Evolutionary Hirota Type (2+1)-Dimensional Equations: Lax Pairs, Recursion Operators and Bi-Hamiltonian Structures

We show that evolutionary Hirota type Euler-Lagrange equations in (2+1) dimensions have a symplectic Monge-Amp\`ere form. We consider integrable equations of this type in the sense that they admit infinitely many hydrodynamic reductions and determine Lax pairs for them. For two seven-parameter families of integrable equations converted to two-component form we have constructed Lagrangians, recursion operators and bi-Hamiltonian representations. We have also presented a six-parameter family of tri-Hamiltonian systems.


Introduction
We study recursion operators, Lax pairs and bi-Hamiltonian representations for (2 + 1)-dimensional equations of the evolutionary Hirota type u tt = f (u t1 , u t2 , u 11 , u 12 , u 22 ). (1.1) Here u = u(t, z 1 , z 2 ) and the subscripts denote partial derivatives of u, namely, u ij = ∂ 2 u/∂z i ∂z j , u ti = ∂ 2 u/∂t∂z i . Equations of this type arise in a wide range of applications including non-linear physics, general relativity, differential geometry and integrable systems. Some examples are the Khokhlov-Zabolotskaya (dKP) equation in non-linear acoustics and the theory of Einstein-Weyl structures and the Boyer-Finley equation in the theory of self-dual gravity. A lot of work has been done by E. Ferapontov et al. for studying integrability of equation (1.1) which is understood as the existence of infinitely many hydrodynamic reductions (see [2,3,4,5,6] and references therein). Ferapontov et al. in [3] derived integrability condition which is equivalent to the property of equation (1.1) to be linearizable by contact transformations. This does not mean that there is a straightforward way to obtain bi-Hamiltonian structures of the nonlinear equations (1.1) by a direct transfer of such structures from the linear equation, because an arbitrary transformation in the space of second partial derivatives of the unknown will not preserve the bi-Hamiltonian structure of the equation.
Our goal here is to study bi-Hamiltonian structures of the integrable equations of the form (1.1) together with Lax pairs and recursion operators. We utilize the method which we used earlier [13,17,18] for constructing a degenerate Lagrangian for two-component evolutionary form of the equation and using Dirac's theory of constraints [1] in order to obtain Hamiltonian form of the system.
Our approach here starts with description of all equations (1.1) which have the Euler-Lagrange form [14]. We do not consider here the equations of the form (1.1) which become Lagrangian only after multiplication by an integrating factor of variational calculus, postponing the appropriate generalization for a further publication. We find that the Lagrangian evolutionary Hirota-type equations have a symplectic Monge-Ampère form and we determine their Lagrangians. Then we study recursion relations for symmetries and Lax pairs for the Lagrangian equations. Here our starting point is to convert the symmetry condition into a "skew-factorized" form. This approach extends the method of A. Sergyeyev [16] for constructing recursion operators. According to [16] one constructs the recursion operator from a certain Lax pair which, in turn, is typically built from the original Lax pair for the equation under study. On the other hand, below we construct such a special Lax pair using the skew-factorized form of the linearized equation (symmetry condition) rather than a previously known Lax pair, and then apply the construction from [16] to obtain the recursion operator.
The next step is to transform Lagrangian equations converted into a two-component form to a Hamiltonian system. Finally, composing a recursion operator with a Hamiltonian operator we obtain the second Hamiltonian operator and also find the corresponding Hamiltonian density. Thus, we end up with a bi-Hamiltonian representation of an integrable equation (1.1) in a twocomponent form. In this way, we obtain two seven-parameter families of bi-Hamiltonian systems and a six-parameter family of tri-Hamiltonian systems.
The paper is organized as follows. In Section 2, we show that all equations (1.1) of the Euler-Lagrange form have the symplectic Monge-Ampère form and we derive a Lagrangian for such equations. In Section 3, we analyze the symmetry condition for the Lagrangian equations. In Section 4, using an integrability condition, we convert the symmetry condition into a "skewfactorized" form and immediately extract Lax pair and recursion relations for symmetries from the symmetry condition in this form. In Section 5, we convert our equation in a two-component form and derive a degenerate Lagrangian for this system. In Section 6, we transform the Lagrangian system into Hamiltonian system using the Dirac's theory of constraints. We obtain the Hamiltonian operator J 0 and corresponding Hamiltonian density H 1 . In Section 7, we derive a recursion operator R in a 2 × 2 matrix form using recursion relations for the two-component form of the equation. In Section 8, by composing the recursion operator with the Hamiltonian operator J 0 we obtain the second Hamiltonian operator J 1 = RJ 0 and the corresponding Hamiltonian density H 0 with one additional "Hamiltonian" constraint for coefficients. Thus, we obtain a seven-parameter family of bi-Hamiltonian systems because of the two constraints on nine coefficients: integrability condition and Hamiltonian condition.
We consider also an alternative skew-factorized representation of the symmetry condition which implies different Lax pair, another recursion operator and a different seven-parameter family of bi-Hamiltonian systems under a different additional constraint on the coefficients. If, in addition, we require that both additional constraints coincide and are compatible with the integrability condition, we obtain a six-parameter family of tri-Hamiltonian systems.
In Sections 4, 7, and 8, we treat separately each of the two generic cases of equation (2.2) when none of the coefficients c 1 , c 2 , c 3 vanishes, particular cases when either c 1 = 0 or c 2 = 0 which are obtained as a specialization of one of the generic cases, and the special case c 3 = 0 which cannot be obtained from the generic cases but should be treated independently.

Lagrangian equations of evolutionary Hirota type
We start with equation (1.1) in the form The Fréchet derivative operator (linearization) of equation (2.1) reads where D i ≡ D z i , D t denote operators of total derivatives. The adjoint Fréchet derivative operator has the form According to Helmholtz conditions [14], equation (2.1) is an Euler-Lagrange equation for a variational problem iff its Fréchet derivative is self-adjoint, D * F = D F , or explicitly, equating to zero coefficients of D t , D 1 , D 2 and the term without operators of total derivatives, we obtain four equations on f The general solution of these equations for f implies the Lagrangian evolutionary Hirota equation (1.1) to have symplectic Monge-Ampère form (2.2) A Lagrangian for the equation (2.2) is readily obtained by applying the homotopy formula [14] for

Symmetry condition
In the following it will be useful to introduce operator of derivative in the direction of the vector 2) may be written as Symmetry condition is the differential compatibility condition of (2.2) and the Lie equation u τ = ϕ, where ϕ is the symmetry characteristic and τ is the group parameter. It has the form of Fréchet derivative (linearization) of equation (3.1) It is convenient to introduce the following differential operators so that the symmetry condition (3.2) becomes

Recursion relations and Lax pairs
E. Ferapontov et al. in [3] have derived an integrability condition for the symplectic Monge-Ampère equation of the following general form (equation (21) in [3]) with constant coefficients. Here integrability means that the equation (4.1) admits infinitely many hydrodynamic reductions [5]. For our evolutionary equation (2.2) the coefficients in (4.1) are The integrability condition given in [3, formula (22)] for the equation (4.1) has the form For the equation (2.2) due to identifications (4.2), the integrability condition (4.3) becomes We show explicitly that any equation of the form (2.2) satisfying (4.4) is also integrable in the traditional sense by constructing Lax pair for such an equation.

Generic case
In the integrability condition (4.4) we assume The following procedure extends A. Sergyeyev's method for constructing recursion operators [16]. Namely, unlike [16], we start with the skew-factorized form of the symmetry condition and extract from there a Lax pair for symmetries instead of building it from a previously known Lax pair. After that we construct a recursion operator from this newly found Lax pair using Proposition 1 from [16]. The linear operator of the symmetry condition (3.3) for integrable equations of the form (2.2) can be presented in the "skew-factorized" form If we introduce two-dimensional vector operators R = (A 1 , A 2 ) and S = (B 1 , B 2 ), then the skew-factorized form (4.5) becomes the cross (vector) product ( R × S)ϕ = 0. Here differential operators A i and B i are defined as These operators satisfy the commutator relations where the last equation is satisfied on solutions of the equation (2.2). It immediately follows that the following two operators also commute on solutions and therefore constitute Lax representation for equation (2.2) with λ being a spectral parameter. Symmetry condition in the form (4.5) not only provides the Lax pair for equation (2.2) but also leads directly to recursion relations for symmetries whereφ is a potential for ϕ. This follows from a special case of Proposition 1 of [16] which we recapitulate here together with its proof for the readers' convenience. Indeed, equations (4.9) together with (4.7) imply ( Moreover, due to (4.9) which shows thatφ satisfies the symmetry condition (4.5) and hence is also a symmetry. Thus, ϕ is a symmetry whenever so isφ and vice versa. The equations (4.9) define an auto-Bäcklund transformation between the symmetry conditions written for ϕ andφ. Hence, the auto-Bäcklund transformation for the symmetry condition is nothing else than a recursion operator. Finally, we must remark that this approach to recursion operators originated from the much older work published in 90th [8,12,15]. The skew-factorized form of the symmetry condition is by no means unique. We can derive another version by using the discrete symmetry transformation while the operator L 12(t) is not changed. Applying (4.10) to operators (4.6) we obtain a new set of operators 11) which also satisfy the skew-factorized form (4.5) of the symmetry condition (3.3) and the same commutator relations (4.7). Using these operators in (4.8) and (4.9) we obtain the second Lax pair and another set of recursion relations for symmetries, respectively. It may be interesting to note algebraic relations between the two recursion operators, namely, determined by the set (4.6), marked below with the superscript (1) , and by the set (4.11), marked with the superscript (2) 2 .
In the first case integrability condition (4.4) reads If we set c 1 = 0 in our first set of operators (4.6) from the generic case, these operators become linearly dependent and the skew-factorized representation (4.5) does not reproduce the symmetry condition (3.2). Therefore, we have to put c 1 = 0 in the second set of operators (4.11) with the result Operators (4.13) satisfy skew-factorized form (4.5) of the symmetry condition (3.3) and the commutator relations (4.7). Therefore, as is shown above, equations (4.9) yield the recursion relations for symmetries and the operators commute on solutions and so constitute Lax pair for the equation (2.2) at c 1 = 0.
In the second case c 2 = 0, integrability condition (4.4) has the form (4.14) We set c 2 = 0 in our first set of operators (4.6) in the generic case to obtain Operators (4.15) also satisfy skew-factorized form (4.5) of the symmetry condition (3.3) and the commutator relations (4.7). Therefore, as is shown above, using these operators in (4.8) and (4.9) we obtain the Lax pair and recursion relations for symmetries in this case. We could not use the second family of operators (4.11) at c 2 = 0 because skew-factorized form (4.5) would give identical zero instead of reproducing the symmetry condition.
We note again that the skew-factorized form of the invariance condition (4.5) is still not unique. One could obtain such a form at c 2 = 0 with a different choice of the operators A i and B i

Special case c 3 = 0
In the integrability condition (4.4) we assume Then the linear operator of the symmetry condition (3.3) can be presented in the skew-factorized form (4.5) with the following operators A i and B i Operators (4.17) satisfy the commutator relations (4.7) and hence, as is shown above, the equations (4.9) produce the recursion relations for symmetries and the operators commute on solutions and so constitute Lax representation for the equation (2.2) at c 3 = 0. We note that if we considered this case as a particular case at c 3 = 0 of the operators (4.6) or (4.11) of the generic case, then operators A i , B i would be linearly dependent and the skewfactorized form (4.5) would not yield the symmetry condition (3.3).

Two-component form
Introducing the additional dependent variable v = u t , we convert equation (3.1) into the evolutionary system (5.1) Lie equations become u τ = ϕ, v τ = ψ, so that u t = v implies the first invariance condition ϕ t = ψ. The second invariance condition is obtained by differentiating the second equation in (5.1) with respect to the group parameter τ

Hamiltonian representation
To transform from Lagrangian to Hamiltonian description, we define canonical momenta which satisfy canonical Poisson brackets where u 1 = u, u 2 = v, z = (z 1 , z 2 ), the only nonzero Poisson bracket being [π u , u] = δ(z 1 − z 1 )δ(z 2 − z 2 ). The Lagrangian (5.2) is degenerate because the momenta cannot be inverted for the velocities. Therefore, following the theory of Dirac's constraints [1], we impose (6.1) as constraints and calculate Poisson bracket for the constraints We obtain the following matrix of Poisson brackets, which for convenience we multiply by the overall factor (−1) The Hamiltonian operator is the inverse to the symplectic operator J 0 = K −1 Operator J 0 is Hamiltonian if and only if its inverse K is symplectic [7], which means that the volume integral Ω = V ωdV of ω = (1/2)du i ∧K ij du j should be a symplectic form, i.e., at appropriate boundary conditions dω = 0 modulo total divergence. Another way of formulation is to say that the vertical differential of ω should vanish [9]. In ω summations over i, j run from 1 to 2 and u 1 = u, u 2 = v. Using (6.2), we obtain Taking exterior derivative of (6.4) and skipping total divergence terms, we have checked that dω = 0 which proves that operator K is symplectic and hence J 0 defined in (6.3) is indeed a Hamiltonian operator. The first Hamiltonian form of this system is where we still need to determine the corresponding Hamiltonian density H 1 . We convert L from (5.2) to the form and apply the formula 7 Recursion operators in 2 × 2 matrix form 7.1 Generic case: c 1 · c 2 · c 3 = 0 We define two-component symmetry characteristic (ϕ, ψ) T (where T means transposed matrix) with ψ = ϕ t and (φ,ψ) T withψ =φ t for the original and transformed symmetries, respectively. The recursion relations (4.9) with the use of (4.6) for the operators A i , B i take the form Combining these two equations to eliminate firstψ and thenφ, we obtain where the subscripts of ϕ and ψ denote partial derivatives. Applying the inverse operator ∇ −1 c , which is defined to satisfy the relations ∇ −1 c ∇ c = 1, we obtain the explicit form of recursion relations Here an important remark is due. The operator ∇ −1 c can make sense merely as a formal inverse of ∇ c . Thus, the relations (7.2) are formal as well. The proper interpretation of the quantities like ∇ −1 c and of (7.2) requires the language of differential coverings, see the original papers [8,12] and the recent survey [9].
In a two component form, the recursion relations (7.1) read with the recursion operator R in the 2 × 2 matrix form with the matrix elements Next, we use the alternative set (4.11) of operators A i , B i in the recursion relations (4.9) presented in a two-component form Combining these two equations to eliminate firstψ and thenφ, we obtain the explicit form of recursion relations and we immediately extract the second recursion operator R in the 2 × 2 matrix form where

Particular cases
We consider again particular cases c 1 = 0, c 2 = 0 and c 2 = 0, c 1 = 0. As we know from Section 4, the first case, c 1 = 0, c 2 = 0 should be considered as a particular case of the second recursion operator R from (7.6) and (7.7) with the result and the integrability condition (4.12) has been used. The second case, c 2 = 0, c 1 = 0 should be considered as a particular case of the first recursion operator R from (7.4) and (7.5) with the result with the matrix elements 7.3 Special case: c 1 · c 2 = 0, c 3 = 0 Recursion relations (4.18) in a two-component form become The explicit two-component form of the recursion relations (7.8) is In the matrix form (7.3), the recursion operator arising from (7.9) reads where we have used an alternative equivalent expression for the matrix element J 22 0 , with the final result Here operator J 1 is manifestly skew symmetric. A check of the Jacobi identities and compatibility of the two Hamiltonian structures J 0 and J 1 is straightforward but too lengthy to be presented here. The method of the functional multi-vectors for checking the Jacobi identity and the compatibility of the Hamiltonian operators is developed by P. Olver in [14, Chapter 7] and has been applied recently for checking bi-Hamiltonian structure of the general heavenly equation [18] and the first heavenly equation of Plebański [17] under the well-founded conjecture that this method is applicable for nonlocal Hamiltonian operators as well. The next problem is to derive the Hamiltonian density H 0 corresponding to the second Hamiltonian operator J 1 such that implies the bi-Hamiltonian representation of the system (5.1) where v t should be replaced by the right-hand side of the second equation in (5.1). Then we could conclude that our system (5.1) is also integrable in the sense of Magri [10,11]. Proof . We will need the following simple relation between J 22 1 and the operator B 1 from (4.6), which is involved in the recursion relations (4.9) and Lax pair (4.8) u + γ 0 .
We have now completely determined the Hamiltonian density H 0 in (8.12) to be (8.4), where we have skipped the additive constant γ 0 .
With the Hamiltonian density H 0 from (8.4), corresponding to the second Hamiltonian operator J 1 , the system (5.1) admits bi-Hamiltonian representation (8.2), provided that the integrability condition (4.4) and bi-Hamiltonian constraint (8.3) are satisfied in the generic case. Thus, we obtain the first seven-parameter family of bi-Hamiltonian systems.

Second family of bi-Hamiltonian systems
Composing an alternative recursion operator R , determined by (7.6) and (7.7), with the first Hamiltonian operator J 0 we obtain an alternative second Hamiltonian operator J 1 = R J 0 with the resulting expression where B 1 is defined in (4.11). Next task is to determine the Hamiltonian density H 0 corresponding to J 1 in the bi-Hamiltonian representation A check of the Jacobi identities for J 1 and compatibility of the two Hamiltonian structures J 0 and J 1 is straightforward and too lengthy to be presented here.
where B 1 is defined in (4.15), so that we have explicitly