential operators:

C(Δ_{2} ∘ Δ_{1}) = CΔ_{2} ∘ CΔ_{1},  (3)

where Δ_{1} : Γ(π_{1}) → Γ(π_{2}), Δ_{2} : Γ(π_{2}) → Γ(π_{3}) are linear differential operators, π_{3} : E_{3} → M being a third vector bundle.

As a corollary of Proposition 1, we get

Proposition 2. The bundle π_{E} possesses a natural flat connection.

Proof. It suffices to take the trivial bundle 1_{M}: M × ℝ → M for the bundles π_{1} and π_{2} and an arbitrary vector field X for the operator Δ. Flatness is a consequence of (3).

Let us denote by D(N) the C^{∞}(N)-module of vector fields on a manifold N.

Definition 1. The connection C: D(M) → D(E^{∞}) is called the Cartan connection on E^{∞}.

Denote by CD(E^{∞}) ⊂ D(E^{∞}) the horizontal distribution on E^{∞} with respect to the Cartan connection (the Cartan distribution) and by D_{C}(E^{∞}) the normalizer of CD(E^{∞}) in D(E^{∞}). Then, since C is flat, CD(E^{∞}) is integrable in a formal Frobenius sense and thus CD(E^{∞}) is an ideal in D_{C}(E^{∞}).

Definition 2. The quotient Lie algebra sym E = D_{C}(E^{∞})/CD(E^{∞}) is called the algebra of higher symmetries of the equation E.

The Cartan connection in π_{E} determines the splitting

D(E^{∞}) = D^{v}(E^{∞}) ⊕ CD(E^{∞}),  (4)

where D^{v}(E^{∞}) denotes the module of π_{E}-vertical vector fields, and for any coset S ∈ sym E there exists a unique vertical representative. In its turn, such a representative is uniquely determined by a section φ ∈ Γ(π_{E}^{∗}(π)) (generating section) satisfying the defining equation

ℓ_{E}φ = 0  (5)

and vice versa. Here ℓ_{E} is the universal linearization operator for E restricted to E^{∞}. Due to this fact, we shall identify the solutions of (5) with higher symmetries of E.

The connection form U_{E} (the structural element of E) of the Cartan connection C is an element of the module D^{v}(Λ^{1}(E^{∞})) of Λ^{1}(E^{∞})-valued vertical derivations. Thus, we can introduce an operator ∂_{E}: D^{v}(Λ^{i}(E^{∞})) → D^{v}(Λ^{i+1}(E^{∞})) defined by

∂_{E}Ω = [[U_{E}, Ω]], Ω ∈ D^{v}(Λ^{i}(E^{∞})),

where [[·, ·]] is the Frölicher–Nijenhuis bracket. Since the Cartan connection is flat, one has [[U_{E}, U_{E}]] = 0, from where it follows that ∂_{E} ∘∂_{E} = 0. Thus we obtain a complex (D^{v}(Λ^{i}(E^{∞})), ∂_{E}), whose cohomology is called the C-cohomology of E and is denoted by H_{C}^{•}(E). It is easy to see that H_{C}^{0}(E) = sym E, while H_{C}^{1}(E), by standard reasons, is identified with classes of nontrivial infinitesimal deformations of U_{E} (or, which is the same, of the equation structure).

If Ω and Θ are elements of D^{v}(Λ^{i}(E^{∞})) and D^{v}(Λ^{j}(E^{∞})) respectively, their contraction Ω Θ is defined as an element of (D^{v}(Λ^{i+j−1}(E^{∞})) This operation is inherited by the C-cohomology groups. In particular, if φ ∈ sym E and R ∈ H_{C}^{1}(E), then φ R = Rφ is a symmetry again. In other words, the module H_{C}^{1}(E) acts on the Lie algebra of higher symmetries.

Let CΛ^{1}(E^{∞}) ⊂ Λ^{1}(E^{∞}) be the submodule of one-forms on E^{∞} vanishing on the Cartan distribution. Then one has the direct sum decomposition

Λ^{1}(E^{∞}) = CΛ^{1}(E^{∞}) ⊕ Λ_{h}^{1}(E^{∞})

dual to (4), where Λ_{h}^{1}(E^{∞}) is the submodule of horizontal forms. This splitting is also inherited by H_{C}^{1}(E^{∞}) and an element R ∈ H_{C}^{1}(E^{∞}) acts nontrivially on sym E if only it corresponds to a derivation from D^{v}(CΛ^{1}(E^{∞})). Moreover, it can be shown that im ∂_{E}∩D^{v}(CΛ^{1}(E^{∞})) = 0 and consequently nontrivial actions can be found by solving the equation

∂_{E}(R) = 0.  (6)

Solutions of (6) are called recursion operators for symmetries. Any recursion operator R is uniquely determined by an element ω_{R} ∈ CΛ^{1}(E^{∞}) ⊗ Γ(π_{E}(π)) satisfying the defining equation

ℓ_{E}^{[1]}(ω_{R}) = 0,  (7)

where ℓ_{E}^{[1]} is the extension of the operator ℓ_{E} to the module CΛ^{1}(E^{∞}) ⊗ Γ(π_{E}(π)). If φ ∈ Γ(π_{E}^{∗}(π)) is a symmetry and S_{φ} ∈ D^{v}(E^{∞}) is the corresponding vertical vector field, then the action of R on φ is given by

Rφ = S_{φ} ω_{R}.  (8)

Local coordinates. Let U ⊂ M be a coordinate neighborhood in M such that the bundle π trivializes over U, x_{1}, …, x_{n} be local coordinates in U and u^{1}, …, u^{m} be coordinates along the fiber in a given trivialization. Then the adapted coordinates in π_{∞}^{−1}(U) ⊂ J^{∞}(π) are given by the functions u_{σ}^{j}, j = 1, …, m, σ = i_{1} … i_{k}, 1 ≤ i_{α} ≤ n, uniquely defined by

j_{∞}(f)^{∗}(u_{σ}^{j}) = \frac{∂^{|σ|}f^{j}}{∂x_{i_{1}} … ∂x_{i_{k}}}

for any local section f = (f^{1}, …, f^{m}) ∈ Γ(π|_{U}). Then the Cartan connection in J^{∞}(π) is given by

C (\frac{∂}{∂x_{i}} ) = D_{i} = \frac{∂}{∂x_{i}} + ∑_{j,σ} u_{σi}^{j} \frac{∂}{∂u_{σ}^{j}} ,  (9)

where D_{i} are the so-called total derivatives. The structural element in this case is given by the formula

U_{π} = ∑_{j,σ} ω_{σ}^{j} ⊗ \frac{∂}{∂u_{σ}^{j}} ,  (10)

where ω_{σ}^{j} = du_{σ}^{j} − ∑_{i} u_{σi}^{j} dx_{i} are the Cartan forms constituting a basis in the module CΛ^{1}(J^{∞}(π)). The forms ω_{σ}^{j} may be rewritten as ω_{σ}^{j} = d_{C}u_{σ}^{j}, where d_{C} is the Cartan differential acting on f ∈ C^{∞}(J^{∞}(π)) by

d_{C}f = ∑ \frac{∂f}{∂u_{σ}^{j}} ω_{σ}^{j}.

This differential restricts to any submanifold E^{∞} ⊂ J^{∞}(π) and for infinite prolongations (10) transforms to

U_{E} = ∑_{I} d_{C}u_{I} ⊗ \frac{∂}{∂u_{I}} ,  (11)

where {u_{I}} spans the set of internal coordinates in E^{∞}.

If an equation E ⊂ J^{k}(π) is given by the system of equalities

\begin{cases}F^{1}(x_{1}, …, x_{n}, …, u_{σ}^{j}, …) = 0, \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ F^{r}(x_{1}, …, x_{n}, …, u_{σ}^{j}, …) = 0, \end{cases}

then its infinite prolongation is defined by

D_{σ}F^{α} = 0, α = 1, …, r, 0 ≤ |σ| < ∞,

where D_{σ} = D_{i_{1}} ∘ ⋯ ∘ D_{i_{k}} for σ = i_{1} … i_{k}. Then the universal linearization operator corresponding to this system is of the form

ℓ_{F} = ∥∑_{σ} \frac{∂F^{α}}{∂u_{σ}^{j}} D_{σ}∥ .  (12)

The operator ℓ_{E} is obtained from (12) by rewriting it in internal coordinates.

Example (Evolutionary 1 + 1 equations). Consider a system E of evolutionary equations

\begin{cases}u_{t}^{1} = F^{1}(x, t, u^{1}, …, u_{k}^{m}), \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ u_{t}^{m} = F^{r}(x, t, u^{1}, …, u_{k}^{m}), \end{cases}  (13)

where x = x_{1}, t = x_{2} and u_{l}^{j} = ∂^{l}u^{j}/∂x^{l}. Then the functions x, t, …, u_{l}^{j}, ⋯ can be taken for internal coordinates on E^{∞}. The total derivatives are written down as

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

in these coordinates. The Cartan forms on E^{∞} are

ω_{l}^{j} = du_{l}^{j} − u_{l+1}^{j} dx − D_{x}^{l}(F^{j}) dt,

while the structural element for E is given by

U_{E} = ∑_{l,j} ω_{l}^{j} ⊗ \frac{∂}{∂u_{l}^{j}} .  (14)

The restriction ℓ_{E} of the universal linearization operator to E^{∞} is of the form

ℓ_{E} = ∥∑_{l,j} \frac{∂F^{α}}{∂u_{l}^{j}} D_{x}^{l} − ED_{t}∥ ,

where E is the identity matrix. So, a vector function φ = (φ^{1}, …, φ^{m}), φ^{j} = φ^{j}(x, t, …, u_{l}^{j}, …), is a symmetry of E if and only if

∑_{l,j} \frac{∂F^{α}}{∂u_{l}^{j}} D_{x}^{l}φ^{j} = D_{t}φ^{α}  (15)

for all α = 1, …, m. The corresponding vector field is given by the formula

�_{φ} = ∑_{l,j} D_{x}^{l}(φ^{j})\frac{∂}{∂u_{l}^{j}} .

In a similar way, recursion operators are determined by vectorvalued forms ω_{R} = (ω_{R}^{1}, …, ω_{R}^{m}), where ω_{R}^{j} = ∑_{l,α} ψ_{α}^{jl}ω_{l}^{α}, ψ_{α}^{jl} ∈ C^{∞}(E^{∞}), satisfying the equations

∑_{l,j} \frac{∂F^{α}}{∂u_{l}^{j}} D_{x}^{l}ω_{R}^{j} = D_{t}ω_{R}^{α}.  (16)

To compute the left and right sides of (16), it suffices to note that

D_{x}ω_{l}^{j} = ω_{l+1}^{j}, D_{t}ω_{l}^{j} = D_{x}^{l}d_{C}F^{j} = D_{x}^{l} ∑_{α,be} \frac{∂F^{j}}{∂u_{β}^{α}} ω_{β}^{α}.

Remark 1. Let ω = (ω^{1}, …, ω^{m}), ω^{j} = ∑_{l,α} ψ_{α}^{jl}ω_{l}^{α}, be a vectorvalued Cartan form on E^{∞}. Then for any vector-valued function φ = (φ^{1}, …, φ^{m}) on E^{∞} the action R_{ω}: φ ↦ R_{ω}φ is defined by R_{ω}φ = �_{φ} ω. In local coordinates, this action is expressed by the formula

(R_{ω}φ)^{j} = ∑_{l,α} ψ_{α}^{jl}D_{x}^{l}(φ^{α}).  (17)

Operators of this type (i.e., expressed in terms of total derivatives) are called C-differential (or total differential) operators.

Remark 2. As it was mentioned above, operators of the form (17), provided ω satisfies (16), take symmetries of the equation at hand to symmetries of the same equation. In other words, one has R_{ω}: ker ℓ_{E} → ker ℓ_{E}. Under not very restrictive conditions on the equation E, this is equivalent to the operator equality

ℓ_{E} ∘ R_{ω} = A ∘ ℓ_{E},

where A is a C-differential operator. Taking formally adjoint, one obtains

R_{ω}^{∗} ∘ ℓ_{E}^{∗} = ℓ_{E}^{∗} ∘ A^{∗},

which means, that A^{∗} is a recursion operator for generating functions of conservation laws [10].

Nonlocal setting. In practice, when solving (7), one usually finds no nontrivial solutions, though the equation E may possess recursion operators.

Example (The Burgers equation). Consider the equation

u_{t} = u_{xx} + uu_{x}.

It is known to possess a recursion operator of the form

R = D_{x} + \frac{1}{2}u_{0} + \frac{1}{2}u_{1}D_{x}^{−1}.  (18)

Nevertheless, the only solution of the equation

ℓ_{E}^{[1]}ω ≡ (D_{x}^{2} + u_{0}D_{x} + u_{1} − D_{t})ω = 0  (19)

for ω = ψ_{0}ω_{0} +… ψ_{k}ω_{k} is αω_{0}, α ∈ ℝ, which provides the trivial action R_{ω}: φ ↦ αφ.

To resolve this apparent contradiction, let us extend the algebra C^{∞}(E^{∞}) with an additional element u_{−1} and set

D_{x}u_{−1} = u_{0},
D_{t}u_{−1} = u_{1} + \frac{1}{2}u_{0}^{2},
d_{C}u_{−1} ≡ ω_{−1} = du_{−1} − u_{0} dx − (u_{1} + \frac{1}{2}u_{0}^{2}) dt.  (20)

Then, solving (19) for ω = ψ_{−1}ω_{−1} + ψ_{0}ω_{0} + … ψ_{k}ω_{k}, we obtain a twoparametric solution

ω = αω_{0} + βΩ, Ω = ω_{1} + \frac{1}{2}u_{0}ω_{0} + \frac{1}{2}u_{1}ω_{−1}.

Then the action φ ↦ R_{Ω}φ = �_{φ} Ω coincides exactly with (18).

This example reflects a general scheme of computations which arises in a lot of applications [5] and is used in Section 2. Namely, in search of recursion operators for symmetries we extend the algebra C^{∞}(E^{∞}) with a new set of variables w^{1}, …, w^{r}, … (so-called nonlocal variables) and respectively extend the total derivatives to vector fields

D̄_{i} = D_{i} + ∑_{α} X_{i}^{α} \frac{∂}{∂w^{α}} , i = 1, …, n,  (21)

in such a way that

[D̄_{i}, D̄_{j}] ≡ [D_{i}, X_{j}] + [X_{i}, D_{j}] + [X_{i}, X_{j}] = 0,  (22)

where X_{l} = ∑_{α} X_{l}^{α}∂/∂w^{α}.

Having a solution X_{1}, …, X_{n} of (21), we obtain an integrable distribution on the space E^{∞} × ℝ^{N}, where N is the number of nonlocal variables (the case N = ∞ is included). The projection τ: \overline{E^{∞}} = E^{∞} × ℝ^{N} → E^{∞} is called a covering over E and N is called its dimension (for an invariant geometrical definition see [7]). Similar to the local case, we define the Lie algebra sym_{τ} E = D_{C}(\overline{E^{∞}})/CD(\overline{E^{∞}}) of nonlocal τ-symmetries.

We introduce Cartan forms

θ^{j} = dw^{j} − ∑_{i=1}^{n} X_{i}^{j} dx_{i}, j = 1, …, N,

corresponding to nonlocal variables on \overline{E^{∞}}. The module of all Cartan forms on \overline{E^{∞}} is denoted by CΛ^{1}(\overline{E^{∞}}). We also extend the universal linearization operator ℓ_{E} to \overline{E^{∞}} just by changing the total derivatives D_{i} to D̄_{i}. Let us now consider two equations, associated to this extension:

ℓ̄_{E}φ = 0 and ℓ̄_{E}^{[1]}Ω = 0,

where φ ∈ Γ((π_{E} ∘τ)^{∗}π) and Ω ∈ Γ((π_{E} ∘τ)^{∗}π)⊗CΛ^{1}(\overline{E^{∞}}). Solutions of the first equation are called τ-shadows of nonlocal symmetries, while solutions of the second one are said to be τ-shadows of recursion operators in the covering τ. The following result establishes relations of shadows to symmetries and recursion operators:

Theorem 1 (see [5, 7]). Let τ: \overline{E^{∞}} → E^{∞} be a covering. Then:

1. If φ is a τ-shadow, then there exists a covering

τ̄: \overline{\overline{E^{∞}}} → \overline{E^{∞}} ⟶^{τ} E^{∞}

such that φ reconstructs up to a nonlocal τ̄-symmetry.

2. If φ is a nonlocal τ-symmetry and R is a recursion operator shadow, then Rφ is a symmetry shadow.

Remark 3. Among all one-dimensional coverings over a given equation there exists a special class consisting of those ones, for which the fields X_{1}, …, X_{n} in (22) are independent of nonlocal variables (socalled abelian coverings). To any such a covering, one can put into correspondence a differential form on E^{∞}:

ω_{τ} = ∑_{l=1}^{n} X_{l}dx_{l}.

This form is closed with respect to the so-called horizontal differential d_{h} = Cd (cf. Proposition 1). Vice versa, to any such a form there corresponds a covering of the above mentioned type. Moreover, two closed forms determine the same class in the cohomology group H^{1}(d_{h}) if and only if the corresponding coverings are equivalent. In particular, if n = 2, the group H^{1}(d_{h}) coincides with the group of conservation laws of the equation E [10]. Thus, to construct a covering under consideration is the same as to find a conservation law. This fact is used in the computations below.

2. Basic computations and results

In this section we shall discuss the complete integrability of the KdV–mKdV system given in (1), i.e.,

u_{t} = −u_{3} + 6uu_{1} − 3ww_{3} − 3w_{1}w_{2} + 3u_{1}w^{2} + 6uww_{1},
w_{t} = −w_{3} + 3w^{2}w_{1} + 3uw_{1} + 3u_{1}w.  (23)

In order to demonstrate the complete integrability of this system, we shall construct the recursion operator for symmetries of this coupled system, leading to infinite hierarchies of symmetries and, most probably, of conservation laws. Due to the very special form of the final results, it seems that integrability of this system, which looks quite ordinary, has not been discussed before or elsewhere. In order to do this, we shall discuss conservation laws in Subsection 2.1 leading to the necessary nonlocal variables.

In Subsection 2.2 we shall discuss local and nonlocal symmetries of the system, while in Subsection 2.3 we construct the recursion operator or deformation of the equation structure (14).

2.1. Conservation laws and nonlocal variables. Here we shall construct conservation laws for (23) in order to arrive at an abelian covering of the coupled KdV–mKdV system as was shown for Burgers equation (20). So we construct X = X(x, t, u, …, w …), T = T (x, t, u, …, w …) such that

D_{x}(T) = D_{t}(X)  (24)

and in a similar way we construct nonlocal conservation laws by the requirement

D̄_{x}(T̄) = D̄_{t}(X̄),  (25)

where D̄_{∗} is defined by (9); moreover X̄, T̄ are dependent on local variables x, t, u, …, w, … as well as the already determined nonlocal variables, denoted here by p_{∗} or p_{∗,∗}, which are associated to the conservation laws (X, T) by the formal definition

D_{x}(p_{∗}) = (p_{∗})_{x} = X,
D_{t}(p_{∗}) = (p_{∗})_{t} = T.

Proceeding in this way, we obtained the following set of nonlocal variables

p_{0,1}, p_{0,2}, p_{1}, p_{1,1}, p_{1,2}, p_{2,1}, p_{3}, p_{3,1}, p_{3,2}, p_{4,1}, p_{5},  (26)

where their defining equations are given by

(p_{1})_{x} = u,
(p_{1})_{t} = 3u^{2} + 3uw^{2} − u_{2} − 3ww_{2},
(p_{0,1})_{x} = w,

10 Paul Kersten, Joseph Krasil′shchik

(p_{0,1})_{t} = 3uw + w^{3} − w_{2},
(p_{0,2})_{x} = p_{1},
(p_{0,2})_{t} = −6p_{3} − u_{1},
(p_{1,1})_{x} = cos(2p_{0,1})p_{1}w + sin(2p_{0,1})w^{2},
(p_{1,1})_{t} = cos(2p_{0,1})(3p_{1}uw + p_{1}w^{3} − p_{1}w_{2} + uw_{1} − u_{1}w − w^{2}w_{1})

+ sin(2p_{0,1})(4uw^{2} + w^{4} − 2ww_{2} + w_{1}^{2}),

(p_{1,2})_{x} = cos(2p_{0,1})w^{2} − sin(2p_{0,1})p_{1}w,
(p_{1,2})_{t} = cos(2p_{0,1})(4uw^{2} + w^{4} − 2ww_{2} + w_{1}^{2})
+ sin(2p_{0,1})(−3p_{1}uw − p_{1}w^{3} + p_{1}w_{2} − uw_{1} + u_{1}w + w^{2}w_{1}),
(p_{2,1})_{x} = (4 cos(2p_{0,1})p_{1,1}w^{2} − 4 sin(2p_{0,1})p_{1}p_{1,1}w + w(p_{1}^{2} − 2u + w^{2}))/2,
(p_{2,1})_{t} = (4 cos(2p_{0,1})p_{1,1}(4uw^{2} + w^{4} − 2ww_{2} + w_{1}^{2})

+ 4 sin(2p_{0,1})p_{1,1}(−3p_{1}uw − p_{1}w^{3} + p_{1}w_{2} − uw_{1} + u_{1}w + w^{2}w_{1}) + 3p_{1}^{2}uw + p_{1}^{2}w^{3} − p_{1}^{2}w_{2} + 2p_{1}uw_{1} − 2p_{1}u_{1}w − 2p_{1}w^{2}w_{1} − 8u^{2}w − uw^{3} + 2uw_{2} − 2u_{1}w_{1} + 2u_{2}w + w^{5} + 3w^{2}w_{2})/2,

(p_{3})_{x} = (−u^{2} − uw^{2} + ww_{2})/2,
(p_{3})_{t} = (−4u^{3} − 9u^{2}w^{2} + 2uu_{2} − 3uw^{4} + 11uww_{2} − uw_{1}^{2} − u_{1}^{2} + u_{1}ww_{1}

+ 4u_{2}w^{2} + 6w^{3}w_{2} + 3w^{2}w_{1}^{2} − ww_{4} + w_{1}w_{3} − w_{2}^{2})/2,

(p_{3,1})_{x} = (cos(2p_{0,1})w(p_{1}^{3} − 6p_{1}u + 39p_{1}w^{2} − 24p_{1,1}p_{1,2}w + 12p_{3} + 6u_{1})

+ 2 sin(2p_{0,1})w(12p_{1}p_{1,1}p_{1,2} + 18p_{1}w_{1} + 2w^{3} + 3w_{2}) + 6p_{1,2}w(−p_{1}^{2} + 2u − w^{2}))/12,

(p_{3,2})_{x} = (2 cos(2p_{0,1})w(12p_{1}p_{1,1}p_{1,2} − 18p_{1}w_{1} − 2w^{3} − 3w_{2})

+ sin(2p_{0,1})w(p_{1}^{3} − 6p_{1}u + 39p_{1}w^{2} + 24p_{1,1}p_{1,2}w + 12p_{3} + 6u_{1}) + 6p_{1,1}w(−p_{1}^{2} + 2u − w^{2}))/12,

(p_{4,1})_{x} = (8 cos(2p_{0,1})w(p_{1}^{3}p_{1,2} + 12p_{1}p_{1,1}^{2}p_{1,2} − 6p_{1}p_{1,2}u + 3p_{1}p_{1,2}w^{2}

− 12p_{1,1}p_{1,2}^{2}w + 18p_{1,1}uw − 4p_{1,1}w^{3} − 6p_{1,1}w_{2} + 12p_{1,2}p_{3} + 6p_{1,2}u_{1}) + 8 sin(2p_{0,1})w(p_{1}^{3}p_{1,1} + 12p_{1}p_{1,1}p_{1,2}^{2} − 6p_{1}p_{1,1}u + 3p_{1}p_{1,1}w^{2} + 12p_{1,1}^{2}p_{1,2}w + 12p_{1,1}p_{3} + 6p_{1,1}u_{1} − 18p_{1,2}uw + 4p_{1,2}w^{3} + 6p_{1,2}w_{2}) + w(−p_{1}^{4} − 24p_{1}^{2}p_{1,1}^{2} − 24p_{1}^{2}p_{1,2}^{2} + 12p_{1}^{2}u − 6p_{1}^{2}w^{2} − 48p_{1}p_{3} − 24p_{1}u_{1} + 48p_{1,1}^{2}u − 24p_{1,1}^{2}w^{2} + 48p_{1,2}^{2}u − 24p_{1,2}^{2}w^{2} − 60u^{2} + 44uw^{2} + 24u_{2} − 13w^{4} + 6ww_{2}))/48,

(p_{5})_{x} = (12u^{3} + 24u^{2}w^{2} − 6uu_{2} + 6uw^{4} − 30uww_{2} − 3u_{2}w^{2} − 8w^{3}w_{2} + 6ww_{4})/6.

In the previous equations, we skipped explicit formulas for (p_{3,1})_{t}, (p_{3,2})_{t}, (p_{4,1})_{t}, and (p_{5})_{t}, because they are too massive, though quite important for the setting to be well defined and in order to avoid ambiguities. The reader is referred to the Appendix for them.

It is quite a striking result that functions cos(2p_{0,1}), sin(2p_{0,1}) appear in the presentation of the conservation laws and their associated nonlocal variables.

We should note that p_{1}, p_{0,1}, p_{3}, p_{5} arise from local conservation laws and we shall call p_{1}, p_{0,1}, p_{3}, p_{5} nonlocalities of first order.

In a similar way we see that p_{0,2}, p_{1,1}, p_{1,2} arise from nonlocal conservation laws, where their x- and t-derivatives are dependent on the first order nonlocalities. For this reason p_{0,2}, p_{1,1}, p_{1,2} are called nonlocalities of second order.

Proceeding in this way p_{2,1}, p_{3,1}, p_{3,2}, p_{4,1} constitute nonlocalities of third order.

2.2. Local and nonlocal symmetries. In this section we shall present results for the construction of local and nonlocal symmetries of system (23). In order to construct these symmetries, we consider the system of partial differential equations obtained by the infinite prolongation of (23) together with the covering by the nonlocal variables

p_{0,1}, p_{0,2}, p_{1}, p_{1,1}, p_{1,2}, p_{2,1}, p_{3}, p_{3,1}, p_{3,2}, p_{4,1}, p_{5}.

So, in the augmented setting governed by (23), their total derivatives and the equations given in Subsection 2.1 we construct symmetries Y = (Y^{u}, Y^{w}) which have to satisfy the symmetry condition

ℓ̄_{E}Y = 0.

From this condition we obtained the following symmetries

Y_{0,1}, Y_{1,1}, Y_{1,2}, Y_{1,3}, Y_{2,1}, Y_{3,1}, Y_{3,2}, Y_{3,3},

where generating functions Y_{∗,∗}^{u}, Y_{∗,∗}^{w} are given as

Y_{0,1}^{u} = 3t(6uu_{1} + 6uww_{1} + 3u_{1}w^{2} − u_{3} − 3ww_{3} − 3w_{1}w_{2}) + xu_{1} + 2u,
Y_{0,1}^{w} = 3t(3uw_{1} + 3u_{1}w + 3w^{2}w_{1} − w_{3}) + xw_{1} + w,
Y_{1,1}^{u} = u_{1},
Y_{1,1}^{w} = w_{1},
Y_{1,2}^{u} = cos(2p_{0,1})(2uw − w_{2}) + sin(2p_{0,1})(u_{1} + 2ww_{1}),
Y_{1,2}^{w} = − cos(2p_{0,1})u − sin(2p_{0,1})w_{1},
Y_{1,3}^{u} = cos(2p_{0,1})(u_{1} + 2ww_{1}) + sin(2p_{0,1})(−2uw + w_{2}),
Y_{1,3}^{w} = − cos(2p_{0,1})w_{1} + sin(2p_{0,1})u,
Y_{2,1}^{u} = (2 cos(2p_{0,1})(p_{1,1}u_{1} + 2p_{1,1}ww_{1} − 2p_{1,2}uw + p_{1,2}w_{2})
+ 2 sin(2p_{0,1})(−2p_{1,1}uw + p_{1,1}w_{2} − p_{1,2}u_{1} − 2p_{1,2}ww_{1})
+ 2p_{1}uw − p_{1}w_{2} + 2uw_{1} + 3u_{1}w + 2w^{2}w_{1} − w_{3})/2,
Y_{2,1}^{w} = (2 cos(2p_{0,1})(−p_{1,1}w_{1} + p_{1,2}u) + 2 sin(2p_{0,1})(p_{1,1}u + p_{1,2}w_{1})

12 Paul Kersten, Joseph Krasil′shchik

− p_{1}u + u_{1} + ww_{1})/2,

Y_{3,1}^{u} = (6uu_{1} + 6uww_{1} + 3u_{1}w^{2} − u_{3} − 3ww_{3} − 3w_{1}w_{2})/3,
Y_{3,1}^{w} = (3uw_{1} + 3u_{1}w + 3w^{2}w_{1} − w_{3})/3,
Y_{3,2}^{u} = (cos(2p_{0,1})(−2p_{1}^{2}uw + p_{1}^{2}w_{2} − 4p_{1}uw_{1} − 6p_{1}u_{1}w − 4p_{1}w^{2}w_{1} + 2p_{1}w_{3}

+ 8p_{1,1}p_{1,2}u_{1} + 16p_{1,1}p_{1,2}ww_{1} − 8p_{1,2}^{2}uw + 4p_{1,2}^{2}w_{2} − 4p_{2,1}u_{1} − 8p_{2,1}ww_{1} + 10u^{2}w + 6uw^{3} − 8uw_{2} − 14u_{1}w_{1} − 8u_{2}w − 11w^{2}w_{2} − 14ww_{1}^{2} + 2w_{4}) + 2 sin(2p_{0,1})(−8p_{1,1}p_{1,2}uw + 4p_{1,1}p_{1,2}w_{2} − 2p_{1,2}^{2}u_{1} − 4p_{1,2}^{2}ww_{1} + 4p_{2,1}uw − 2p_{2,1}w_{2} + 6uu_{1} + 10uww_{1} + 3u_{1}w^{2} − u_{3} + 2w^{3}w_{1} − 3ww_{3} − 5w_{1}w_{2}) + 4p_{1,2}(2p_{1}uw − p_{1}w_{2} + 2uw_{1} + 3u_{1}w + 2w^{2}w_{1} − w_{3}))/8,

Y_{3,2}^{w} = (cos(2p_{0,1})(p_{1}^{2}u − 2p_{1}u_{1} − 2p_{1}ww_{1} − 8p_{1,1}p_{1,2}w_{1} + 4p_{1,2}^{2}u + 4p_{2,1}w_{1}

− 4u^{2} − 3uw^{2} + 2u_{2} + 4ww_{2} + 2w_{1}^{2}) + 2 sin(2p_{0,1})(4p_{1,1}p_{1,2}u + 2p_{1,2}^{2}w_{1} − 2p_{2,1}u − 3uw_{1} − 3u_{1}w − 3w^{2}w_{1} + w_{3}) + 4p_{1,2}(−p_{1}u + u_{1} + ww_{1}))/8,

Y_{3,3}^{u} = (2 cos(2p_{0,1})(2p_{1,1}^{2}u_{1} + 4p_{1,1}^{2}ww_{1} − 4p_{2,1}uw + 2p_{2,1}w_{2} − 6uu_{1}

− 10uww_{1} − 3u_{1}w^{2} + u_{3} − 2w^{3}w_{1} + 3ww_{3} + 5w_{1}w_{2}) + sin(2p_{0,1})(−2p_{1}^{2}uw + p_{1}^{2}w_{2} − 4p_{1}uw_{1} − 6p_{1}u_{1}w − 4p_{1}w^{2}w_{1} + 2p_{1}w_{3} − 8p_{1,1}^{2}uw + 4p_{1,1}^{2}w_{2} − 4p_{2,1}u_{1} − 8p_{2,1}ww_{1} + 10u^{2}w + 6uw^{3} − 8uw_{2} − 14u_{1}w_{1} − 8u_{2}w − 11w^{2}w_{2} − 14ww_{1}^{2} + 2w_{4}) + 4p_{1,1}(2p_{1}uw − p_{1}w_{2} + 2uw_{1} + 3u_{1}w + 2w^{2}w_{1} − w_{3}))/8,

Y_{3,3}^{w} = (2 cos(2p_{0,1})(−2p_{1,1}^{2}w_{1} + 2p_{2,1}u + 3uw_{1} + 3u_{1}w + 3w^{2}w_{1} − w_{3})

+ sin(2p_{0,1})(p_{1}^{2}u − 2p_{1}u_{1} − 2p_{1}ww_{1} + 4p_{1,1}^{2}u + 4p_{2,1}w_{1} − 4u^{2} − 3uw^{2} + 2u_{2} + 4ww_{2} + 2w_{1}^{2}) + 4p_{1,1}(−p_{1}u + u_{1} + ww_{1}))/8.

2.3. Recursion operator. Here we present the recursion operator R for symmetries for this case obtained as a higher symmetry in the Cartan covering of system of equations (1) augmented by equations governing the nonlocal variables (26). As explained in the previous section, the recursion operator is in effect the a deformation of the equation structure (11).

As demonstrated there, this deformation is a form-valued vector field (or a vectorfield-valued one-form) and has to satisfy

ℓ̄_{E}^{[1]}R = 0.  (27)

In order to arrive at a nontrivial result as was explained for Burgers’ equation too (c.f. Example 1), we have to introduce nonlocal variables

p_{0,1}, p_{0,2}, p_{1}, p_{1,1}, p_{1,2}, p_{2,1}, p_{3}, p_{3,1}, p_{3,2}, p_{4,1}, p_{5} and their associated Cartan contact forms

ω_{p_{0,1}}, ω_{p_{0,2}}, ω_{p_{1}}, ω_{p_{1,1}}, ω_{p_{1,2}}, ω_{p_{2,1}}, ω_{p_{3}}, ω_{p_{3,1}}, ω_{p_{3,2}}, ω_{p_{4,1}}, ω_{p_{5}}.

The final result, which is dependent on the nonlocal Cartan forms

ω_{p_{0,1}}, ω_{p_{1}}, ω_{p_{1,1}}, ω_{p_{1,2}},

is given by

R = R^{u} \frac{∂}{∂u} + R^{w} \frac{∂}{∂w} + …,  (28)

where the components R^{u}, R^{w} are given by

R_{u} = ω_{u_{2}}(−1) + ω_{u}(4u + w^{2}) + ω_{w_{2}}(−2w) + ω_{w_{1}}(−w_{1}) + ω_{w}(3uw − 2w_{2})

+ ω_{p_{1,2}}(− cos(2p_{0,1})(u_{1} + 2ww_{1}) + sin(2p_{0,1})(2uw − w_{2}))

+ ω_{p_{1,1}}(cos(2p_{0,1})(−2uw + w_{2}) − sin(2p_{0,1})(u_{1} + 2ww_{1}))

+ ω_{p_{1}}(2u_{1} + ww_{1}) + ω_{p_{0,1}}(2p_{1}uw − p_{1}w_{2} + 2uw_{1} + 3u_{1}w + 2w^{2}w_{1} − w_{3}),

R_{w} = ω_{w_{2}}(−1) + ω_{w}(2u + w^{2}) + ω_{u}(2w)

+ ω_{p_{1,2}}(cos(2p_{0,1})w_{1} − sin(2p_{0,1})u)

+ ω_{p_{1,1}}(cos(2p_{0,1})u + sin(2p_{0,1})w_{1})

+ ω_{p_{1}}(w_{1}) + ω_{p_{0,1}}(−p_{1}u + u_{1} + ww_{1}).  (29)

We shall now present this result in a more conventional form which appeals to expressions using operators of the form D_{x} and D_{x}^{−1}. In order to do this, we first split (29) into the so-called local part and nonlocal parts, consisting of terms associated to ω_{u_{2}}, ω_{u}, ω_{w_{2}}, ω_{w_{1}}, ω_{w} and those associated to ω_{p_{1,2}}, ω_{p_{1,1}}, ω_{p_{1}}, ω_{p_{0,1}} respectively. The first part will account for D_{x} presentation, while the second one accounts for the D_{x}^{−1} part.

Due to the action of contraction �_{φ} R, the local part is given by the following matrix operator:

[\begin{matrix}−D_{x}^{2} + 4u + w^{2} & −2wD_{x}^{2} − w_{1}D_{x} + 3uw − 2w_{2} \\ 2w & −D_{x}^{2} + 2u + w^{2} \end{matrix}] .

The nonlocal part will be split into parts associated to ω_{p_{1}}, ω_{p_{0,1}} and ω_{p_{1,2}}, ω_{p_{1,1}}, respectively. The first one is given as

[\begin{matrix}(2u_{1} + ww_{1})D_{x}^{−1} & (2p_{1}uw − p_{1}w_{2} + 2uw_{1} + 3u_{1}w + 2w^{2}w_{1} − w_{3})D_{x}^{−1} \\ w_{1}D_{x}^{−1} & (−p_{1}u + u_{1} + ww_{1})D_{x}^{−1} \end{matrix}] .

To deal with the last part, let us introduce the notation:

A_{1} = cos(2p_{0,1})(−2uw + w_{2}) − sin(2p_{0,1})(u_{1} + 2ww_{1}),
A_{2} = cos(2p_{0,1})u + sin(2p_{0,1})w_{1},
B_{1} = − cos(2p_{0,1})(u_{1} + 2ww_{1}) + sin(2p_{0,1})(2uw − w_{2}),
B_{2} = cos(2p_{0,1})w_{1} − sin(2p_{0,1})u,

being the coefficients at ω_{p_{1,1}} and ω_{p_{1,2}} in (29).

According to the presentations of (p_{1,1})_{x} and (p_{1,2})_{x}, i.e.,

(p_{1,1})_{x} = cos(2p_{0,1})p_{1}w + sin(2p_{0,1})w^{2}
(p_{1,2})_{x} = cos(2p_{0,1})w^{2} − sin(2p_{0,1})p_{1}w,

we introduce their partial derivatives with respect to p_{0,1}, p_{1}, and w as

α_{1} = −2p_{1}w sin(2p_{0,1}) + 2w^{2} cos(2p_{0,1}),
α_{2} = w cos(2p_{0,1}),
α_{3} = p_{1} cos(2p_{0,1}) + 2w sin(2p_{0,1}),
β_{1} = −2w^{2} sin(2p_{0,1}) − 2p_{1}w cos(2p_{0,1}),
β_{2} = −w sin(2p_{0,1}),
β_{3} = 2w cos(2p_{0,1}) − p_{1} sin(2p_{0,1}).

From this we arrive in a straightforward way at the last nonlocal part of the recursion operator, i.e.,

[\begin{matrix}A_{1}D_{x}^{−1}α_{2}D_{x}^{−1} & A_{1}D_{x}^{−1}(α_{1}D_{x}^{−1} + α_{3}) \\ A_{2}D_{x}^{−1}α_{2}D_{x}^{−1} & A_{2}D_{x}^{−1}(α_{1}D_{x}^{−1} + α_{3}) \end{matrix}] + [\begin{matrix}B_{1}D_{x}^{−1}β_{2}D_{x}^{−1} & B_{1}D_{x}^{−1}(β_{1}D_{x}^{−1} + β_{3}) \\ B_{2}D_{x}^{−1}β_{2}D_{x}^{−1} & B_{2}D_{x}^{−1}(β_{1}D_{x}^{−1} + β_{3}) \end{matrix}]

So, in the final form we obtain the recursion operator as

R = [\begin{matrix}−D_{x}^{2} + 4u + w^{2} & −2wD_{x}^{2} − w_{1}D_{x} + 3uw − 2w_{2} \\ 2w & −D_{x}^{2} + 2u + w^{2} \end{matrix}]

+ [\begin{matrix}(2u_{1} + ww_{1})D_{x}^{−1} & (2p_{1}uw − p_{1}w_{2} + 2uw_{1} + 3u_{1}w + 2w^{2}w_{1} − w_{3})D_{x}^{−1} \\ w_{1}D_{x}^{−1} & (−p_{1}u + u_{1} + ww_{1})D_{x}^{−1} \end{matrix}]

+ [\begin{matrix}A_{1}D_{x}^{−1}α_{2}D_{x}^{−1} & A_{1}D_{x}^{−1}(α_{1}D_{x}^{−1} + α_{3}) \\ A_{2}D_{x}^{−1}α_{2}D_{x}^{−1} & A_{2}D_{x}^{−1}(α_{1}D_{x}^{−1} + α_{3}) \end{matrix}]

+ [\begin{matrix}B_{1}D_{x}^{−1}β_{2}D_{x}^{−1} & B_{1}D_{x}^{−1}(β_{1}D_{x}^{−1} + β_{3}) \\ B_{2}D_{x}^{−1}β_{2}D_{x}^{−1} & B_{2}D_{x}^{−1}(β_{1}D_{x}^{−1} + β_{3}) \end{matrix}] .

3. Conclusion

We gave an outline of the theory of deformations of the equation structure of differential equations, leading to the construction of recursion operators for symmetries of such equations. The extension of this theory to the nonlocal setting of differential equations is essential for getting nontrivial results. The theory has been applied to the construction of the recursion operator for symmetries for a coupled KdV–mKdV system, leading to a highly nonlocal result for this system. Moreover the appearance of nonpolynomial nonlocal terms in all results, e.g., conservation laws, symmetries and recursion operator is striking and reveals some unknown and intriguing underlying structure of the equations.

Complete integrability of the coupled KdV–mKdV system 15

Appendix

Here we present explicit formulas for (p_{3,1})_{t}, (p_{3,2})_{t}, (p_{4,1})_{t}, and (p_{5})_{t}:

(p_{3,1})_{t} = (cos(2p_{0,1})(3p_{1}^{3}uw + p_{1}^{3}w^{3} − p_{1}^{3}w_{2} + 3p_{1}^{2}uw_{1} − 3p_{1}^{2}u_{1}w − 3p_{1}^{2}w^{2}w_{1}

− 24p_{1}u^{2}w + 105p_{1}uw^{3} + 6p_{1}uw_{2} − 6p_{1}u_{1}w_{1} + 6p_{1}u_{2}w + 39p_{1}w^{5} − 27p_{1}w^{2}w_{2} − 96p_{1,1}p_{1,2}uw^{2} − 24p_{1,1}p_{1,2}w^{4} + 48p_{1,1}p_{1,2}ww_{2} − 24p_{1,1}p_{1,2}w_{1}^{2} + 36p_{3}uw + 12p_{3}w^{3} − 12p_{3}w_{2} − 12u^{2}w_{1} + 48uu_{1}w + 39uw^{2}w_{1} + 3u_{1}w^{3} − 6u_{1}w_{2} + 6u_{2}w_{1} − 6u_{3}w − 9w^{4}w_{1} − 12w^{2}w_{3} − 18ww_{1}w_{2} + 6w_{1}^{3}) + 2 sin(2p_{0,1})(36p_{1}p_{1,1}p_{1,2}uw + 12p_{1}p_{1,1}p_{1,2}w^{3} − 12p_{1}p_{1,1}p_{1,2}w_{2} + 54p_{1}uww_{1} + 54p_{1}u_{1}w^{2} + 54p_{1}w^{3}w_{1} − 18p_{1}ww_{3} + 12p_{1,1}p_{1,2}uw_{1} − 12p_{1,1}p_{1,2}u_{1}w − 12p_{1,1}p_{1,2}w^{2}w_{1} − 9u^{2}w^{2} + 18uw^{4} + 27uww_{2} − 9uw_{1}^{2} + 9u_{1}ww_{1} + 3u_{2}w^{2} + 2w^{6} − 11w^{3}w_{2} + 12w^{2}w_{1}^{2} − 3ww_{4} + 3w_{1}w_{3} − 3w_{2}^{2}) + 6p_{1,2}(−3p_{1}^{2}uw − p_{1}^{2}w^{3} + p_{1}^{2}w_{2} − 2p_{1}uw_{1} + 2p_{1}u_{1}w + 2p_{1}w^{2}w_{1} + 8u^{2}w + uw^{3} − 2uw_{2} + 2u_{1}w_{1} − 2u_{2}w − w^{5} − 3w^{2}w_{2}))/12,

(p_{3,2})_{t} = (2 cos(2p_{0,1})(36p_{1}p_{1,1}p_{1,2}uw + 12p_{1}p_{1,1}p_{1,2}w^{3} − 12p_{1}p_{1,1}p_{1,2}w_{2}

− 54p_{1}uww_{1} − 54p_{1}u_{1}w^{2} − 54p_{1}w^{3}w_{1} + 18p_{1}ww_{3} + 12p_{1,1}p_{1,2}uw_{1} − 12p_{1,1}p_{1,2}u_{1}w − 12p_{1,1}p_{1,2}w^{2}w_{1} + 9u^{2}w^{2} − 18uw^{4} − 27uww_{2} + 9uw_{1}^{2} − 9u_{1}ww_{1} − 3u_{2}w^{2} − 2w^{6} + 11w^{3}w_{2} − 12w^{2}w_{1}^{2} + 3ww_{4} − 3w_{1}w_{3} + 3w_{2}^{2}) + sin(2p_{0,1})(3p_{1}^{3}uw + p_{1}^{3}w^{3} − p_{1}^{3}w_{2} + 3p_{1}^{2}uw_{1} − 3p_{1}^{2}u_{1}w − 3p_{1}^{2}w^{2}w_{1} − 24p_{1}u^{2}w + 105p_{1}uw^{3} + 6p_{1}uw_{2} − 6p_{1}u_{1}w_{1} + 6p_{1}u_{2}w + 39p_{1}w^{5} − 27p_{1}w^{2}w_{2} + 96p_{1,1}p_{1,2}uw^{2} + 24p_{1,1}p_{1,2}w^{4} − 48p_{1,1}p_{1,2}ww_{2} + 24p_{1,1}p_{1,2}w_{1}^{2} + 36p_{3}uw + 12p_{3}w^{3} − 12p_{3}w_{2} − 12u^{2}w_{1} + 48uu_{1}w + 39uw^{2}w_{1} + 3u_{1}w^{3} − 6u_{1}w_{2} + 6u_{2}w_{1} − 6u_{3}w − 9w^{4}w_{1} − 12w^{2}w_{3} − 18ww_{1}w_{2} + 6w_{1}^{3}) + 6p_{1,1}(−3p_{1}^{2}uw − p_{1}^{2}w^{3} + p_{1}^{2}w_{2} − 2p_{1}uw_{1} + 2p_{1}u_{1}w + 2p_{1}w^{2}w_{1} + 8u^{2}w + uw^{3} − 2uw_{2} + 2u_{1}w_{1} − 2u_{2}w − w^{5} − 3w^{2}w_{2}))/12,

(p_{4,1})_{t} = (8 cos(2p_{0,1})(3p_{1}^{3}p_{1,2}uw + p_{1}^{3}p_{1,2}w^{3} − p_{1}^{3}p_{1,2}w_{2} + 3p_{1}^{2}p_{1,2}uw_{1}

− 3p_{1}^{2}p_{1,2}u_{1}w − 3p_{1}^{2}p_{1,2}w^{2}w_{1} + 36p_{1}p_{1,1}^{2}p_{1,2}uw + 12p_{1}p_{1,1}^{2}p_{1,2}w^{3} − 12p_{1}p_{1,1}^{2}p_{1,2}w_{2} − 24p_{1}p_{1,2}u^{2}w − 3p_{1}p_{1,2}uw^{3} + 6p_{1}p_{1,2}uw_{2} − 6p_{1}p_{1,2}u_{1}w_{1} + 6p_{1}p_{1,2}u_{2}w + 3p_{1}p_{1,2}w^{5} + 9p_{1}p_{1,2}w^{2}w_{2} + 12p_{1,1}^{2}p_{1,2}uw_{1} − 12p_{1,1}^{2}p_{1,2}u_{1}w − 12p_{1,1}^{2}p_{1,2}w^{2}w_{1} − 48p_{1,1}p_{1,2}^{2}uw^{2} − 12p_{1,1}p_{1,2}^{2}w^{4} + 24p_{1,1}p_{1,2}^{2}ww_{2} − 12p_{1,1}p_{1,2}^{2}w_{1}^{2} + 72p_{1,1}u^{2}w^{2} + 18p_{1,1}uw^{4} − 54p_{1,1}uww_{2} + 18p_{1,1}uw_{1}^{2} − 18p_{1,1}u_{1}ww_{1} − 24p_{1,1}u_{2}w^{2} − 4p_{1,1}w^{6} − 32p_{1,1}w^{3}w_{2} − 24p_{1,1}w^{2}w_{1}^{2} + 6p_{1,1}ww_{4} − 6p_{1,1}w_{1}w_{3} + 6p_{1,1}w_{2}^{2} + 36p_{1,2}p_{3}uw + 12p_{1,2}p_{3}w^{3} − 12p_{1,2}p_{3}w_{2} − 12p_{1,2}u^{2}w_{1} + 48p_{1,2}uu_{1}w + 39p_{1,2}uw^{2}w_{1} + 3p_{1,2}u_{1}w^{3}

− 6p_{1,2}u_{1}w_{2} + 6p_{1,2}u_{2}w_{1} − 6p_{1,2}u_{3}w − 9p_{1,2}w^{4}w_{1} − 12p_{1,2}w^{2}w_{3}

− 18p_{1,2}ww_{1}w_{2} + 6p_{1,2}w_{1}^{3})

+ 8 sin(2p_{0,1})(3p_{1}^{3}p_{1,1}uw + p_{1}^{3}p_{1,1}w^{3} − p_{1}^{3}p_{1,1}w_{2} + 3p_{1}^{2}p_{1,1}uw_{1} − 3p_{1}^{2}p_{1,1}u_{1}w

− 3p_{1}^{2}p_{1,1}w^{2}w_{1} + 36p_{1}p_{1,1}p_{1,2}^{2}uw + 12p_{1}p_{1,1}p_{1,2}^{2}w^{3} − 12p_{1}p_{1,1}p_{1,2}^{2}w_{2}

− 24p_{1}p_{1,1}u^{2}w − 3p_{1}p_{1,1}uw^{3} + 6p_{1}p_{1,1}uw_{2} − 6p_{1}p_{1,1}u_{1}w_{1} + 6p_{1}p_{1,1}u_{2}w

+ 3p_{1}p_{1,1}w^{5} + 9p_{1}p_{1,1}w^{2}w_{2} + 48p_{1,1}^{2}p_{1,2}uw^{2} + 12p_{1,1}^{2}p_{1,2}w^{4} − 24p_{1,1}^{2}p_{1,2}ww_{2}

+ 12p_{1,1}^{2}p_{1,2}w_{1}^{2} + 12p_{1,1}p_{1,2}^{2}uw_{1} − 12p_{1,1}p_{1,2}^{2}u_{1}w − 12p_{1,1}p_{1,2}^{2}w^{2}w_{1}

+ 36p_{1,1}p_{3}uw + 12p_{1,1}p_{3}w^{3} − 12p_{1,1}p_{3}w_{2} − 12p_{1,1}u^{2}w_{1} + 48p_{1,1}uu_{1}w

+ 39p_{1,1}uw^{2}w_{1} + 3p_{1,1}u_{1}w^{3} − 6p_{1,1}u_{1}w_{2} + 6p_{1,1}u_{2}w_{1} − 6p_{1,1}u_{3}w

− 9p_{1,1}w^{4}w_{1} − 12p_{1,1}w^{2}w_{3} − 18p_{1,1}ww_{1}w_{2} + 6p_{1,1}w_{1}^{3} − 72p_{1,2}u^{2}w^{2}

− 18p_{1,2}uw^{4} + 54p_{1,2}uww_{2} − 18p_{1,2}uw_{1}^{2} + 18p_{1,2}u_{1}ww_{1} + 24p_{1,2}u_{2}w^{2}

+ 4p_{1,2}w^{6} + 32p_{1,2}w^{3}w_{2} + 24p_{1,2}w^{2}w_{1}^{2} − 6p_{1,2}ww_{4}

+ 6p_{1,2}w_{1}w_{3} − 6p_{1,2}w_{2}^{2}) − 3p_{1}^{4}uw − p_{1}^{4}w^{3} + p_{1}^{4}w_{2} − 4p_{1}^{3}uw_{1} + 4p_{1}^{3}u_{1}w

+ 4p_{1}^{3}w^{2}w_{1} − 72p_{1}^{2}p_{1,1}^{2}uw − 24p_{1}^{2}p_{1,1}^{2}w^{3} + 24p_{1}^{2}p_{1,1}^{2}w_{2} − 72p_{1}^{2}p_{1,2}^{2}uw

− 24p_{1}^{2}p_{1,2}^{2}w^{3} + 24p_{1}^{2}p_{1,2}^{2}w_{2} + 48p_{1}^{2}u^{2}w + 6p_{1}^{2}uw^{3} − 12p_{1}^{2}uw_{2} + 12p_{1}^{2}u_{1}w_{1}

− 12p_{1}^{2}u_{2}w − 6p_{1}^{2}w^{5} − 18p_{1}^{2}w^{2}w_{2} − 48p_{1}p_{1,1}^{2}uw_{1} + 48p_{1}p_{1,1}^{2}u_{1}w

+ 48p_{1}p_{1,1}^{2}w^{2}w_{1} − 48p_{1}p_{1,2}^{2}uw_{1} + 48p_{1}p_{1,2}^{2}u_{1}w + 48p_{1}p_{1,2}^{2}w^{2}w_{1} − 144p_{1}p_{3}uw

− 48p_{1}p_{3}w^{3} + 48p_{1}p_{3}w_{2} + 48p_{1}u^{2}w_{1} − 192p_{1}uu_{1}w − 156p_{1}uw^{2}w_{1}

− 12p_{1}u_{1}w^{3} + 24p_{1}u_{1}w_{2} − 24p_{1}u_{2}w_{1} + 24p_{1}u_{3}w + 36p_{1}w^{4}w_{1} + 48p_{1}w^{2}w_{3}

+ 72p_{1}ww_{1}w_{2} − 24p_{1}w_{1}^{3} + 192p_{1,1}^{2}u^{2}w + 24p_{1,1}^{2}uw^{3} − 48p_{1,1}^{2}uw_{2}

+ 48p_{1,1}^{2}u_{1}w_{1} − 48p_{1,1}^{2}u_{2}w − 24p_{1,1}^{2}w^{5} − 72p_{1,1}^{2}w^{2}w_{2} + 192p_{1,2}^{2}u^{2}w

+ 24p_{1,2}^{2}uw^{3} − 48p_{1,2}^{2}uw_{2} + 48p_{1,2}^{2}u_{1}w_{1} − 48p_{1,2}^{2}u_{2}w − 24p_{1,2}^{2}w^{5}

− 72p_{1,2}^{2}w^{2}w_{2} − 48p_{3}uw_{1} + 48p_{3}u_{1}w + 48p_{3}w^{2}w_{1} − 252u^{3}w − 36u^{2}w^{3}

+ 60u^{2}w_{2} − 144uu_{1}w_{1} + 240uu_{2}w − 7uw^{5} + 342uw^{2}w_{2} + 24uww_{1}^{2}

+ 144u_{1}^{2}w + 228u_{1}w^{2}w_{1} + 94u_{2}w^{3} − 24u_{2}w_{2} + 24u_{3}w_{1} − 24u_{4}w

− 13w^{7} + 47w^{4}w_{2} − 4w^{3}w_{1}^{2} − 78w^{2}w_{4} − 60ww_{1}w_{3}

− 120ww_{2}^{2} + 60w_{1}^{2}w_{2})/48,

(p_{5})_{t} = (54u^{4} + 180u^{3}w^{2} − 72u^{2}u_{2} + 126u^{2}w^{4} − 282u^{2}ww_{2} − 12u^{2}w_{1}^{2}

− 84uu_{1}ww_{1} − 174uu_{2}w^{2} + 6uu_{4} + 18uw^{6} − 300uw^{3}w_{2} − 90uw^{2}w_{1}^{2}

+ 66uww_{4} + 6uw_{1}w_{3} + 48uw_{2}^{2} + 42u_{1}^{2}w^{2} − 6u_{1}u_{3} + 12u_{1}w^{3}w_{1}

+ 42u_{1}ww_{3} − 48u_{1}w_{1}w_{2} + 6u_{2}^{2} − 39u_{2}w^{4} + 162u_{2}ww_{2} − 48u_{2}w_{1}^{2}

+ 48u_{3}ww_{1} + 21u_{4}w^{2} − 42w^{5}w_{2} − 12w^{4}w_{1}^{2} + 35w^{3}w_{4} + 120w^{2}w_{1}w_{3} + 195w^{2}w_{2}^{2} + 120ww_{1}^{2}w_{2} − 6ww_{6} − 30w_{1}^{4} + 6w_{1}w_{5} − 6w_{2}w_{4})/6.

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. American Mathematical Society, Providence, RI, 1999. Edited and with a preface by Krasil′shchik and Vinogradov.

[2] R. K. Dodd, J. C. Eilbeck, J. D. Gibbons, and H. C. Morris. Solitons and Nonlinear Wave Equations. Academic Press, 1982.

[3] P. H. M. Kersten. Supersymmetries and recursion operators for N = 2 supersymmetric KdV-equation, RIMS Kokyuroku, 1150, Kyoto Uniniv., Kyoto, Japan, 2000, pp. 153–161.

[4] I. S. Krasil′shchik. Some new cohomological invariants for nonlinear differential equations. Differential Geom. Appl., 2 (1992) no. 4, 307–350.

[5] I. S. Krasil’shchik and P. H. M. Kersten, Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations, Kluwer Acad. Publ., Dordrecht/Boston/London, 2000.

[6] I. S. Krasil′shchik, V. V. Lychagin, and A. M. Vinogradov, Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach, New York, 1986.

[7] I. S. Krasil′shchik and A. M. Vinogradov. Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations. Acta Appl. Math., 15 (1989) no. 1-2, 161–209.

[8] S. Krivonos, A. Sorin. Extended N = 2 supersymmetric matrix (1, s)-KdV hierarchies. Phys. Lett. A251 (1999) 109.

[9] P. Mathieu. Open problems for the super KdV equation. AARMS-CRM Workshop on Baecklund and Darboux transformations. The Geometry of Soliton Theory. June 4–9, 1999. Halifax, Nova Scotia.

[10] A. M. Vinogradov. Local symmetries and conservation laws, Acta Appl. Math. 3 (1984) 21–78.

University of Twente, Faculty of Mathematical Sciences, P.O.Box 217, 7500 AE Enschede, The Netherlands

E-mail address: kersten@math.utwente.nl

Independent University of Moscow Correspondence to: 1st Tverskoy-Yamskoy per. 14, Apt. 45, 125047 Moscow, Russia

E-mail address: josephk@online.ru