The Diffiety Institute Preprint Series

Preprint DIPS–2/2005 November 3, 2005

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

The Diffiety Institute

A geometric study of the dispersionless Boussinesq type equation

by

Paul Kersten, I. Krasil′shchik, A. Verbovetsky

A geometric study of the dispersionless Boussinesq type equation

PAUL KERSTEN, I. KRASIL′SHCHIK, A. VERBOVETSKY

Abstract. We discuss the dispersionless Boussinesq type equation, which is equivalent to the Benney–Lax equation, being a system of equations of hydrodynamical type. This equation was discussed in [4].The results include: a description of local and nonlocal Hamiltonian and symplectic structures, hierarchies of symmetries, hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws (cosymmetries). Highly interesting are the appearances of operators that send conservation laws and symmetries to each other but are neither Hamiltonian, nor symplectic. These operators give rise to a noncommutative infinite-dimensional algebra of recursion operators.

Introduction

Below we deal with the dispersionless Boussinesq type equation (the dB-equation), which is the system

w_{t} = u_{x},
u_{t} = ww_{x} + v_{x},  (1)
v_{t} = −uw_{x} − 3wu_{x},

2 PAUL KERSTEN, I. KRASIL′SHCHIK, A. VERBOVETSKY

being equivalent to the Benney–Lax equation, and which is known to be integrable, [4]. In particular, it possesses a bi-Hamiltonian structure. System (1) is of hydrodynamical type and can be obtained as a reduction of the Khokhlov–Zabolotskaya equation. Using the methods developed in [6], we rediscover the above mentioned bi-Hamiltonian structure and show that it is only a part of the infinite-dimensional space of operators that take conservation laws of (1) (their generating functions, more precisely) to symmetries. These operators, in a standard way, generate an infinite associative (but not commutative) algebra of recursion operators for symmetries. Every recursion operator, applied to a known symmetry (e.g., a point one), gives rise to an infinite family of symmetries (both local and nonlocal ones). Contrary to the known examples, these families are not hierarchies in the usual sense, because their jet order does not grow infinitely but remains at level 1 for all symmetries we found. Dually, there exists an infinite-dimensional space of operators that take symmetries of the dB-equation to generating functions (or cosymmetries) and only some of these operators determine symplectic (or inverse Noether) structures on the equation. In a similar way, we obtain an infinite algebra of recursion operators for cosymmetries and infinite families of conservation laws (also of 1st order). Below we present a detailed analysis of all these structures. In Section 1 a very informal introduction to the theoretical background is given. Section 2 contains preparatory material needed to achieve the main results. These results are exposed in Section 3 and discussed in concluding remarks (Sections 4 and 5).

2000 Mathematics Subject Classification. 37K05, 35Q53.

Key words and phrases. Symmetry, conservation law, Hamiltonian structure, symplectic structure.

This work was supported in part by the NWO grant 047017015.

1. Background

As it was mentioned in the Introduction, our computations are based on the results of paper [6] (see also [7]). For the general theoretical background we also refer the reader to books [1, 10, 12]. Here we shall give an informal description of the computational scheme we use in subsequent sections. We consider a system E of evolution equations

u_{t} = F (x, t, u, u_{1}, …, u_{k}),  (2)

where both u = (u^{1}, …, u^{m}) and F = (F^{1}, …, F^{m}) are vectors and u_{t} = ∂u/∂t, u_{s} = ∂^{s}u/∂x^{s}. For simplicity, we restrict ourselves to the case of one-dimensional space variable x, though everything works in the general situation as well. We are interested in symmetries and conservation laws of system (2) and in various operators that relate these objects to each other (recursion operators, Hamiltonian and symplectic structures).

1.1. Symmetries and conservation laws. A symmetry of equation (2) is a vector field (an evolutionary field)

�_{φ} = ∑_{i≥0} ∑_{j=1}^{m} D_{x}^{i}(φ^{j})\frac{∂}{∂u_{i}^{j}} ,  (3)

where φ = (φ^{1}, …, φ^{m}) is a vector function depending on x, t, u, u_{1}, …, u_{s} and satisfying the equations

D_{t}(φ^{j}) = ∑_{i,l} \frac{∂F^{j}}{∂u_{i}^{l}} D_{x}^{i}(φ^{l}),  j = 1, …, m.  (4)

Here and below

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

are total derivatives with respect to x and t. We identify fields �_{φ} with functions φ (the generating functions). Thus, a symmetry is a function that satisfies the equation

ℓ_{E}(φ) = 0,  (5)

where

ℓ_{E} = D_{t} − ℓ_{F}  (6)

and

ℓ_{F} = (\begin{matrix}∑_{i} ∂F^{1}/∂u_{i}^{1}D_{x}^{i} & & … & & & ∑_{i} ∂F^{1}/∂u_{i}^{m}D_{x}^{i} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ ∑_{i} ∂F^{m}/∂u_{i}^{1}D_{x}^{i} & & … & & & ∑_{i} ∂F^{m}/∂u_{i}^{m}D_{x}^{i} \end{matrix}) .  (7)

Operator (6) is called the linearization operator for the equation E. The set of symmetries is a Lie algebra denoted by sym(E). A conservation law for equation (2) is a horizontal 1-form

ω = X dx + T dt

closed with respect to the horizontal de Rham differential

d_{h} = dx ∧ D_{x} + dt ∧ D_{t},

i.e., such that

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

where T = T (x, t, u, u_{1}, …), X = X(x, t, u, u_{1}, …). A conservation law is trivial if it is of the from ω = d_{h}f, i.e.,

X = D_{x}(f), T = D_{t}(f), f = f(x, t, u, u_{1}, …).

The space of equivalence classes of conservation laws modulo trivial ones coincides with the 1st horizontal de Rham cohomology group for E and is denoted by H_{h}^{1}(E).

Remark 1. If the number of the space variables x equals n, then this space coincides with H_{h}^{n}(E).

To any conservation law ω = X dx + T dt there corresponds its generating function

ψ_{ω} = δX = (\frac{δX}{δu^{1}} , …, \frac{δX}{δu^{m}} ),  (8)

where δ denotes the Euler operator and

\frac{δ}{δu^{j}} = ∑_{i≥0}(−1)^{i}D_{x}^{i} ∘ \frac{∂}{∂u_{i}^{j}}  (9)

is the variational derivative with respect to u^{j}. Any generating function (8) satisfies the equation

ℓ_{E}^{∗}(ψ_{ω}) = 0,  (10)

where

ℓ_{E}^{∗} = −D_{t} + ℓ_{F}^{∗}

= −D_{t} + (\begin{matrix}∑_{i}(−1)^{i}D_{x}^{i} ∘ ∂F^{1}/∂u_{i}^{1} & & … & & & ∑_{i}(−1)^{i}D_{x}^{i} ∘ ∂F^{m}/∂u_{i}^{1} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ ∑_{i}(−1)^{i}D_{x}^{i} ∘ ∂F^{1}/∂u_{i}^{m} & & … & & & ∑_{i}(−1)^{i}D_{x}^{i} ∘ ∂F^{m}/∂u_{i}^{m} \end{matrix}) (11)

is the operator adjoint to ℓ_{E}. Solutions of equation (10) are called cosymmetries and in general not all of them are generating functions of conservation laws. The space of cosymmetries will be denoted by sym^{∗}(E). The Euler operator determines the embedding

δ: H_{h}^{1}(E) → sym^{∗}(E).  (12)

Symmetries and cosymmetries may be understood as vector fields and differential 1-forms, respectively, on the equation E and there is a natural pairing between them: if φ = (φ^{1}, …, φ^{m}) ∈ sym(E) and ψ = (ψ^{1}, …, ψ^{m}) ∈ sym^{∗}(E), we set

⟨ψ, φ⟩ = ψ^{1}φ^{1} + ⋯ + ψ^{m}φ^{m}.  (13)

At first glance, the right-hand side of (13) looks like a function, but the “physical meaning” of ⟨ψ, φ⟩ is quite different. Namely, applying D_{t} to ⟨ψ, φ⟩ we have

D_{t}⟨ψ, φ⟩ = ⟨D_{t}(ψ), φ⟩ + ⟨ψ, D_{t}(φ)⟩ = −⟨ℓ_{F}^{∗} (ψ), φ⟩ + ⟨ψ, ℓ_{F} (φ)⟩

and, consequently,

D_{t}⟨ψ, φ⟩ = D_{x}(T_{ψ,φ})

for some T_{ψ,φ}. Though the conservation law

⟨ψ, φ⟩ dx + T_{ψ,φ} dt  (14)

is not defined uniquely, its cohomology class depends on φ and ψ only and we obtain

⟨· , ·⟩: sym^{∗}(E) × sym(E) → H_{h}^{1}(E).

Note that for any ω ∈ H_{h}^{1}(E) one has

⟨δω, φ⟩ = �_{φ}(ω).

1.2. Recursion operators, Hamiltonian and symplectic structures. In this subsection we shall discuss a local theory and shall explain how nonlocal components are incorporated in all construction in Subsection 1.4. All operators considered below are matrix operators in total derivatives of the form

Δ = (∑_{i,s} a_{jl}^{is}D_{x}^{i}D_{t}^{s}) .  (15)

We call such operators C-differential operators. In particular, we shall deal with operators

Δ = (∑_{i} a_{jl}^{i}D_{x}^{i})  (16)

in D_{x} only. In the case of equation (2) we may consider the operator ℓ_{E} to act from the space κ of vector functions φ = (φ^{1}, …, φ^{m}) to the same space. Then ℓ_{E}^{∗} acts from κ^{∗} to κ^{∗}, where κ^{∗} is the dual space.

Remark 2. A coordinate-free description of actions of ℓ_{E} and ℓ_{E}^{∗} may be found in [1].

From Subsection 1.1 we have

sym(E) = ker ℓ_{E} ⊂ κ, sym^{∗}(E) = ker ℓ_{E}^{∗} ⊂ κ^{∗}

and we are looking for C-differential operators of the form (16) acting as follows

R: κ → κ,  (17)
H: κ^{∗} → κ,  (18)
S: κ → κ^{∗},  (19)
R^{∗}: κ^{∗} → κ^{∗},  (20)

and such that

R(sym(E)) ⊂ sym(E),  (21)
H(sym^{∗}(E)) ⊂ sym(E),  (22)
S(sym(E)) ⊂ sym^{∗}(E),  (23)
R^{∗}(sym^{∗}(E)) ⊂ sym^{∗}(E).  (24)

To find such operators makes the first step in constructing the structures we are interested in. Namely,

• operators R are recursion operators for symmetries;

• operators H satisfying conditions described in Remark 3 below are Hamiltonian structures on the equation E;

• operators S satisfying conditions described in Remark 4 below are symplectic structures on the equation E;

• operators R^{∗}.

Remark 3 (Hamiltonianity conditions). Let H be an operator of type (18). Then for any two conservation laws ω_{1} and ω_{2} of equation (2) (or their equivalence classes in H_{h}^{1}(E)) one can define the bracket

{ω_{1}, ω_{2}}_{H} = ⟨δω_{1}, H(δω_{2})⟩ = �_{H(δω_{2})}(δω_{1}).  (25)

H is a Hamiltonian structure if this bracket is skew-symmetric and satisfies the Jacobi identity. The fist condition means that H is a skew-adjoint operator (H = −H^{∗}) and thus may be understood as a bivector on E, while the second one amounts to

[[H, H]] = 0,  (26)

where [[·, ·]] is the variational Schouten bracket, [5]. Recall also that two Hamiltonian structures are compatible (or constitute a Hamiltonian pair) if and only if

[[H_{1}, H_{2}]] = 0.  (27)

Efficient coordinate formulas to check conditions (26) and (27) will be given in Subsection 1.5.

Remark 4 (symplectic conditions). Let S be an operator of type (19). Then it can be understood as a map

S: sym(E) × sym(E) → H_{h}^{1}(E)  (28)

by setting

S(φ_{1}, φ_{2}) = ⟨S(φ_{1}), φ_{2}⟩.  (29)

We say that S is a two-form on E if

S(φ_{1}, φ_{2}) = −S(φ_{2}, φ_{1})  (30)

for all φ_{1}, φ_{2} ∈ sym(E) (this is equivalent to S = −S^{∗}). For such a form one can define its differential

δS: sym(E) × sym(E) × sym(E) → H_{h}^{1}(E)

by

δS(φ_{1}, φ_{2}, φ_{3}) = �_{φ_{1}}S(φ_{2}, φ_{3}) − �_{φ_{2}}S(φ_{1}, φ_{3}) − �_{φ_{3}}S(φ_{1}, φ_{2}) − S({φ_{1}, φ_{2}}, φ_{3}) + S({φ_{1}, φ_{3}}, φ_{2}) − S({φ_{2}, φ_{3}}, φ_{3}), (31)

where the bracket {φ, φ′} is uniquely defined by the formula

�_{{φ,φ′}} = [�_{φ}, �_{φ′}].

We say that S is a symplectic structure if the corresponding two-form is closed with respect to δ, i.e.,

δS = 0.  (32)

Again, computational formulas to check conditions (30) and (32) will be given in Subsection 1.5.

Remark 5 (Nijenhuis conditions). Any operator (17) satisfying (21) can be considered as a recursion on symmetries, i.e., it takes a symmetry of the equation E to a symmetry. Similarly, any operator (24) is a recursion operator for cosymmetries. Thus, applying R to a given symmetry φ one obtains the whole family

φ_{0} = φ, φ_{1} = Rφ, …, φ_{i} = R^{i}φ, …  (33)

Commutativity of such families closely relates to integrability of E. For any recursion operator R: sym(E) → sym(E) its Nijenhuis torsion

N_{R}: sym(E) × sym(E) → sym(E)  (34)

is defined by

N_{R}(φ_{1}, φ_{2}) = {Rφ_{1}, Rφ_{1}} − (R({Rφ_{1}, φ_{2}}) − {φ_{1}, Rφ_{2}} + R({φ_{1}, φ_{2}})),

while for a symmetry φ its action on R is defined by

(�_{φ}R)(φ′) = {φ, Rφ′} − R{φ, φ′}.

Both operations can be expressed in terms of the Nijenhuis bracket and as it was shown in [9] the conditions

N_{R} = 0,  �_{φ}R = 0  (35)

imply commutativity of family (33).

1.3. Δ-coverings. To construct operators possessing properties (21)–(24), we solve the following operator equations

ℓ_{E} ∘ R = A_{R} ∘ ℓ_{E}  (36)
ℓ_{E} ∘ H = A_{H} ∘ ℓ_{E}^{∗}  (37)
ℓ_{E}^{∗} ∘ S = A_{S} ∘ ℓ_{E}  (38)
ℓ_{E}^{∗} ∘ R^{∗} = A_{R^{∗}} ∘ ℓ_{E}^{∗}  (39)

with respect to R, H, S, and R^{∗} for some C-differential operators A_{R}, A_{H}, A_{S}, and A_{R^{∗}}.

Remark 6. It is easily shown that

A_{R} = R, A_{H} = −H, A_{S} = −S, A_{R^{∗}} = R^{∗}

and for any solution R of equation (36) its adjoint R^{∗} is a solution of (39) and vice versa, while for any solution H of equation (37) its inverse H^{−1} (if it makes sense) is a solution of (38) and vice versa. In addition, for any solution H its adjoint H^{∗} is a solution as well and the same for S.

All four problems (36)–(39) can be formulated in the following general form. Let P, Q, P′, Q′ be some spaces of vector functions on the equation E and Δ: P → P′, ∇: Q → Q′ be C-differential operators. How to find all C-differential operators X : P → Q such that the diagram

P ⟶^{X} Q
Δ↓  ↓^{∇}
P′ ⟶^{A_{X}} Q′

is commutative for some C-differential operator A_{X}? For simplicity, we shall assume that X is an operator in D_{x} only, i.e., of the form (16).

Remark 7. This problem has an obvious set of trivial solutions. Namely, any X of the form X = X′ ∘ Δ. We shall look for equivalence classes of solutions modulo trivial ones.

To this end, we use the following construction. Assume that the operator Δ acts on functions v = (v^{1}, …, v^{r}) and consider the system of equations

\begin{cases}u_{t} = F (x, t, u, u_{1}, …, u_{k}), \\ Δ(v) = 0 \end{cases}  (40)

(recall that the coefficient of Δ may depend on the function u and its derivatives). We say that system (40) is the Δ-covering (or the Δ-extension) of the equation E.

Example 1 (the ℓ_{E}-covering). Let us take the operator ℓ_{E} = D_{t} − ℓ_{F} for Δ. Then the corresponding extension is called the ℓ_{E}-covering of E. For example, if E is the Burgers equation

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

then ℓ_{E} = D_{t} − D_{x}^{2} − uD_{x} − u_{x} and the ℓ_{E} covering is of the form

\begin{cases}u_{t} = u_{xx} + uu_{x}, \\ v_{t} = v_{xx} + uv_{x} + vu_{x}. \end{cases}

Example 2 (the ℓ_{E}^{∗}-covering). If we take the operator ℓ_{E}^{∗} = −D_{t} − ℓ_{F}^{∗} for Δ, then this extension is called the ℓ_{E}^{∗}-covering. For the Burgers equation, ℓ_{E}^{∗} = −D_{t} − D_{x}^{2} + D_{x} ∘ u − u_{x} = −D_{t} − D_{x}^{2} + uD_{x} and thus

\begin{cases}u_{t} = u_{xx} + uu_{x}, \\ v_{t} = −v_{xx} + uv_{x} \end{cases}

is the ℓ_{E}^{∗}-covering.

Denote by D̃_{x} and D̃_{t} the total derivatives in the space of the Δ-covering. Then, since ∇ is a C-differential operator, we can consider the operator ˜∇ which is obtained from ∇ by changing D_{x} and D_{t} to D̃_{x} and D̃_{t}, respectively. Consider now the linear differential equation

˜∇(a) = 0,  (41)

where a = (a^{1}, …, a^{s}) is a vector function linear in the variables v, v_{1}, …, v_{i}, …, i.e.,

a^{j} = ∑_{i,l} a_{il}^{j}v_{i}^{l},  (42)

a_{il}^{j} being functions on E.

Theorem 1. Equivalence classes of solutions of the equation

∇ ∘ X = A_{X} ∘ Δ

modulo trivial solutions are in one-to-correspondence with the solutions of equation (41) of the form (42). The operator X corresponding to solution (42) is of the form

X = (∑_{i} a_{il}^{j}D_{x}^{i}) .  (43)

1.4. Nonlocal theory. Δ-coverings are a particular case of the general geometric construction that allows to introduce nonlocal objects related to the initial differential equation (see [11]). Here we shall present all needed constructions in the coordinate form.

Let E be an equation of the form (2) and w = (w^{1}, …, w^{r}) be a new unknown vector function in x and t (the case of infinite number of w’s is included). A system Ẽ

\begin{cases}w_{x}^{j} = X^{j}(x, t, w, u, u_{1}, …, u_{p}), \\ w_{t}^{j} = T^{j}(x, t, w, u, u_{1}, …, u_{p}), \end{cases}  (44)

j = 1, …, r, is called a covering over E if its compatibility is provided by E. The number r of new variables is called the dimension of the covering while w’s are said to be nonlocal variables in this covering.

Example 3. The system

w_{x} = u, w_{t} = u_{xx} + \frac{1}{2}u^{2}

is a covering over the KdV equation.

Total derivatives on Ẽ are denote D̃_{x} and D̃_{t} and they are of the form

D̃_{x} = D_{x} + ∑_{j=1}^{m} X^{j} \frac{∂}{∂w^{j}} ,  D̃_{t} = D_{t} + ∑_{j=1}^{m} T^{j} \frac{∂}{∂w^{j}} .

Thus, if Δ is a C-differential operator on E, we can define the operator Δ̃ on Ẽ substituting D_{x} by D̃_{x} and D_{t} by D̃_{t}. Compatibility conditions for (44) read

D̃_{t}X^{j} = D̃_{x}T^{j} mod Ẽ.

If X^{j} and T^{j} do not depend on w, then these conditions mean that the forms

ω^{1} = X^{1} dx + T^{1} dt, …, ω^{j} = X^{j} dx + T^{j} dt, …

are conservation laws for the equation E. Conversely, to any system of conservation laws we can put into correspondence a covering. The system Ẽ is a differential equation itself and thus we can consider its symmetries, cosymmetries, conservation laws, etc. These objects will be referred to as nonlocal with respect to the initial equation E. In particular, nonlocal symmetries are determined by the equation

ℓẼ(φ̃) = 0,

while nonlocal cosymmetries are defined by

ℓ∗ ; Ẽ(ψ̃) = 0.

On the other hand, since ℓ_{E} is a C-differential operator, the operator ℓ̃_{E} is well defined. In general, ℓẼ ≠ ℓ̃_{E} and the equations

ℓ̃_{E}(φ̃) = 0

and

ℓ̃_{E}^{∗}(ψ̃) = 0

can be considered. Their solutions are called shadows of symmetries and cosymmetries, respectively.

Remark 8. Δ-coverings are (usually, infinite-dimensional) coverings in the sense of this subsection. Consider, for example, the ℓ_{E}-covering which can be presented in the form

u_{t} = F (x, t, u, u_{1}, …, u_{k}), v_{t} = ℓ_{F} (v)

and set

ω^{0} = v, ω^{1} = v_{x}, ω^{2} = v_{xx}, …

Then the ℓ_{E}-covering is equivalent to the system

\begin{cases}ω_{x}^{j} = ω^{j+1}, \\ ω_{t}^{j} = D̃_{x}^{j}(ℓ_{F} (ω^{0})), \end{cases}

j = 0, 1, 2, … In a similar way, the ℓ_{E}^{∗}-covering is equivalent to

\begin{cases}ρ_{x}^{j} = ρ^{j+1}, \\ ρ_{t}^{j} = −D̃_{x}^{j}(ℓ_{F}^{∗} (ρ^{0})). \end{cases}

From now on, the notation ω and ρ is used for nonlocal variables in the ℓ_{E}- and ℓ_{E}^{∗}- coverings, respectively.

Comparing the results of this and the previous subsections, we can formulate the statement fundamental for all subsequent computations.

Theorem 2. Let E be an equation of the form (2). Then equivalence classes of solutions R, H, S, R^{∗} of equations (36), (37), (38), (39) are in one-to-one correspondence with

• shadows of symmetries in the ℓ_{E}-covering for (36),

• shadows of symmetries in the ℓ_{E}^{∗}-covering for (37),

• shadows of cosymmetries in the ℓ_{E}-covering for (38),

• shadows of cosymmetries in the ℓ_{E}^{∗}-covering for (39). All shadows are taken to be linear with respect to nonlocal variables in the corresponding covering.

Since Ẽ is an equation, we can consider its coverings. In this way, coverings over coverings, etc., arise. Our next remark is concerned with special coverings over ℓ_{E}- and ℓ_{E}^{∗}-coverings and with interpretation of the corresponding nonlocal variables.

Remark 9 (nonlocal vectors and covectors). Consider an equation E and its ℓ_{E}^{∗}-covering. Let φ be a symmetry of E. Then, since φ satisfies the equation D_{t}φ = ℓ_{F} φ and ρ is defined by D_{t}ρ = −ℓ_{F}^{∗} ρ, we have

D_{t}(ρφ) = D_{x}(T_{φ})

for some function T_{φ} on the ℓ_{E}^{∗}-covering (see the end of Subsection 1.1). Consequently,

ρφ dx + T_{φ} dt

is a conservation law on the ℓ_{E}^{∗}-covering. Denote by ρ^{φ} the nonlocal variable in the corresponding covering, i.e.,

ρ_{x}^{φ} = ρφ,  ρ_{t}^{φ} = T_{φ}.  (45)

We call ρ^{φ} the nonlocal vector corresponding to φ. In a similar way, any cosymmetry ψ of E determines the conservation law

ψω dx + T_{ψ} dt

on the ℓ_{E}-covering. The nonlocal variable ω^{ψ} defined by

ω_{x}^{ψ} = ψω,  ω_{t}^{ψ} = T_{ψ}  (46)

is called the nonlocal covector (or nonlocal form) corresponding to ψ. Take a set of symmetries φ^{1}, …, φ^{r} and the covering corresponding to this set by (45). Then we can consider shadows of symmetries and cosymmetries in this covering linear with respect to the variables ρ and ρ^{φ}, i.e., shadows of the form a = (a^{1}, …, a^{m}), where

a^{j} = ∑_{l=1}^{m} ∑_{i≥0} a_{i}^{jl}ρ_{i}^{l} + ∑_{s=1}^{r} a_{s}^{j}ρ^{φ_{s}}.  (47)

Proposition 1. The operators corresponding to functions (47) are of the form

Δ = (∑_{i≥0} a_{i}^{jl}D_{x}^{i} + ∑_{s=1}^{r} a_{s}^{j}D_{x}^{−1} ∘ φ_{s}^{l}) ,  j, l = 1, …, m,  (48)

and they enjoy properties (37) and (39) for shadows of symmetries and cosymmetries, respectively. In addition, if (47) is a shadow of symmetry, then the vector functions

a_{s} = (a_{s}^{1}, …, a_{s}^{m}), s = 1, …, r,

are symmetries of equation E, and if (47) is a shadow of cosymmetry, then these functions are cosymmetries.

In the same way, one can consider a set ψ_{1}, …, ψ_{r} of cosymmetries and the covering over the ℓ_{E}-covering given by (46). Let us take a shadow b = (b^{1}, …, b^{m}),

b^{j} = ∑_{l=1}^{m} ∑_{i≥0} b_{i}^{jl}ω_{i}^{l} + ∑_{s=1}^{r} b_{s}^{j}ω^{ψ_{s}}  (49)

in this covering.

Proposition 2. The operators corresponding to functions (49) are of the form

Δ = (∑_{i≥0} b_{i}^{jl}D_{x}^{i} + ∑_{s=1}^{r} b_{s}^{j}D_{x}^{−1} ∘ ψ_{s}^{l}) ,  j, l = 1, …, m,  (50)

and they enjoy properties (36) and (38) for shadows of symmetries and cosymmetries, respectively. In addition, if (49) is a shadow of symmetry, then the vector functions

b_{s} = (b_{s}^{1}, …, b_{s}^{m}), s = 1, …, r,

are symmetries of E, and if (49) is a shadow of cosymmetry, then these functions are cosymmetries.

Note that all operators (48) and (50) are obtained in the weakly nonlocal form (cf. [2]).

Remark 10. In the sequel, we shall use a short notation

⟨a_{1}, …, a_{r} | D_{x}^{−1} | φ_{1}, …, φ_{r}⟩ = ∑_{s=1}^{r} a_{s}^{j}D_{x}^{−1} ∘ φ_{s}^{l}

and

⟨b_{1}, …, b_{r} | D_{x}^{−1} | ψ_{1}, …, ψ_{r}⟩ = ∑_{s=1}^{r} b_{s}^{j}D_{x}^{−1} ∘ ψ_{s}^{l}

for the nonlocal part of operators (48) and (50), resp.

1.5. General outline of the computational scheme. The computations described in Sections 2 and 3 are divided into two parts. The first one plays an auxiliary role and consists of the following steps:

• Construction of a number of conservation laws and the corresponding nonlocal variables for the initial equation.

• Construction of local and nonlocal symmetries that will serve as seed symmetries for infinite families and to be used to construct necessary nonlocal vectors.

• Construction of local and nonlocal cosymmetries that will serve as seed cosymmetries for infinite families and to be used to construct necessary nonlocal covectors.

• Construction of nonlocal vectors and covectors. In the second part we reveal the main structures associated to the dB-equation. Namely,

• We solve the equation

ℓ̃_{E}(b) = 0

in the ℓ_{E}-covering to construct recursion operators for symmetries of the form (50).

• We solve the equation

ℓ̃_{E}^{∗}(a) = 0

in the ℓ_{E}^{∗}-covering to construct recursion operators for cosymmetries of the form (48).

• We solve the equation

ℓ̃_{E}(a) = 0  (51)

in the ℓ_{E}^{∗}-covering to find operators of the form (48) that take cosymmetries to symmetries. Hamiltonian structures on E are distinguished by Theorem 3 below.

• We solve the equation

ℓ̃_{E}^{∗}(b) = 0  (52)

in the ℓ_{E}-covering to find operators of the form (50) that take symmetries to cosymmetries. Symplectic structures on E are distinguished by Theorem 4.

We shall now present two criteria that allow to distinguish Hamiltonian and symplectic structures between local solutions of (51) and (52).

Theorem 3. Let a = (a^{1}, …, a^{m}) be a solution of (51), where a^{j} = ∑_{l,i} a_{i}^{jl}ρ_{i}^{l}. Consider the function A = ∑_{j} a^{j}ρ^{j}. Then the corresponding C-differential operator is a Hamiltonian structure on the equation E if and only if

∑_{j} \frac{δA}{δρ^{j}} ρ^{j} = −2A  (53)

and

δ ∑_{j} \frac{δA}{δu^{j}} · \frac{δA}{δρ^{j}} = 0.  (54)

In addition, two Hamiltonian structures with the functions A and A′ are compatible if and only if

δ ∑_{j} (\frac{δA}{δu^{j}} · \frac{δA′}{δρ^{j}} + \frac{δA′}{δu^{j}} · \frac{δA}{δρ^{j}} ) = 0.  (55)

Theorem 4. Let b = (b^{1}, …, b^{m}) be a solution of (52), where b^{j} = ∑_{l,i} b_{i}^{jl}ω_{i}^{l}. Consider the function B = ∑_{j} b^{j}ω^{j}. Then the corresponding C-differential operator is a symplectic structure on the equation E if and only if

∑_{j} \frac{δB}{δω^{j}} ω^{j} = −2B  (56)

and

δ ∑_{j} \frac{δB}{δu^{j}} ω^{j} = 0.  (57)

Remark 11. To use these two results correctly, note that (1) The variables ρ_{i}^{j} and ω_{i}^{j} should be considered as odd ones.

(2) The operator δ is the total Euler operator in the corresponding covering equation, i.e.,

δ = (\frac{δ}{δu^{1}} , …, \frac{δ}{δu^{m}} , \frac{δ}{δρ^{1}} , …, \frac{δ}{δρ^{m}})

for the ℓ_{E}^{∗}-covering and

δ = (\frac{δ}{δu^{1}} , …, \frac{δ}{δu^{m}} , \frac{δ}{δω^{1}} , …, \frac{δ}{δω^{m}})

for the ℓ_{E}-covering.

Remark 12. Conditions (53) and (56) guarantee that the corresponding operators are skew-adjoint. Conditions (54), (55), and (57) are equivalent to (26), (27), and (32), respectively. An amazing property of evolution equations in “general position” was established in [3] and [8] and is the following one.

Theorem 5. Let E be an evolution equation of the form (2) and assume that it is not a first-order scalar equation. Then if the symbol of the right-hand side is nondegenerate, one has

• any skew-adjoint C-differential operator satisfying (37) satisfies (54) as well (any bivector is Poissonian),

• any two skew-adjoint C-differential operators satisfying (37) satisfy (55) as well (any two Hamiltonian structures are compatible),

• any skew-adjoint C-differential operator satisfying (38) satisfies (57) as well (any two-form is closed).

2. Preparatory computations

We now come back to the dB-equation

w_{t} = u_{x}, u_{t} = ww_{x} + v_{x}, v_{t} = −uw_{x} − 3wu_{x}

and first of all note that it is homogeneous with respect to the following gradings (weights)

|x| = −1, |t| = −2, |w| = 2, |u| = 3, |v| = 4

and corresponding weights of all variables and their polynomial functions. All constructions used in subsequent computations become homogeneous and we can restrict ourselves to homogeneous components.

2.1. Nonlocal variables. We looked for the coverings associated with conservation laws of the dB-equation and found the variables p_{1}, p_{2}, p_{3}, p_{5}, …, defined by

(p_{1})_{x} = w,  (p_{1})_{t} = u,
(p_{2})_{x} = u,  (p_{2})_{t} = \frac{1}{2}(2v + w^{2}),
(p_{3})_{x} = v + w^{2},  (p_{3})_{t} = −uw,
(p_{5})_{x} = (u^{2} + 2vw + 2w^{3})/2,  (p_{5})_{t} = uv.

The subscript refers to the grading of the corresponding variable. Thus,

|p_{1}| = 1, |p_{2}| = 2, |p_{3}| = 3, |p_{5}| = 5.

Further computations show that exactly one new conservation law arises at each grading level except for gradings 4, 8, 12, … We use the corresponding nonlocal variables to construct nonlocal symmetries and cosymmetries in Subsections 2.2 and 2.3.

Remark 13. Void positions with gradings divisible by 4 are occupied by nonlocal variables of the next level of nonlocality.

(q_{0})_{x} = p_{1},  (q_{0})_{t} = p_{2},
(q_{4})_{x} = p_{3}w − 3p_{5},  (q_{4})_{t} = p_{3}u − 2p_{6},

etc. Moreover, we found two independent nonlocal conservation laws for each of gradings 8 and 12.

2.2. Symmetries. The defining equations for symmetries (or their shadows in the nonlocal case) of the dB-equation are

D̃_{t}(φ^{w}) = D̃_{x}(φ^{u}),
D̃_{t}(φ^{u}) = wD̃_{x}(φ^{w}) + w_{1}φ^{w} + D̃_{x}(φ^{v}),  (58)
D̃_{t}(φ^{v}) = −uD̃_{x}(φ^{w}) − 3u_{1}φ^{w} − 3wD̃_{x}(φ^{u}) − w_{1}φ^{u}.

The grading of a symmetry is understood as the grading of the corresponding vector field and hence

|φ| = |φ^{w}| − |w| = |φ^{u}| − |u| = |φ^{v}| − |v| .

2.2.1. (x, t)-independent local symmetries. Solving equations (58) we found a number of local symmetries independent of x and t. Here are the results for grading ≤ 7:

φ_{−4}^{w} = 0,  φ_{1}^{w} = w_{1},  φ_{2}^{w} = u_{1},
φ_{−4}^{u} = 0,  φ_{1}^{u} = u_{1},  φ_{2}^{u} = w_{1}w + v_{1},
φ_{−4}^{v} = 1,  φ_{1}^{v} = v_{1},  φ_{2}^{v} = −w_{1}u − 3u_{1}w,

and

φ_{3}^{w} = 2w_{1}w + v_{1},
φ_{3}^{u} = −w_{1}u − u_{1}w,
φ_{3}^{v} = −3w_{1}w^{2} − u_{1}u − 2v_{1}w,
φ_{5}^{w} = w_{1}(v + 3w^{2}) + u_{1}u + v_{1}w,
φ_{5}^{u} = u_{1}v + v_{1}u,
φ_{5}^{v} = w_{1}(−u^{2} − 3w^{3}) − 4u_{1}uw + v_{1}(v − w^{2}),
φ_{6}^{w} = 2w_{1}uw + u_{1}(2v + w^{2}) + 2v_{1}u,
φ_{6}^{u} = w_{1}(−2u^{2} + 2vw + w^{3}) − 4u_{1}uw + v_{1}(2v + w^{2}),
φ_{6}^{v} = w_{1}u(−2v − 7w^{2}) + u_{1}(−2u^{2} − 6vw − 3w^{3}) − 6v_{1}uw,
φ_{7}^{w} = w_{1}(−3u^{2} + 12vw + 14w^{3}) − 6u_{1}uw + 6v_{1}(v + w^{2}),

φ_{7}^{u} = 6w_{1}u(−v − 2w^{2}) + u_{1}(−3u^{2} − 6vw − 4w^{3}) − 6v_{1}uw,
φ_{7}^{v} = 6w_{1}w(u^{2} − 3vw − 3w^{3}) + 6u_{1}u(−v + 2w^{2}) + v_{1}(−3u^{2} − 12vw − 10w^{3}).

The subscript in the notation above equals the grading of the corresponding symmetry. We computed this kind of symmetries up to grading 15 and found that they exist at all levels except for 4, 8, 12.

Remark 14. Note that only the first three symmetries are classical ones. All other symmetries are higher despite the fact that their jet order equals 1.

2.2.2. Nonlocal symmetries. Nonlocal symmetries (or, to be more precise, their shadows) arise at gradings 4, 8, 12, etc. We computed these symmetries up to grading 16 and found that at level 4 there exists one symmetry, at level 8 two of them, at level 12 three symmetries arise, etc. For example, at level 4 we have the symmetry φ_{4,1} with the components

φ_{4,1}^{w} = −20p_{3}w_{1} − 12p_{2}u_{1} − 6p_{1}(2w_{1}w + v_{1}) + 9u^{2} + 16vw + 16w^{3}
φ_{4,1}^{u} = −20p_{3}u_{1} − 12p_{2}(w_{1}w + v_{1}) + 6p_{1}(w_{1}u + u_{1}w) + 8u(3v + w^{2}),
φ_{4,1}^{v} = −20p_{3}v_{1} + 12p_{2}(w_{1}u + 3u_{1}w) + 6p_{1}(3w_{1}w^{2} + u_{1}u + 2v_{1}w)
− 39u^{2}w + 16v^{2} − 12w^{4},

while at level 8 we have two symmetries φ_{8,1} and φ_{8,2} with the components

φ_{8,1}^{w} = −1764p_{7}w_{1} − 980p_{6}u_{1} + 1176p_{5}(−2w_{1}w − v_{1})
+ 280p_{3}(w_{1}v + 3w_{1}w^{2} + u_{1}u + v_{1}w) + 42p_{2}(2w_{1}u + 2u_{1}v + u_{1}w^{2} + 2v_{1}u)
+ 14(54u^{2}v + 9u^{2}w^{2} + 48v^{2}w + 96vw^{3} + 52w^{5}),
φ_{8,1}^{u} = −1764p_{7}u_{1} − 980p_{6}(w_{1}w + v_{1}) + 1176p_{5}(w_{1}u + u_{1}w) + 280p_{3}(u_{1}v + v_{1}u)
+ 42p_{2}(−2w_{1}u^{2} + 2w_{1}vw + w_{1}w^{3} − 4u_{1}uw + 2v_{1}v + v_{1}w^{2})
+ 28u(−27u^{2}w + 36v^{2} + 24vw^{2} − w^{4}),
φ_{8,1}^{v} = −1764p_{7}v_{1} + 980p_{6}(w_{1}u + 3u_{1}w) + 1176p_{5}(3w_{1}w^{2} + u_{1}u + 2v_{1}w)
+ 280p_{3}(−w_{1}u^{2} − 3w_{1}w^{3} − 4u_{1}uw + v_{1}v − v_{1}w^{2})
+ 42p_{2}(−2w_{1}uv − 7w_{1}uw^{2} − 2u_{1}u^{2} − 6u_{1}vw − 3u_{1}w^{3} − 6v_{1}uw)
+ 7(−27u^{4} − 468u^{2}vw − 190u^{2}w^{3} + 64v^{3} − 144vw^{4} − 96w^{6})

and

φ^{w}(8, 2) = −2184p_{7}w_{1} − 1456p_{6}u_{1} + 2184p_{5}(−2w_{1}w − v_{1})
+ 1040p_{3}(w_{1}v + 3w_{1}w^{2} + u_{1}u + v_{1}w)
+ 312p_{2}(2w_{1}uw + 2u_{1}v + u_{1}w^{2} + 2v_{1}u)
+ 52p_{1}(−3w_{1}u^{2} + 12w_{1}vw + 14w_{1}w^{3} − 6u_{1}uw + 6v_{1}v + 6v_{1}w^{2}),

φ^{u}(8, 2) = −2184p_{7}u_{1} − 1456p_{6}(w_{1}w + v_{1}) + 2184p_{5}(w_{1}u + u_{1}w)
+ 1040p_{3}(u_{1}v + v_{1}u)
+ 312p_{2}(−2w_{1}u^{2} + 2w_{1}vw + w_{1}w^{3} − 4u_{1}uw + 2v_{1}v + v_{1}w^{2})
+ 52p_{1}(−6w_{1}uv − 12w_{1}uw^{2} − 3u_{1}u^{2} − 6u_{1}vw − 4u_{1}w^{3} − 6v_{1}uw),
φ^{v}(8, 2) = −2184p_{7}v_{1} + 1456p_{6}(w_{1}u + 3u_{1}w) + 2184p_{5}(3w_{1}w^{2} + u_{1}u + 2v_{1}w)
+ 1040p_{3}(−w_{1}u^{2} − 3w_{1}w^{3} − 4u_{1}uw + v_{1}v − v_{1}w^{2})
+ 312p_{2}(−2w_{1}uv − 7w_{1}uw^{2} − 2u_{1}u^{2} − 6u_{1}vw − 3u_{1}w^{3} − 6v_{1}uw)
+ 52p_{1}(6w_{1}u^{2}w − 18w_{1}vw^{2} − 18w_{1}w^{4} − 6u_{1}uv
+ 12u_{1}uw^{2} − 3v_{1}u^{2} − 12v_{1}vw − 10v_{1}w^{3}).

2.2.3. (x, t)-dependent symmetries. These symmetries arise at levels 0, 4, 8, etc., and at each level we have two new symmetries φ̄_{0,1}, φ̄_{0,2}, φ̄_{4,1}, φ̄_{4,2}, … For example, we have

φ̄_{0,1}^{w} = xw_{1} − 2w,  φ̄_{0,2}^{w} = tu_{1} + 2w,
φ̄_{0,1}^{u} = xu_{1} − 3u,  φ̄_{0,2}^{u} = t(w_{1}w + v_{1}) + 3u,
φ̄_{0,1}^{v} = xv_{1} − 4v,  φ̄_{0,2}^{v} = −t(w_{1}u + 3u_{1}w) + 4v

and

φ̄_{4,1}^{w} = x(w_{1}v + 3w_{1}w^{2} + u_{1}u + v_{1}w) − 3p_{1}(2w_{1}w + v_{1}) − 4p_{2}u_{1} − 5p_{3}w_{1},
φ̄_{4,1}^{u} = x(u_{1}v + v_{1}u) + 3p_{1}(w_{1}u + u_{1}w) − 4p_{2}(w_{1}w + v_{1}) − 5p_{3}u_{1},
φ̄_{4,1}^{v} = −x(w_{1}u^{2} + 3w_{1}w^{3} + 4u_{1}uw − v_{1}v + v_{1}w^{2}) + 3p_{1}(3w_{1}w^{2} + u_{1}u + 2v_{1}w)
+ 4p_{2}(w_{1}u + 3u_{1}w) − 5p_{3}v_{1},
φ̄_{4,2}^{w} = t(2w_{1}uw + 2u_{1}v + u_{1}w^{2} + 2v_{1}u) + 4p_{1}(2w_{1}w + v_{1}) + 6p_{2}u_{1} + 8p_{3}w_{1},
φ̄_{4,2}^{u} = −t(2w_{1}u^{2} − 2w_{1}vw − w_{1}w^{3} + 4u_{1}uw − 2v_{1}v − v_{1}w^{2})
− 4p_{1}(w_{1}u + u_{1}w) + 6p_{2}(w_{1}w + v_{1}) + 8p_{3}u_{1},
φ̄_{4,2}^{v} = −t(2w_{1}uv + 7w_{1}uw^{2} + 2u_{1}u^{2} + 6u_{1}vw + 3u_{1}w^{3} + 6v_{1}uw)
− 4p_{1}(3w_{1}w^{2} + u_{1}u + 2v_{1}w) − 6p_{2}(w_{1}u + 3u_{1}w) + 8p_{3}v_{1}.

Note that only the two symmetries are local.

2.2.4. Distribution of symmetries. We present the distribution of symmetries described above in Table 1.

2.3. Cosymmetries. The defining equations for cosymmetries (or their shadows) of the dB-equation are

D̃_{t}(ψ^{w}) = wD̃_{x}(ψ^{u}) − uD̃_{x}(ψ^{v}) + 2u_{1}ψ^{v},

-4 -3 -2 -1 0  1 2 3  4  5 6 7  8  9 10 11  12
φ_{−4}  φ_{1} φ_{2} φ_{3}  φ_{5} φ_{6} φ_{7}  φ_{9} φ_{10} φ_{11}
φ_{4,1}  φ_{8,1}  φ_{12,1}
φ_{8,2}  φ_{12,2}
φ_{12,3}
φ̄_{0,1}  φ̄_{4,1}  φ̄_{8,1}  φ̄_{12,1}
φ̄_{0,2}  φ̄_{4,2}  φ̄_{8,2}  φ̄_{12,2}

Table 1. Distribution of symmetries along gradings

D̃_{t}(ψ^{u}) = D̃_{x}(ψ^{w}) − 3wD̃_{x}(ψ^{v}) − 2w_{1}ψ^{v},  (59)
D̃_{t}(ψ^{v}) = D̃_{x}(ψ^{u}).

The grading of a cosymmetry is understood as the grading of the corresponding differential form and thus

|ψ| = |ψ^{w}| + |w| = |ψ^{u}| + |u| = |ψ^{v}| + |v| .

2.3.1. (x, t)-independent local cosymmetries. We computed these cosymmetries up to grading 16 and found that they exist at all levels except for 1, 5, 9, etc. Here we present the results for gradings ≤ 8:

ψ_{2}^{w} = 1,  ψ_{3}^{w} = 0,  ψ_{4}^{w} = w,
ψ_{2}^{u} = 0,  ψ_{3}^{u} = 1,  ψ_{4}^{u} = 0,
ψ_{2}^{v} = 0,  ψ_{3}^{v} = 0,  ψ_{4}^{v} = \frac{1}{2},
ψ_{6}^{w} = v + 3w^{2},  ψ_{7}^{w} = uw,  ψ_{8}^{w} = \frac{1}{14}(−3u^{2} + 12vw + 14w^{3})
ψ_{6}^{u} = u,  ψ_{7}^{u} = \frac{1}{2}(2v + w^{2}),  ψ_{8}^{u} = −\frac{3}{7}uw,
ψ_{6}^{v} = w,  ψ_{7}^{v} = u,  ψ_{8}^{v} = \frac{3}{7}(v + w^{2}).

2.3.2. Nonlocal cosymmetries. Nonlocal cosymmetries independent of x and t were found at levels 9 (one cosymmetry), 13 (two cosymmetries), and 17 (three cosymmetries). For example, we have

ψ_{9,1}^{w} = \frac{1}{14}(−42p_{7} − 84p_{5}w + 20p_{3}(v + 3w^{2}) + 12p_{2}uw
+ p_{1}(−3u^{2} + 12vw + 14w^{3})),
ψ_{9,1}^{u} = \frac{1}{7}(−14p_{6} + 10p_{3}u + 3p_{2}(2v + w^{2}) − 3p_{1}uw),
ψ_{9,1}^{v} = \frac{1}{7}(−21p_{5} + 10p_{3}w + 6p_{2}u + 3p_{1}(v + w^{2})),
ψ_{13,1}^{w} = \frac{1}{3}(195p_{11} + 484p_{9}w − 126p_{7}(v − 3w^{2}) − 70p_{6}uw

+ 14p_{5}(3u^{2} − 12vw − 14w^{3}) + 5p_{3}(2v^{2} + 12vw^{2} + 11w^{4})

+ p_{2}u(−2u^{2} + 6vw + 3w^{3})),

ψ_{13,1}^{u} = \frac{1}{12}(297p_{10} − 504p_{7}u − 140p_{6}(2v + w^{2}) + 336p_{5}uw + 80p_{3}uv

+ 3p_{2}(−8u^{2}w + 4v^{2} + 4vw^{2} + w^{4})),

ψ_{13,1}^{v} = \frac{1}{3}(242p_{9} − 126p_{7}w − 70p_{6}u − 84p_{5}(v + w^{2}) + 10p_{3}(u^{2} + 2vw + 2w^{3})

+ 3p_{2}u(2v + w^{2})),

and

ψ_{13,2}^{w} = \frac{1}{9}(−351p_{11} − 847p_{9}w + 210p_{7}(v + 3w^{2}) + 112p_{6}uw

+ 21p_{5}(−3u^{2} + 12vw + 14w^{3}) − 5p_{3}(2v^{2} + 12vw^{2} + 11w^{4})

+ p_{1}(−3u^{2}v − 6u^{2}w^{2} + 6v^{2}w + 14vw^{3} + 9w^{5})),

ψ_{13,2}^{u} = \frac{1}{9}(−132p_{10} + 210p_{7}u + 56p_{6}(2v + w^{2}) − 126p_{5}uw − 20p_{3}uv

− p_{1}u(u^{2} + 6vw + 4w^{3})),

ψ_{13,2}^{v} = \frac{1}{18}(−847p_{9} + 420p_{7}w + 224p_{6}u + 252p_{5}(v + w^{2})

− 20p_{3}(u^{2} + 2vw + 2w^{3}) + p_{1}(−6u^{2}w + 6v^{2} + 12vw^{2} + 7w^{4})).

2.3.3. (x, t)-dependent cosymmetries. These cosymmetries exist at levels 1, 5, 9, etc. The first one,

ψ̄_{1,1}^{w} = x, ψ̄_{1,1}^{u} = t, ψ̄_{1,1}^{v} = 0,

is local, all others are nonlocal: two at level 5,

ψ̄_{5,1}^{w} = x(v + 3w^{2}) − 6p_{1}w − 5p_{3},  ψ̄_{5,2}^{w} = tuw + 4p_{1}w + 4p_{3},
ψ̄_{5,1}^{u} = xu − 4p_{2},  ψ̄_{5,2}^{u} = \frac{1}{2}(t(2v + w^{2}) + 6p_{2}),
ψ̄_{5,1}^{v} = xw − 3p_{1},  ψ̄_{5,2}^{v} = tu + 2p_{1},

two at level 9,

ψ̄_{9,1}^{w} = \frac{1}{11}(x(2v^{2} + 12vw^{2} + 11w^{4}) + 8p_{2}uw + 20p_{3}(v + 3w^{2})

− 112p_{5}w − 63p_{7}),

ψ̄_{9,1}^{u} = \frac{4}{11}(xuv + p_{2}(2v + w^{2}) + 5p_{3} − 10p_{6}),
ψ̄_{9,1}^{v} = \frac{2}{11}(x(u^{2} + 2vw + 2w^{3}) + 4p_{2}u + 10p_{3}w − 28p_{5}),

ψ̄_{9,2}^{w} = \frac{1}{3}(tu(−2u^{2} + 6vw + 3w^{3}) − 6p_{2}uw − 16p_{3}(v + 3w^{2}) + 96p_{5}w + 56p_{7}),
ψ̄_{9,2}^{u} = \frac{1}{12}(3t(−8u^{2}w + 4v^{2} + 4vw^{2} + w^{4}) − 12p_{2}(2v + w^{2}) − 64p_{3}u + 140p_{6}),
ψ̄_{9,2}^{w} = \frac{1}{3}(3tu(2v + w^{2}) − 6p_{2}u − 16p_{3}w + 48p_{5}),

etc.

2.3.4. Distribution of cosymmetries. We present the distribution of cosymmetries described above in Table 2.

1  2 3 4  5  6 7 8  9  10 11 12  13  14 15 16  17
ψ_{2} ψ_{3} ψ_{4}  ψ_{6} ψ_{7} ψ_{8}  ψ_{10} ψ_{11} ψ_{12}  ψ_{14} ψ_{15} ψ_{16}
ψ_{9,1}  ψ_{13,1}  ψ_{17,1}
ψ_{13,2}  ψ_{17,2}
ψ_{17,3}
ψ̄_{1,1}  ψ̄_{5,1}  ψ̄_{9,1}  ψ̄_{13,1}  ψ̄_{17,1}
ψ̄_{5,2}  ψ̄_{9,2}  ψ̄_{13,2}  ψ̄_{17,2}

Table 2. Distribution of cosymmetries along gradings

2.4. Nonlocal vectors. We construct nonlocal vectors on the ℓ_{E}^{∗}-covering (see Remark 9) associated with symmetries φ_{−4}, …, φ_{7} and denote them by ρ^{−4}, …, ρ^{7}, respectively. Thus, by definition, we have

D̃_{x}(ρ^{i}) = ρ^{w}φ_{i}^{w} + ρ^{u}φ_{i}^{u} + ρ^{v}φ_{i}^{v}, i = −4, 1, 2, 3, 5, 6, 7.

The corresponding t-components are given by the following relations:

D̃_{t}(ρ^{i}) = ρ^{w}φ_{i}^{u} + ρ^{u}(φ_{i}^{v} + φ_{i}^{w}w) − ρ^{v}(3φ_{i}^{u}w + φ_{i}^{w}u).

Here ρ^{w}, ρ^{u}, ρ^{v} are the coordinates in the ℓ_{E}^{∗}-covering (see equations (61) below).

2.5. Nonlocal forms. We also constructed nonlocal forms associated to the cosymmetries ψ_{2}, ψ_{3}, ψ_{4}, and ψ_{6}, ψ_{7}, ψ_{8}. We denote these forms by ω^{2}, …, ω^{8} and have by definition

D̃_{x}(ω^{i}) = ψ_{i}^{w}ω^{w} + ψ_{i}^{u}ω^{u} + ψ_{i}^{v}ω^{v}, i = 2, 3, 4, 6, 7, 8,

where ω^{w}, ω^{u}, ω^{v} are the coordinates in the ℓ_{E}-covering (see equations (60)). The t-components of the nonlocal forms under consideration are given by the equations

D̃_{t}(ω^{i}) = (ψ_{i}^{u}w − ψ_{i}^{v}u)ω^{w} + (ψ_{i}^{w} − 3ψ_{i}^{v}w)ω^{u} + ψ_{i}^{u}ω^{v}.

3. The main results

We begin with the defining equations for the ℓ_{E}- and ℓ_{E}^{∗}-coverings of the dB-equation. In the first case we have

ω_{t}^{w} = ω_{x}^{u},
ω_{t}^{u} = wω_{x}^{w} + w_{x}ω^{w} + ω_{x}^{v},  (60)
ω_{t}^{v} = −uω_{x}^{w} − w_{x}ω^{u} − 3wω_{x}^{u} − 3u_{x}ω^{w},

while the equations of the ℓ_{E}^{∗}-covering are

ρ_{t}^{w} = wρ_{x}^{u} − uρ_{x}^{v} + 2u_{x}ρ^{v},
ρ_{t}^{u} = ρ_{x}^{w} − 3wρ_{x}^{v} − 2w_{x}ρ^{v},  (61)
ρ_{t}^{v} = ρ_{x}^{u}.

The total derivatives on the ℓ_{E}-covering are of the form

D̃_{x} = D_{x} + ∑_{i≥0} (ω_{i+1}^{w} \frac{∂}{∂ω_{i}^{w}} + ω_{i+1}^{u} \frac{∂}{∂ω_{i}^{u}} + ω_{i+1}^{v} \frac{∂}{∂ω_{i}^{v}} ),

D̃_{t} = D_{t} + ∑_{i≥0} (ω_{i+1}^{u} \frac{∂}{∂ω_{i}^{w}} + D̃_{x}^{i}(wω_{1}^{w} + w_{1}ω^{w} + ω_{1}^{v})\frac{∂}{∂ω_{i}^{u}}  (62)

− D̃_{x}^{i}(uω_{1}^{w} + w_{1}ω^{u} + 3wω_{1}^{u} + 3u_{1}ω^{w})\frac{∂}{∂ω_{i}^{v}} ),

where D_{x} and D_{t} are total derivatives on the dB-equation, while on the ℓ_{E}^{∗}-covering they are presented by

D̃_{x} = D_{x} + ∑_{i≥0} (ρ_{i+1}^{w} \frac{∂}{∂ρ_{i}^{w}} + ρ_{i+1}^{u} \frac{∂}{∂ρ_{i}^{u}} + ρ_{i+1}^{v} \frac{∂}{∂ρ_{i}^{v}} ),

D̃_{t} = D_{t} + ∑_{≥0} (D̃_{x}(wρ_{1}^{u} − uρ_{1}^{v} + 2u_{1}ρ^{v})\frac{∂}{∂ρ_{i}^{w}}  (63)

+ D̃_{x}(ρ_{1}^{w} − 3wρ_{1}^{v} − 2w_{1}ρ^{v})\frac{∂}{∂ρ_{i}^{u}} + ρ_{i+1}^{u} \frac{∂}{∂ρ_{i}^{v}} ).

3.1. Recursion operators for symmetries. We construct recursion operators R = (R^{w}, R^{u}, R^{v}) for symmetries of the dB-equation solving the system

D̃_{t}(R^{w}) = D̃_{x}(R^{u}),
D̃_{t}(R^{u}) = wD̃_{x}(R^{w}) + w_{1}R^{w} + D̃_{x}(R^{v}),
D̃_{t}(R^{v}) = −uD̃_{x}(R^{w}) − 3u_{1}R^{w} − 3wD̃_{x}(R^{u}) − w_{1}R^{u},

where the total derivatives are defined by formulas (62). Recall that we look for solutions linear in ω_{i}’s and admit their dependency on nonlocal forms. Besides the trivial solution

R_{0}^{w} = ω^{w}, R_{0}^{u} = ω^{u}, R_{0}^{v} = ω^{v}

that corresponds to the identity operator, we found the following two nontrivial solutions of grading 4:

R_{4,1}^{w} = (2ω_{4}w_{1} + ω^{3}u_{1} + ω^{2}(2w_{1}w + v_{1}))/2,
R_{4,1}^{u} = (2ω^{4}u_{1} + ω^{3}(w_{1}w + v_{1}) − ω^{2}(w_{1}u + u_{1}w))/2,
R_{4,1}^{v} = (2ω^{4}v_{1} + ω^{3}(−w_{1}u − 3u_{1}w) + ω^{2}(−3w_{1}w^{2} − u_{1}u − 2v_{1}w))/2,
R_{4,2}^{w} = (−4ω^{4}w_{1} − ω^{3}u_{1} + 2ω^{v}w + 3ω^{u}u + 4ω^{w}(v + 2w^{2}))/8,
R_{4,2}^{u} = (−4ω^{4}u_{1} − ω^{3}(w_{1}w + v_{1}) + 3ω^{v}u + 2ω^{u}(2v + w^{2}) + ω^{w}uw)/8,
R_{4,2}^{v} = (−4ω^{4}v_{1} + ω^{3}(w_{1}u + 3u_{1}w) + 4ω^{v}v − 11ω^{u}uw + 3ω^{w}(−u^{2} − 2w^{3}))/8.

Remark 15. We also found three solutions at level 8, but they are too cumbersome to present them here.

The operators corresponding to the above solutions are

R_{4,1} = \frac{1}{2}⟨2φ_{1}, φ_{2}, φ_{3} | D_{x}^{−1} | ψ_{4}, ψ_{3}, ψ_{2}⟩  (64)

and

R_{4,2} = −\frac{1}{8}⟨4φ_{1}, φ_{2} | D_{x}^{−1} | ψ_{4}, ψ_{3}⟩

+ \frac{1}{8} (\begin{matrix}4(v + 2w^{2}) & 3u & 2w \\ uw & 2(2v + w^{2}) & 3u \\ −3(u^{2} + 2w^{3}) & −11uw & 4v \end{matrix}) (65)

(here and below we use the notation introduced in Remark 10). The explicit expressions for symmetries φ_{i} and cosymmetries ψ_{j} are given in Subsections 2.2.1 and 2.3.1.

3.2. Recursion operators for cosymmetries. Recursion operators for cosymmetries are of the form R^{∗} = (R^{∗,u}, R^{∗,u}, R^{∗,v}) and can be constructed from the equation

D̃_{t}(R^{∗,w}) = wD̃_{x}(R^{∗,u}) − uD̃_{x}(R^{∗,v}) + 2u_{1}R^{∗,v},
D̃_{t}(R^{∗,u}) = D̃_{x}(R^{∗,w}) − 3wD̃_{x}(R^{∗,v}) − 2w_{1}R^{∗,v},
D̃_{t}(R^{∗,v}) = D̃_{x}(R^{∗,u}),

where the total derivatives are given by formulas (63). The first two nontrivial solutions (both of grading 4) are

R_{4,1}^{∗,w} = (ρ^{3} + 2ρ^{1}w)/2,
R_{4,1}^{∗,u} = ρ^{2}/2,

THE D-BOUSSINESQ EQUATION 23

R_{4,1}^{∗,v} = ρ^{1}/2,
R_{4,2}^{∗,w} = (−2ρ^{3} + 3ρ^{v}(−2w^{3} − u^{2}) + ρ^{u}wu + 4ρ^{w}(2w^{2} + v))/8,
R_{4,2}^{∗,u} = (−ρ^{2} − 11ρ^{v}wu + 2ρ^{u}(w^{2} + 2v) + 3ρ^{w}u)/8,
R_{4,2}^{∗,v} = (4ρ^{v}v + 3ρ^{u}u + 2ρ^{w}w)/8.

The operators corresponding to these solutions are

R_{4,1}^{∗} = \frac{1}{2}⟨2ψ_{4}, ψ_{3}, ψ_{2} | D_{x}^{−1} | φ_{1}, φ_{2}, φ_{3}⟩  (66)

and

R_{4,2}^{∗} = −\frac{1}{8}⟨ψ_{3}, 2ψ_{2} | D_{x}^{−1} | φ_{2}, φ_{3}⟩

+ \frac{1}{8} (\begin{matrix}4(2w^{2} + v) & wu & −3(2w^{3} + u^{2}) \\ 3u & 2(w^{2} + 2v) & −11wu \\ 2w & 3u & 4v \end{matrix}) . (67)

3.3. Hamiltonian structures. The first step to construct Hamiltonian structures H = (H^{w}, H^{u}, H^{v}) is to solve the equation

D̃_{t}(H^{w}) = D̃_{x}(H^{u}),
D̃_{t}(H^{u}) = wD̃_{x}(H^{w}) + w_{1}H^{w} + D̃_{x}(H^{v}),
D̃_{t}(H^{v}) = −uD̃_{x}(H^{w}) − 3u_{1}H^{w} − 3wD̃_{x}(H^{u}) − w_{1}H^{u}

with total derivatives defined by formulas (63). The first three solutions are

H_{0,1}^{w} = ρ_{1}^{v},
H_{0,1}^{u} = ρ_{1}^{u},
H_{0,1}^{v} = −4ρ_{1}^{v}w + ρ_{1}^{w} − 2ρ^{v}w_{1},
H_{4,1}^{w} = (4ρ_{1}^{v}v + 3ρ_{1}^{u}u + 2ρ_{1}^{w}w + 2ρ^{v}v_{1} + ρ^{u}u_{1})/2,
H_{4,1}^{u} = (−11ρ_{1}^{v}wu + 2ρ_{1}^{u}(w^{2} + 2v) + 3ρ_{1}^{w}u − 2ρ^{v}(wu_{1} + 4uw_{1})
+ ρ^{u}(ww_{1} + v_{1}))/2,
H_{4,1}^{v} = (−ρ_{1}^{v}(6w^{3} + 16wv + 3u^{2}) − 11ρ_{1}^{u}wu + 4ρ_{1}^{w}v
− 2ρ^{v}(3w^{2}w_{1} + 2wv_{1} + uu_{1} + 4vw_{1})
+ ρ^{u}(−3wu_{1} − uw_{1}))/2,
H_{4,2}^{w} = ρ^{v}v_{1} + ρ^{u}u_{1} + ρ^{w}w_{1},
H_{4,2}^{u} = ρ^{v}(3wu_{1} + uw_{1}) + ρ^{u}(ww_{1} + v_{1}) + ρ^{w}u_{1},
H_{4,2}^{v} = ρ^{v}(−3w^{2}w_{1} − 4wv_{1} − uu_{1}) + ρ^{u}(−3wu_{1} − uw_{1}) + ρ^{w}v_{1}.

The operators corresponding to these solutions are

H_{0,1} = H_{1} = (\begin{matrix}0 & 0 & D_{x} \\ 0 & D_{x} & 0 \\ D_{x} & 0 & −4D_{x}w − 2w_{1} \end{matrix})

and

H_{4,1} = \frac{1}{2} (\begin{matrix}2wD_{x} & 3uD_{x} + u_{1} & 2(2vD_{x} + v_{1}) \\ 3uD_{x} & 2(w^{2} + 2v)D_{x} + ww_{1} + v_{1} & −11wuD_{x} − 2(wu_{1} + 4uw_{1}) \\ 4vD_{x} & −11wuD_{x} − 3wu_{1} − uw_{1} & h_{2,2}^{1}D_{x} + h_{2,2}^{0} \end{matrix}) ,

H_{4,2} = (\begin{matrix}w_{1} & u_{1} & v_{1} \\ u_{1} & ww_{1} + v1 & −3wu_{1} − uw_{1} \\ v_{1} & −3wu_{1} − uw_{1} & −3w^{2}w_{1} − 4wv_{1} − uu_{1} \end{matrix}) ,

where

h_{2,2}^{1} = −(6w^{3} + 16wv + 3u^{2}), h_{2,2}^{0} = −2(3w^{2}w_{1} + 2wv_{1} + uu_{1} + 4vw_{1}).

The operator H_{1} is skew-adjoint and by Theorem 5 is a Hamiltonian structure for the dB-equation, but neither of the last two operators is Hamiltonian. Nevertheless, their linear combination

H_{2} = H_{4,1} + \frac{1}{2}H_{4,2}

is skew-adjoint and consequently Hamiltonian. Again, by Theorem 5 the structures H_{1} and H_{2} are compatible.

3.4. Symplectic structures. To find symplectic structures S = (S^{w}, S^{u}, S)^{v}, we solve the equation

D̃_{t}(S^{w}) = wD̃_{x}(S^{u}) − uD̃_{x}(S^{v}) + 2u_{1}S^{v},
D̃_{t}(S^{u}) = D̃_{x}(S^{w}) − 3wD̃_{x}(S^{v}) − 2w_{1}S^{v},
D̃_{t}(S^{v}) = D̃_{x}(S^{u}),

where the total derivatives are given by formulas (62). We obtained one solution

S_{0,1}^{w} = ω^{4} + ω^{2}w,
S_{0,1}^{u} = ω^{3}/2,
S_{0,1}^{v} = ω^{2}/2

of grading 0 and to solutions of grading 4:

S_{4,1}^{w} = −14ω^{8} − 8ω^{6}w + 4ω^{4}(3w^{2} + v) + ω^{3}wu,
S_{4,1}^{u} = (−10ω^{7} + 8ω^{4}u + ω^{3}(w^{2} + 2v))/2,
S_{4,1}^{v} = −4ω^{6} + 4ω^{4}w + ω^{3}u,
S_{4,2}^{w} = (70ω^{8} + 36ω^{6}w − 12ω^{4}(3w^{2} + v) + ω^{2}(14w^{3} + 12wv − 3u^{2}))/14,

S_{4,2}^{u} = (12ω^{7} − 6ω^{4}u − 3ω^{2}wu)/7,
S_{4,2}^{v} = (9ω^{6} − 6ω^{4}w + 3ω^{2}(w^{2} + v))/7.

The operator

S_{1} = S_{0,1} = ⟨ψ_{4}, \frac{1}{2}ψ_{3}, ψ_{2} | D_{x}^{−1} | ψ_{2}, ψ_{3}, ψ_{4}⟩

corresponds to the first solution, while to the second and the third ones the operators

S_{4,1} = ⟨ψ_{7}, 4ψ_{6}, −8ψ_{4}, −5ψ_{3}, −14ψ_{2} | D_{x}^{−1} | ψ_{3}, ψ_{4}, ψ_{6}, ψ_{7}, ψ_{8}⟩

and

S_{4,2} = ⟨ψ_{8}, −\frac{6}{7}ψ_{6}, \frac{18}{7}ψ_{4}, \frac{12}{7}ψ_{3}, 5ψ_{2} | D_{x}^{−1} | ψ_{2}, ψ_{4}, ψ_{6}, ψ_{7}, ψ_{8}⟩

correspond. The operator S_{1} is skew-adjoint and hence is a symplectic structure for the dBequation. In addition, the operator

S_{2} = 2S_{4,1} + 7S_{4,2}

is a symplectic structure as well.

4. Interrelations

We shall now present the basic algebraic relations between the above described structures. They include description of the following compositions:

R ∘ R, R ∘ H, S ∘ R, H ∘ S, S ∘ H, H ∘ R^{∗}, R^{∗} ∘ S, R^{∗} ∘ R^{∗},

together with the action of the operators R on symmetries and of R^{∗} on cosymmetries and as well as commutator relations between symmetries. Of course, some of these relations are deducible from the other ones, but we prefer explicit presentation.

4.1. R ∘ R. Compositions of the first two recursion operators look as follows

R_{4,1} ∘ R_{4,1} = \frac{1}{4}R_{8,1} + \frac{7}{12}R_{8,2},
R_{4,1} ∘ R_{4,2} = −\frac{1}{16}R_{8,1},
R_{4,2} ∘ R_{4,1} = \frac{3}{16}R_{8,1} + \frac{7}{12}R_{8,2},
R_{4,2} ∘ R_{4,2} = −\frac{3}{64}R_{8,1} + \frac{13}{16}R_{8,3}.

So we have

R_{4,2} ∘ R_{4,1} = R_{4,1} ∘ R_{4,2} + R_{4,1} ∘ R_{4,1},

or

[R_{4,2}, R_{4,1}] = R_{4,1}^{2}.  (68)

Actually, this relation determines the entire structure of all algebraic invariants related to the dB-equation.

4.2. R ∘ H. We computed the following compositions:

R_{4,1} ∘ H_{0,1} =\frac{1}{2}H_{4,2},
R_{4,1} ∘ H_{4,1} = − H_{8,1} + H_{8,2} + 2H_{8,3},
R_{4,1} ∘ H_{4,2} =H_{8,1},
R_{4,2} ∘ H_{0,1} =\frac{1}{4}H_{4,1} − \frac{1}{4}H_{4,2},
R_{4,2} ∘ H_{4,1} = − \frac{1}{2}H_{8,2} − \frac{1}{2}H_{8,3},
R_{4,2} ∘ H_{4,2} =H_{8,1} + \frac{1}{2}H_{8,2} + \frac{1}{H}_{8,3},
R_{8,1} ∘ H_{0,1} =8H_{8,1} − 4H_{8,2} − 8H_{8,3},
R_{8,2} ∘ H_{0,1} = − \frac{18}{7}H_{8,1} + \frac{12}{7}H_{8,2} + \frac{24}{7}H_{8,3},
R_{8,3} ∘ H_{0,1} =\frac{6}{13}H_{8,1} − \frac{7}{13}H_{8,2} − \frac{12}{13}H_{8,3}.

etc. These relations imply, in particular, the equality

(3R_{4,1} + 4R_{4,2}) ∘ H_{0,1} = H_{4,1} + \frac{1}{2}H_{4,2}.

Denoting 3R_{4,1} + 4R_{4,2} by R, we have

R ∘ H_{1} = H_{2},  (69)

where H_{1} and H_{2} are the Hamiltonian structures introduced in Subsection 3.3.

Proposition 3. The operator

R = 3R_{4,1} + 4R_{4,2}  (70)

satisfies the Nijenhuis conditions (see Remark 5) and generates and infinite family of compatible Hamiltonian structures

H_{n} = R^{n−1} ∘ H_{1}

for the dB-equation.

4.3. S ∘ R. These compositions look as follows

S_{0,1} ∘ R_{4,1} = \frac{1}{4}S_{4,1} + \frac{7}{12}S_{4,2},
S_{0,1} ∘ R_{4,2} = −\frac{1}{16}S_{4,1},

S_{0,1} ∘ R_{8,1} = \frac{11}{2}S_{8,1} + \frac{1}{4}S_{8,2},
S_{0,1} ∘ R_{8,2} = −\frac{33}{28}S_{8,1} + \frac{9}{28}S_{8,3},
S_{0,1} ∘ R_{8,3} = \frac{11}{104}S_{8,1},
S_{4,1} ∘ R_{4,1} = −\frac{11}{2}S_{8,1} − \frac{3}{4}S_{8,2} − 3S_{8,3},
S_{4,1} ∘ R_{4,2} = \frac{11}{4}S_{8,1} + \frac{3}{16}S_{8,2},
S_{4,2} ∘ R_{4,1} = \frac{99}{28}S_{8,1} + \frac{3}{7}S_{8,2} + \frac{45}{28}S_{8,3},
S_{4,2} ∘ R_{4,2} = −\frac{99}{56}S_{8,1} − \frac{3}{28}S_{8,2},

etc. 4.4. H ∘ S. These compositions look as follows

H_{0,1} ∘ S_{0,1} =\frac{1}{2}R_{0,1},
H_{0,1} ∘ S_{4,1} = − 8R_{4,2},
H_{0,1} ∘ S_{4,2} =\frac{6}{7}R_{4,1} + \frac{24}{7}R_{4,2},
H_{0,1} ∘ S_{8,1} =\frac{52}{11}R_{8,3},
H_{0,1} ∘ S_{8,2} =2R_{8,1} − 104R_{8,3},
H_{0,1} ∘ S_{8,3} =R_{8,1} + \frac{14}{9}R_{8,2} + \frac{52}{3}R_{8,3},
H_{4,1} ∘ S_{0,1} =R_{4,1} + 2R_{4,2},
H_{4,1} ∘ S_{4,1} =\frac{5}{2}R_{8,1} − 26R_{8,3},
H_{4,1} ∘ S_{4,2} =3R_{8,2} + \frac{78}{7}R_{8,3},
H_{4,2} ∘ S_{0,1} =R_{4,1},
H_{4,2} ∘ S_{4,1} =R_{8,1},
H_{4,2} ∘ S_{4,2} =R_{8,1} + R_{8,2},
H_{8,1} ∘ S_{0,1} =\frac{1}{4}R_{8,1} + \frac{7}{12}R_{8,2},
H_{8,2} ∘ S_{0,1} = − \frac{3}{4}R_{8,1} − \frac{7}{2}R_{8,2} − \frac{13}{2}R_{8,3},

H_{8,3} ∘ S_{0,1} =\frac{9}{16}R_{8,1} + \frac{7}{3}R_{8,2} + \frac{13}{4}R_{8,3},

etc. 4.5. S ∘ H. Similar to the identities of the previous subsection, we have

S_{0,1} ∘ H_{0,1} =\frac{1}{2}R_{0,1}^{∗},
S_{0,1} ∘ H_{4,1} = − R_{4,1}^{∗} + 2R_{4,2}^{∗},
S_{0,1} ∘ H_{4,2} =R_{4,1}^{∗},
S_{0,1} ∘ H_{8,1} =\frac{1}{4}R_{8,1}^{∗} + \frac{7}{12}R_{8,2}^{∗},
S_{0,1} ∘ H_{8,2} = − \frac{1}{4}R_{8,1}^{∗} − \frac{13}{2}R_{8,3}^{∗},
S_{0,1} ∘ H_{8,3} =\frac{1}{16}R_{8,1}^{∗} + \frac{13}{4}R_{8,3}^{∗},
S_{4,1} ∘ H_{0,1} =8R_{4,1}^{∗} − 8R_{4,2}^{∗},
S_{4,1} ∘ H_{4,1} = − \frac{3}{2}R_{8,1}^{∗} − 26R_{8,3}^{∗},
S_{4,1} ∘ H_{4,2} =R_{8,1}^{∗},
S_{4,2} ∘ H_{0,1} = − \frac{18}{7}R_{4,1}^{∗} + \frac{24}{7}R_{4,2}^{∗},
S_{4,2} ∘ H_{4,1} = − R_{8,2}^{∗} + \frac{78}{7}R_{8,3}^{∗},
S_{4,2} ∘ H_{4,2} =R_{8,2}^{∗},
S_{8,1} ∘ H_{0,1} =\frac{20}{11}R_{8,1}^{∗} + \frac{28}{11}R_{8,2}^{∗} + \frac{52}{11}R_{8,3}^{∗},
S_{8,2} ∘ H_{0,1} = − 30R_{8,1}^{∗} − \frac{112}{3}R_{8,2}^{∗} − 104R_{8,3}^{∗},
S_{8,3} ∘ H_{0,1} =4R_{8,1}^{∗} + \frac{14}{3}R_{8,2}^{∗} + \frac{52}{3}R_{8,3}^{∗}

etc. 4.6. H ∘ R^{∗}. We computed the following compositions of this type

H_{0,1} ∘ R_{4,1}^{∗} = \frac{1}{2}H_{4,2},
H_{0,1} ∘ R_{4,2}^{∗} = \frac{1}{4}H_{4,1} + \frac{1}{4}H_{4,2},
H_{0,1} ∘ R_{8,1}^{∗} = −4H_{8,2} − 8H_{8,3},

H_{0,1} ∘ R_{8,2}^{∗} = \frac{6}{7}H_{8,1} + \frac{12}{7}H_{8,2} + \frac{24}{7}H_{8,3},
H_{0,1} ∘ R_{8,3}^{∗} = \frac{1}{13}H_{8,2} + \frac{4}{13}H_{8,3},
H_{4,1} ∘ R_{4,1}^{∗} = H_{8,1} + H_{8,2} + H_{8,3},
H_{4,1} ∘ R_{4,2}^{∗} = \frac{1}{2}H_{8,2} + \frac{3}{2}H_{8,3},
H_{4,2} ∘ R_{4,1}^{∗} = H_{8,1},
H_{4,2} ∘ R_{4,2}^{∗} = \frac{1}{2}H_{8,2} + H_{8,3},

which are in a sense dual to the results of Subsection 4.6. 4.7. R^{∗}∘S. Computing compositions of recursion operators with operators S we obtain

R_{4,1}^{∗} ∘ S_{0,1} =\frac{1}{4}S_{4,1} + \frac{7}{12}S_{4,2},
R_{4,1}^{∗} ∘ S_{4,1} =\frac{11}{2}S_{8,1} + \frac{1}{4}S_{8,2},
R_{4,1}^{∗} ∘ S_{4,2} = − \frac{33}{28}S_{8,1} + \frac{9}{28}S_{8,3},
R_{4,2}^{∗} ∘ S_{0,1} =\frac{3}{16}S_{4,1} + \frac{7}{12}S_{4,2},
R_{4,2}^{∗} ∘ S_{4,1} =\frac{33}{4}S_{8,1} + \frac{7}{16}S_{8,2},
R_{4,2}^{∗} ∘ S_{4,2} = − \frac{99}{56}S_{8,1} + \frac{9}{14}S_{8,3},
R_{8,1}^{∗} ∘ S_{0,1} = − \frac{11}{2}S_{8,1} − \frac{3}{4}S_{8,2} − \frac{3}{S}_{8,3},
R_{8,2}^{∗} ∘ S_{0,1} =\frac{99}{28}S_{8,1} + \frac{3}{7}S_{8,2} + \frac{45}{28}S_{8,3},
R_{8,3}^{∗} ∘ S_{0,1} =\frac{33}{104}S_{8,1} + \frac{3}{52}S_{8,2} + \frac{15}{52}S_{8,3},

etc. In particular, we have the relation

(4R_{4,2}^{∗} − R_{4,1}^{∗}) ∘ S_{1} = \frac{1}{4}S_{2},

where

S_{1} = S_{0,1}, S_{2} = 2S_{4,1} + 7S_{4,2}

are the symplectic structures introduced in Subsection 3.4. Moreover, we have the following result:

Proposition 4. The operator

R^{∗} = 4(4R_{4,2}^{∗} − R_{4,1}^{∗})  (71)

generates an infinite family of symplectic structures for the dB-equation by

S_{n} = (R^{∗})^{n−1} ∘ S_{1}.

4.8. R^{∗} ∘ R^{∗}. The first relations we obtained are

R_{4,1}^{∗} ∘ R_{4,1}^{∗} = \frac{1}{4}R_{8,1}^{∗} + \frac{7}{12}R_{8,2}^{∗},
R_{4,1}^{∗} ∘ R_{4,2}^{∗} = −\frac{1}{16}R_{8,1}^{∗},
R_{4,2}^{∗} ∘ R_{4,1}^{∗} = \frac{3}{16}R_{8,1}^{∗} + \frac{7}{12}R_{8,2}^{∗},
R_{4,2}^{∗} ∘ R_{4,2}^{∗} = −\frac{3}{64}R_{8,1}^{∗} + \frac{13}{16}R_{8,3}^{∗}.

Hence, the relation

R_{4,2}^{∗} ∘ R_{4,1}^{∗} = R_{4,1}^{∗} ∘ R_{4,2}^{∗} + R_{4,1}^{∗} ∘ R_{4,1}^{∗}

takes place, or

[R_{4,2}^{∗}, R_{4,1}^{∗}] = R_{4,1}^{∗} ∘ R_{4,1}^{∗}  (72)

that is similar to the one we had for the operators R (see equation (68)). 4.9. Action of R on symmetries. 4.9.1. Action on local symmetries. The action of the operator R_{4,1} is as follows

R_{4,1}(φ_{1}) = \frac{1}{2}φ_{5},  R_{4,1}(φ_{2}) = \frac{1}{4}φ_{6},  R_{4,1}(φ_{3}) = \frac{1}{12}φ_{7},
R_{4,1}(φ_{5}) = \frac{1}{8}φ_{9},  R_{4,1}(φ_{6}) = \frac{1}{24}φ_{10},  R_{4,1}(φ_{7}) = \frac{1}{4}φ_{11},
R_{4,1}(φ_{9}) = \frac{1}{60}φ_{13},  R_{4,1}(φ_{10}) = \frac{1}{4}φ_{14},  R_{4,1}(φ_{11}) = \frac{1}{28}φ_{15}

and is related with the action of R_{4,2} in the following way

R_{4,1}(φ_{i}) = \frac{4}{i + 1}R_{4,2}(φ_{i}).  (73)

4.9.2. Action on (x, t)-independent nonlocal symmetries. Here we have

R_{4,1}(φ_{4,1}) = −\frac{1}{104}φ_{8,2},
R_{4,1}(φ_{8,1}) = 385φ_{12,1} + 21φ_{12,2},
R_{4,1}(φ_{8,2}) = 1430φ_{12,1} + 156φ_{12,2} + 234φ_{12,3}

and

R_{4,2}(φ_{4,1}) = \frac{1}{56}φ_{8,1} − \frac{1}{104}φ_{8,2}
R_{4,2}(φ_{8,1}) = \frac{1155}{2}φ_{12,1} + \frac{147}{4}φ_{12,2}
R_{4,2}(φ_{8,2}) = 2145φ_{12,1} + 273φ_{12,2} + 468φ_{12,3}

etc. 4.9.3. Action on (x, t)-dependent nonlocal symmetries. We have

R_{4,1}(φ̄_{0,1}) = \frac{1}{2}φ̄_{4,1}
R_{4,1}(φ̄_{0,2}) = \frac{1}{4}φ̄_{4,2}
R_{4,1}(φ̄_{4,1}) = −\frac{1}{208}φ_{8,2} + \frac{1}{8}φ̄_{8,1}
R_{4,1}(φ̄_{4,2}) = \frac{1}{156}φ_{8,2} + \frac{1}{24}φ̄_{8,2}
R_{4,1}(φ̄_{8,1}) = 2φ_{12,2} + \frac{55}{2}φ_{12,1} + \frac{1}{60}φ̄_{12,1}
R_{4,1}(φ̄_{8,2}) = −6φ_{12,2} − 88φ_{12,1} + \frac{1}{4}φ̄_{12,2}

and

R_{4,2}(φ̄_{0,1}) = −\frac{1}{8}φ_{4,1} + \frac{1}{4}φ̄_{4,1},
R_{4,2}(φ̄_{0,2}) = \frac{1}{8}φ_{4,1} + \frac{3}{16}φ̄_{4,2},
R_{4,2}(φ̄_{4,1}) = −\frac{1}{208}φ_{8,2} + \frac{3}{16}φ̄_{8,1},
R_{4,2}(φ̄_{4,2}) = \frac{1}{156}φ_{8,2} + \frac{7}{96}φ̄_{8,2},
R_{4,2}(φ̄_{8,1}) = \frac{7}{2}φ_{12,2} + \frac{165}{4}φ_{12,1} + \frac{1}{24}φ̄_{12,1},
R_{4,2}(φ̄_{8,2}) = −\frac{21}{2}φ_{12,2} − 132φ_{12,1} + \frac{11}{16}φ̄_{12,2}.

4.10. Action of R^{∗} on cosymmetries. In a similar way, we have the following actions of R^{∗}: 4.10.1. Action on local cosymmetries.

R_{4,1}^{∗}(ψ_{2}) = \frac{1}{2}ψ_{6},  R_{4,1}^{∗}(ψ_{3}) = \frac{1}{2}ψ_{7},  R_{4,1}^{∗}(ψ_{4}) = \frac{7}{12}ψ_{8},

R_{4,1}^{∗}(ψ_{6}) = \frac{11}{8}ψ_{10},  R_{4,1}^{∗}(ψ_{7}) = \frac{1}{4}ψ_{11},  R_{4,1}^{∗}(ψ_{8}) = \frac{1}{56}ψ_{12},
R_{4,1}^{∗}(ψ_{10}) = \frac{91}{330}ψ_{14},  R_{4,1}^{∗}(ψ_{11}) = \frac{1}{8}ψ_{15},  R_{4,1}^{∗}(ψ_{12}) = \frac{57}{14}ψ_{16}

and

R_{4,2}^{∗}(ψ_{i}) = \frac{i}{4}R_{4,1}^{∗}(ψ_{i}).  (74)

4.10.2. Action on (x, t)-independent nonlocal cosymmetries.

R_{4,1}^{∗}(ψ_{9,1}) = \frac{3}{14}ψ_{13,1} + \frac{9}{28}ψ_{13,2},
R_{4,1}^{∗}(ψ_{13,1}) = \frac{91}{18}ψ_{17,1} + \frac{1}{8}ψ_{17,2},
R_{4,1}^{∗}(ψ_{13,2}) = −\frac{91}{54}ψ_{17,1} + \frac{19}{84}ψ_{17,3},
R_{4,2}^{∗}(ψ_{9,1}) = \frac{3}{8}ψ_{13,1} + \frac{9}{14}ψ_{13,2},
R_{4,2}^{∗}(ψ_{13,1}) = \frac{455}{36}ψ_{17,1} + \frac{11}{32}ψ_{17,2},
R_{4,2}^{∗}(ψ_{13,2}) = −\frac{455}{108}ψ_{17,1} + \frac{19}{28}ψ_{17,3}.

4.10.3. Action on (x, t)-dependent nonlocal cosymmetries.

R_{4,1}^{∗} ∘ ψ̄_{1,1} = \frac{1}{2}ψ̄_{5,1} + \frac{1}{2}ψ̄_{5,2},
R_{4,1}^{∗} ∘ ψ̄_{5,1} = −\frac{7}{2}ψ_{9,1} + \frac{11}{8}ψ̄_{9,1},
R_{4,1}^{∗} ∘ ψ̄_{5,2} = \frac{7}{3}ψ_{9,1} + \frac{1}{4}ψ̄_{9,2},
R_{4,1}^{∗} ∘ ψ̄_{9,1} = \frac{2}{11}ψ_{13,1} + \frac{91}{330}ψ̄_{13,1},
R_{4,1}^{∗} ∘ ψ̄_{9,2} = −\frac{1}{2}ψ_{13,1} + \frac{1}{8}ψ̄_{13,2},
R_{4,2}^{∗} ∘ ψ̄_{1,1} = \frac{1}{4}ψ̄_{5,1} + \frac{3}{8}ψ̄_{5,2},
R_{4,2}^{∗} ∘ ψ̄_{5,1} = −\frac{7}{2}ψ_{9,1} + \frac{33}{16}ψ̄_{9,1},
R_{4,2}^{∗} ∘ ψ̄_{5,2} = \frac{7}{3}ψ_{9,1} + \frac{7}{16}ψ̄_{9,2},
R_{4,2}^{∗} ∘ ψ̄_{9,1} = \frac{7}{22}ψ_{13,1} + \frac{91}{132}ψ̄_{13,1},

R_{4,2}^{∗} ∘ ψ̄_{9,2} = −\frac{7}{8}ψ_{13,1} + \frac{11}{32}ψ̄_{13,2}.

4.11. Commutators. 4.11.1. [•, φ_{i}]-commutators. First of all we have the relation

[φ_{i}, φ_{j}] = 0,

and besides these relations we obtain:

[φ_{4,1}, φ_{1}] = 0,  [φ_{8,1}, φ_{1}] = 0,  [φ_{8,2}, φ_{1}] = 0,
[φ_{4,1}, φ_{2}] = 0,  [φ_{8,1}, φ_{2}] = 0,  [φ_{8,2}, φ_{2}] = 0,
[φ_{4,1}, φ_{3}] = \frac{1}{3}φ_{7},  [φ_{8,1}, φ_{3}] = \frac{14}{3}φ_{11},  [φ_{8,2}, φ_{3}] = −\frac{26}{3}φ_{11},
[φ_{4,1}, φ_{5}] = 3φ_{9}  [φ_{8,1}, φ_{5}] = \frac{21}{5}φ_{13},  [φ_{8,2}, φ_{5}] = −\frac{26}{5}φ_{13},
[φ_{4,1}, φ_{6}] = \frac{5}{3}φ_{10},  [φ_{8,1}, φ_{6}] = \frac{245}{6}φ_{14},  [φ_{8,2}, φ_{6}] = −\frac{130}{3}φ_{14},
[φ_{4,1}, φ_{7}] = 15φ_{11},  [φ_{8,1}, φ_{7}] = 60φ_{15},  [φ_{8,2}, φ_{7}] = −\frac{390}{7}φ_{15},
[φ_{4,1}, φ_{9}] = \frac{28}{15}φ_{13},
[φ_{4,1}, φ_{10}] = 36φ_{14},
[φ_{4,1}, φ_{11}] = \frac{45}{7}φ_{15},

and

[φ̄_{0,1}, φ_{1}] = φ_{1},  [φ̄_{0,2}, φ_{1}] = 0,
[φ̄_{0,1}, φ_{2}] = 0,  [φ̄_{0,2}, φ_{2}] = φ_{2},
[φ̄_{0,1}, φ_{3}] = −φ_{3},  [φ̄_{0,2}, φ_{3}] = 2φ_{3},
[φ̄_{0,1}, φ_{5}] = −3φ_{5},  [φ̄_{0,2}, φ_{5}] = 4φ_{5},
[φ̄_{0,1}, φ_{6}] = −4φ_{6},  [φ̄_{0,2}, φ_{6}] = 5φ_{6},
[φ̄_{0,1}, φ_{7}] = −5φ_{7},  [φ̄_{0,2}, φ_{7}] = 6φ_{7},
[φ̄_{4,1}, φ_{1}] = φ_{5},  [φ̄_{4,2}, φ_{1}] = 0,
[φ̄_{4,1}, φ_{2}] = 0,  [φ̄_{4,2}, φ_{2}] = φ_{6},
[φ̄_{4,1}, φ_{3}] = −\frac{1}{6}φ_{7},  [φ̄_{4,2}, φ_{3}] = \frac{2}{3}φ_{7},
[φ̄_{4,1}, φ_{5}] = −\frac{3}{4}φ_{9},  [φ̄_{4,2}, φ_{5}] = 2φ_{9},
[φ̄_{4,1}, φ_{6}] = −\frac{1}{3}φ_{10},  [φ̄_{4,2}, φ_{6}] = \frac{5}{6}φ_{1},

[φ̄_{4,1}, φ_{7}] = −\frac{5}{2}φ_{11},  [φ̄_{4,2}, φ_{7}] = \frac{6}{φ}_{11},
[φ̄_{8,1}, φ_{1}] = φ_{9},  [φ̄_{8,2}, φ_{1}] = 0,
[φ̄_{8,1}, φ_{2}] = 0,  [φ̄_{8,2}, φ_{2}] = φ_{10},
[φ̄_{8,1}, φ_{3}] = −\frac{2}{3}φ_{11},  [φ̄_{8,2}, φ_{3}] = \frac{16}{3}φ_{11},
[φ̄_{8,1}, φ_{5}] = −\frac{3}{10}φ_{13},  [φ̄_{8,2}, φ_{5}] = \frac{8}{5}φ_{13},
[φ̄_{8,1}, φ_{6}] = −\frac{7}{3}φ_{14},  [φ̄_{8,2}, φ_{6}] = \frac{35}{3}φ_{14},
[φ̄_{8,1}, φ_{7}] = −\frac{20}{7}φ_{15},  [φ̄_{8,2}, φ_{7}] = \frac{96}{7}φ_{15}.

4.11.2. [•, φ_{i,j}]-commutators. We computed the following ones

[φ_{4,1}, φ_{8,1}] = 9240φ_{12,1} + 840φ_{12,2} + 504φ_{12,3},
[φ_{4,1}, φ_{8,2}] = 34320φ_{12,1} + 6240φ_{12,2} + 13104φ_{12,3}

and

[φ̄_{0,1}, φ_{4,1}] = −4φ_{4,1},
[φ̄_{0,1}, φ_{8,1}] = −8φ_{8,1},
[φ̄_{0,1}, φ_{8,2}] = −8φ_{8,2},
[φ̄_{0,2}, φ_{4,1}] = 4φ_{4,1},
[φ̄_{0,2}, φ_{8,1}] = 8φ_{8,1},
[φ̄_{0,2}, φ_{8,2}] = 8φ_{8,2},
[φ̄_{4,1}, φ_{4,1}] = \frac{1}{26}φ_{8,2} − 3φ̄_{8,1},
[φ̄_{4,1}, φ_{8,1}] = 252φ_{12,3} − 168φ_{12,2} − 2310φ_{12,1} − \frac{21}{5}φ̄_{12,1},
[φ̄_{4,1}, φ_{8,2}] = −2808φ_{12,3} − 1248φ_{12,2} − 8580φ_{12,1} + \frac{26}{5}φ̄_{12,1},
[φ̄_{4,2}, φ_{4,1}] = −\frac{4}{39}φ_{8,2} − \frac{5}{3}φ̄_{8,2},
[φ̄_{4,2}, φ_{8,1}] = −336φ_{12,3} + 420φ_{12,2} + 6160φ_{12,1} − \frac{245}{6}φ̄_{12,2},
[φ̄_{4,2}, φ_{8,2}] = 6240φ_{12,3} + 3120φ_{12,2} + 22880φ_{12,1} + \frac{130}{3}φ̄_{12,2},
[φ̄_{8,1}, φ_{4,1}] = 72φ_{12,3} − 80φ_{12,2} − 880φ_{12,1} − \frac{28}{15}φ̄_{12,1},
[φ̄_{8,2}, φ_{4,1}] = −576φ_{12,3} + 96φ_{12,2} + 2112φ_{12,1} − 36φ̄_{12,2}.

4.11.3. [•, φ̄_{i,j}]-commutators. We computed the following commutators of this type

[φ̄_{0,1}, φ̄_{0,2}] = 0,
[φ̄_{0,1}, φ̄_{4,1}] = −4φ̄_{4,1},
[φ̄_{0,1}, φ̄_{4,2}] = −4φ̄_{4,2},
[φ̄_{0,1}, φ̄_{8,1}] = −8φ̄_{8,1},
[φ̄_{0,1}, φ̄_{8,2}] = −8φ̄_{8,2},
[φ̄_{0,2}, φ̄_{4,1}] = 4φ̄_{4,1},
[φ̄_{0,2}, φ̄_{4,2}] = 4φ̄_{4,2},
[φ̄_{0,2}, φ̄_{8,1}] = 8φ̄_{8,1},
[φ̄_{0,2}, φ̄_{8,2}] = 8φ̄_{8,2},
[φ̄_{4,1}, φ̄_{4,2}] = \frac{1}{39}φ_{8,2} − 2φ̄_{8,1} − \frac{1}{3}φ̄_{8,2},
[φ̄_{4,1}, φ̄_{8,1}] = −36φ_{12,3} − 16φ_{12,2} − 110φ_{12,1} + \frac{1}{15}φ̄_{12,1},
[φ̄_{4,1}, φ̄_{8,2}] = 288φ_{12,3} + 96φ_{12,2} + 528φ_{12,1} − \frac{8}{5}φ̄_{12,1} − 4φ̄_{12,2},
[φ̄_{4,2}, φ̄_{8,1}] = 48φ_{12,3} + 40φ_{12,2} + 352φ_{12,1} + \frac{8}{15}φ̄_{12,1} + \frac{7}{3}φ̄_{12,2},
[φ̄_{4,2}, φ̄_{8,2}] = −384φ_{12,3} − 192φ_{12,2} − 1408φ_{12,1} − \frac{8}{3}φ̄_{12,2}.

5. Conclusions

We studied the dispersionless Boussinesq equation

w_{t} = u_{x},
u_{t} = ww_{x} + v_{x},
v_{t} = −uw_{x} − 3wu_{x}

and found out that it possesses three families of commuting local symmetries, Φ_{1}, Φ_{2} and Φ_{3}. The seed symmetries for these families are φ_{1}, φ_{2} and φ_{3} presented in Subsection 2.2.1. In addition, the equation has a family Φ_{4} of nonlocal (x, t)-independent symmetries generated by the symmetry φ_{4,1} (Subsection 2.2.2) and two families, Φ̄_{1} and Φ̄_{2}, of nonlocal (x, t)-dependent symmetries generated by φ̄_{0,1} and φ̄_{0,2} (Subsection 2.2.3). All nonlocal symmetries act, by the commutator, as hereditary symmetries for the local ones. We also constructed an infinite algebra Rec(E) of recursion operators for symmetries. This is an associative noncommutative algebra with two generators R_{4,1} and R_{4,2} and one relation (see Subsection 3.1 and equation (68)). The operators produce the above mentioned families from the corresponding seed symmetries. This algebra contains an Abelian subalgebra of operators satisfying the Nijenhuis condition. The subalgebra is generated by operator (70). Existence of this operator is the reason for commutativity of the families Φ_{1}, Φ_{2} and Φ_{3}. In complete parallel, one observes five families of cosymmetries and an algebra of the corresponding recursion operator that generate these families. This similarity is due to existence of operators H: sym^{∗}(E) → sym(E) and S: sym(E) → sym^{∗}(E) that relate symmetries and cosymmetries to each other and are described in Subsections 3.3 and 3.4, resp. The operators H form a left module over the algebra Rec(E) with one generator H_{0,1}. This operator is a Hamiltonian structure for the dB-equation and by the action of recursion operator (70) one obtains an infinite family of compatible Hamiltonian structures of which the first two are local. In a similar way, the operators S form a left module over the algebra of recursion operators for cosymmetries, while the recursion operator given by equation (71) generates an infinite family of symplectic structures, all of them being nonlocal. On the other hand, the dB-equation E admits the potential y defined by

y_{x} = w, y_{t} = u

identical to the nonlocal variable p_{1} (see Subsection 2.1). The corresponding covering equation E_{1} is of the form

y_{tt} = y_{x}y_{xx} + v_{t}, v_{t} = −y_{t}y_{xx} − 3y_{x}y_{xt}.

The potential z defined by

z_{x} = u, z_{t} = v + \frac{1}{2}w^{2},

that corresponds to the nonlocal variable p_{2} leads to the covering equation E_{2}

w_{t} = z_{xx}, z_{tt} − 2ww_{t} = −z_{x}w_{x} − 3wz_{xx}.

Finally, the equation E′

y_{ttt} + 2y_{x}y_{xxt} + y_{t}y_{xtt} + 3y_{xx}y_{xt} = 0.

This means that we have the following covering structure

\begin{matrix}E & ⟵ & E_{1} \\ ↑ & & ↓ \\ E_{2} & ⟶ & E′ \end{matrix}

that relates equations E and E′. From this structure it follows that an auto-Bäcklund transformation can be constructed for the dB-equation. Existence of this transformation may be closely related to the algebraic structures described in the preceding sections.

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

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

Iosif Krasil′shchik, Independent University of Moscow, B. Vlasevsky 11, 119002 Moscow, Russia

E-mail address: josephk@diffiety.ac.ru

Alexander Verbovetsky, Independent University of Moscow, B. Vlasevsky 11, 119002 Moscow, Russia

E-mail address: verbovet@mccme.ru