Weakly Nonlocal Hamiltonian Structures: Lie Derivative and Compatibility

We show that under certain technical assumptions any weakly nonlocal Hamiltonian structure compatible with a given nondegenerate weakly nonlocal symplectic structure $J$ can be written as the Lie derivative of $J^{-1}$ along a suitably chosen nonlocal vector field. Moreover, we present a new description for local Hamiltonian structures of arbitrary order compatible with a given nondegenerate local Hamiltonian structure of zero or first order, including Hamiltonian operators of the Dubrovin-Novikov type.


Introduction
Nonlinear integrable systems usually are bihamiltonian, i.e., possess two compatible Hamiltonian structures. This ingenious discovery of Magri [14] has naturally lead to an intense study of pairs of compatible Hamiltonian structures both in finitely and infinitely many dimensions, see e.g. [1,3,5,13,16,24,26,30] and references therein.
Using the ideas from the Lichnerowicz-Poisson cohomology theory [13,30] it can be shown [5,26] that under certain minor technical assumptions all Hamiltonian structures compatible with a given nondegenerate Hamiltonian structure P can be written as the Lie derivatives of P along suitably chosen vector fields. This allows for a considerable reduction in the number of unknown functions: roughly speaking, we deal with components of a vector field rather than with those of a skew-symmetric tensor, and the number of the former is typically much smaller than that of the latter, see e.g. [26] for more details. This idea works well for compatible pairs of finite-dimensional Hamiltonian structures [26,28] and of local Hamiltonian operators of Dubrovin-Novikov type [21,26], when the corresponding vector fields are local as well.
In the present work we extend this approach to the weakly nonlocal [16] Hamiltonian structures using weakly nonlocal vector fields. To this end we first generalize the local homotopy formula (7) to weakly nonlocal symplectic structures in Theorem 2 below. This enables us to characterize large classes of Hamiltonian structures compatible with a given weakly nonlocal symplectic structure using the weakly nonlocal (co)vector fields, i.e., elements ofṼ (resp.Ṽ * ), as presented below in Theorems 3 and 4 and Corollaries 2, 3, 4, and 5.
The paper is organized as follows. In Section 2 we recall some basic features of infinitedimensional Hamiltonian formalism. Section 3 contains the main theoretical results of the paper while Sections 4 and 5 deal with the particular cases of local Hamiltonian structures of zero and first order where important simplifications occur. Finally, in Section 6 we briefly discuss the results of the present work.

Preliminaries
Following [5,24], recall some basic aspects of infinite-dimensional Hamiltonian formalism for the case of one independent variable x ∈ B (usually B = R or B = S 1 ) and n dependent variables.
We start with an algebra A j of smooth functions of x, u, u 1 , . . . , u j , where u k = (u 1 k , . . . , u n k ) T for k > 0 are n-component vectors from R n , u 0 ≡ u ∈ M ⊂ R n , M is an open domain in R n , and the superscript T indicates the transposed matrix. Set A = ∞ j=0 A j . The elements of A are called local functions.
Consider (see e.g. [5] and [24] and references therein) a derivation of A and let Im D be the image of D in A, andĀ = A/Im D. The spaceĀ is the counterpart the algebra of (smooth) functions on a finite-dimensional manifold in the standard de Rham complex.
Informally, x can be thought of as a space variable and D as a total x-derivative, cf. e.g. [24]. The canonical projection π : A →Ā is traditionally denoted by dx, and for any f, g ∈ A we have The quantity F = f dx should not be confused with a nonlocal variable D −1 (f ): these are different objects. Informally, f dx can be thought of as B f dx, i.e., this is, roughly speaking, a definite x-integral, and D −1 (f ) is a formal indefinite x-integral. If f ∈ ImD then D −1 (f ) ∈ A, and we need to augment A to include a nonlocal variable ω such that D(ω) = f and to extend the action of D accordingly, see below for further details.
The generalized Leibniz rule [17,18,19,24]  In what follows we shall often omit the composition sign • (for instance, we shall write KL instead of K • L) wherever this does not lead to a possible confusion.
The [17,18,19,24] the greatest integer m such that h m = 0. If, moreover, det h m = 0 we shall call L nondegenerate, and then there exists a unique formal series for the formal adjoint of L, see e.g. [17,18,19,24]. A formal series L is said to be skew-symmetric if L † = −L. As usual, an L ∈ Mat q (A)[[D −1 ]] is said to be a purely differential (or just differential) operator if L − = 0.
Let A q be the space of q-component functions with entries from A, no matter whether they are interpreted as column or row vectors. For any f ∈ A q define (see e.g. [12]) its directional derivative as We shall also need the operator of variational derivative (see e.g. [1,5,24,2]) Following [16], an Nearly all known today Hamiltonian and symplectic operators in (1+1) dimensions are weakly nonlocal, cf. e.g. [32]. Recall that an operator of the form L = f ⊗ D −1 • g acts on an h ∈ A q as follows: where "·" denotes the standard Euclidean scalar product in A q .
Denote by V the space of n-component columns with entries from A. The commutator [P , Q] = Q ′ [P ] − P ′ [Q] turns V into a Lie algebra, see e.g. [1,12,18,24]. The Lie derivative of R ∈ V along Q ∈ V reads L Q (R) = [Q, R], see e.g. [1,5,31,24]. The natural dual of V is the space V * of n-component rows with entries from A.
The canonical pairing of V and V * is given by the formula (see e.g. [5,32]) where γ ∈ V * , Q ∈ V, and "·" here and below refers to the standard Euclidean scalar product of the n-component vectors.
For weakly nonlocal R : V → V, J : V → V * , P : V * → V, N : V * → V * define [12] their Lie derivatives along a Q ∈ V as follows: Here and below we do not assume R and J to be defined on the whole of V, respectively P and N on the whole of V * .
We shall call an operator J : V → V * (respectively P : V * → V) formally skew-symmetric if it is skew-symmetric when considered as a formal series, i.e., J † = −J (respectively P † = −P ).
Recall that the proper way to extend the concept of the finite-dimensional Hamiltonian structure to evolutionary systems of PDEs in (1+1) dimensions is the following one. A formally skew-symmetric operator P : V * → V is Hamiltonian [5] (or implectic [12]) if its Schouten bracket with itself vanishes: [P, P ] = 0. The Schouten bracket [·, ·] is given by the formula where χ i ∈ V * and , is given by (2), see e.g. [5]. Throughout the rest of the paper [·, ·] will denote the Schouten bracket rather than the commutator. Two Hamiltonian operators are said to be compatible [12] (or to form a Hamiltonian pair [5]) if any linear combination thereof is again a Hamiltonian operator. Note that the Hamiltonian operators are compatible if and only if their Schouten bracket vanishes [5].
The Poisson bracket {, } P associated with a Hamiltonian operator P is (see e.g. [5,24]) a mapping fromĀ ×Ā toĀ given by the formula for any F, G ∈Ā. Here we set δF for any P , Q, R ∈ V.
Following the tradition established in the literature we shall sometimes speak of Hamiltonian (or symplectic) structures rather than of Hamiltonian (or symplectic) operators, even though the latter terms are equivalent with the former.
We shall call a Hamiltonian or symplectic operator nondegenerate if it is nondegenerate as a formal series in powers of D. A nondegenerate operator P : V * → V is Hamiltonian if and only if P −1 is symplectic. Following [12], and in contrast with a number of other references, in what follows we do not assume symplectic operators to be a priori nondegenerate.
We have the following homotopy formula (see [24,Ch. 5] and [5,23] for details): if J : V → V * is a differential symplectic operator and M × B is a star-shaped domain (recall that M and B are domains of values of u and x, respectively) then we have Here J(u) means the result of action of the differential operator J on the vector u, and for any f ∈ A the quantity f [λu] is defined as follows: In what follows we make the blanket assumption that M × B is a star-shaped domain so that (7) is automatically valid. In order to see how (7) works, consider the following simple example. Let J = D. Then we have J(u) = D(u) = u 1 , and therefore (J(u))[λu] = λu 1 . By (7) we obtain ζ = u 1 /2 and indeed the equality J = ζ ′ − ζ ′ † holds, as desired.
Note that the proper geometrical framework for the above results is provided by the formal calculus of variations, and we refer the interested reader to [2,5,24,31] and references therein for further details.
Our immediate goal is to generalize (7) to the case when the matrix operator J is weakly nonlocal rather than purely differential, see Theorem 2 below. However, we shall need a few more definitions and known results in order to proceed.
A symplectic operator J is compatible [12] with a Hamiltonian operatorP if JP J is again symplectic. If the symplectic operator J is an inverse of a Hamiltonian operator P , then the compatibility of J andP is equivalent to that of P andP . In fact, a more general assertion holds.
By equation (4.12) and Proposition 4.3 of [30] which are readily seen to be applicable in the infinite-dimensional case as well, we have and the result follows.
Note also the following easy corollary of Theorem 1 of [15].
We now need to extend A, V and V * to include weakly nonlocal elements. First of all, a qcomponent vector function f is said to be weakly nonlocal if there exist a nonnegative integer s and f 0 ∈ A q , f α ∈ A q , K α ∈ A, α = 1, . . . , s such that f can be written as where f α are linearly independent over A for α = 1, . . . , s, δK α /δu = 0, α = 1, . . . , s, and K α are linearly independent over the constants. We shall denote the space of weakly nonlocal q-component vectors in the sense of above definition byÃ q ;Ṽ (resp.Ṽ * ) will stand for the space of n-component columns (resp. rows) with entries fromÃ ≡Ã 1 . The definition of directional derivative is extended toÃ q as follows: for f of the form (10) we set Moreover, the definitions of directional derivative and the Lie derivative along Q ∈ V readily extend to the elements ofṼ. In the present paper we adopt a relatively informal approach to nonlocal variables in spirit of [11]. For a more rigorous approach to nonlocal symmetries see e.g. [2,25] and references therein. We shall call a weakly nonlocal Hamiltonian operator P normal if for any Q ∈Ṽ the condition L Q (P ) = 0 implies that Q ∈ V.

Main results
We start with the following nonlocal generalization of the homotopy formula (7). Theorem 2. Let J : V → V * be a weakly nonlocal formally skew-symmetric operator. Suppose that there exist ε α and local H α such that ε 2 α = 1 (i.e., ε α = ±1) and we have Then the operator J is symplectic if and only if there exists a local γ 0 ∈ V * such that we have Proof . If there exists γ 0 such that γ (11) satisfies then J is obviously symplectic. Now assume that J is symplectic and construct a suitable γ 0 such that γ (11) satisfies (12). Let γ = γ − γ 0 . We readily see that we have On the other hand, γ ′ − γ ′ † obviously is a symplectic operator and therefore so is By virtue of (13) we haveJ − = 0, i.e.,J is purely differential. Let Clearly, this γ 0 is local [5], and by (7) we haveJ = γ ′ 0 − (γ ′ 0 ) † . Hence γ (11) satisfies (12), and the result follows.
Theorem 2 means that the existence of a (not necessarily globally defined) weakly nonlocal γ such that (12) holds is a necessary and sufficient condition for a weakly nonlocal J to be symplectic. An important feature of this result is that the nonlocal terms in γ are uniquely determined by the structure of nonlocal terms in J, so in fact we only need to determine a local γ 0 .
Combining Lemma 1 and Theorem 2 we arrive at the following results.
Corollary 1. Let P be a nondegenerate Hamiltonian operator andP : V * → V be a formally skew-symmetric operator such that P −1P P −1 is weakly nonlocal and there exist ε α = ±1 and local F α such that Then [P,P ] = 0 if and only if there exists a local γ 0 ∈ V * such that Corollary 2. Under the assumptions of Corollary 1 suppose that P is a normal weakly nonlocal Hamiltonian operator of the form where a m are n × n matrices with entries from A,ǭ ρ are arbitrary nonzero constants, G ρ ∈ V, and we have L G ρ (δF α /δu) = 0, α = 1, . . . , s, ρ = 1, . . . ,p.
On the other hand, if there exists a weakly nonlocal τ such thatP = L τ (P ) then we have [P,P ] = 0, cf. the proof of Proposition 7.8 of [5] or equation (4) of [26], and the result follows.
The above two results are more than a mere test of whether a givenP has a zero Schouten bracket with P (and, in particular, whether the Hamiltonian operators P andP are compatible). In particular, Corollary 2 shows that if P is purely differential and normal then, under certain technical assumptions that appear to hold in all interesting examples, all weakly nonlocal Hamiltonian operators compatible with P can be written in the form L τ (P ) for suitably chosen weakly nonlocal τ . Therefore, we can search for Hamiltonian operators compatible with P by picking a general weakly nonlocal τ and requiring the operator L τ (P ) to be Hamiltonian. Clearly, we have considerably fewer unknown functions to determine than if we would just assume thatP is weakly nonlocal and formally skew-symmetric and then requireP to be a Hamiltonian operator compatible with P .
It is natural to ask under which conditions the operator P −1P P −1 meets the requirements of Corollary 1. To this end consider first a weakly nonlocal operator of the form where b m are n×n matrices with entries from A, ε α are arbitrary nonzero constants, and ψ α ∈ A are local functions.
In what follows we assume without loss of generality that δψ α /δu, α = 1, . . . , q, are linearly independent over the constants. We have the following well-known result.
Lemma 2. Let J : V → V * be a nondegenerate operator of the form (19). If P = J −1 is a purely differential operator then we have In particular, if J is symplectic then ψ α dx are Casimir functionals for the bracket {, } P .
Proof . We have where c α are arbitrary constants. Acting by P = J −1 on the left-and right-hand side of this equation yields q α=1 c α ε α P δψ α δu = 0, and since c α are arbitrary we obtain (20).
Further letP be a weakly nonlocal formally skew-symmetric operator of the form whereã m are n × n matrices with entries from A andε ρ are arbitrary nonzero constants.
Theorem 3. Let J be a weakly nonlocal symplectic operator of the form (19) andP : V * → V be a weakly nonlocal formally skew-symmetric operator of the form (21). Suppose that there exist local functions H ρ and K α such that JP (δψ α /δu) = δK α /δu, α = 1, . . . , q, Then JP J is weakly nonlocal and we have Moreover, the operator JP J is symplectic if and only if there exists a local γ 0 ∈ V * such that The proof is by straightforward computation. Note that imposing the conditions (22) is a very weak restriction, as (22) can be shown to follow from weak nonlocality and symplecticity of JP J under certain minor technical assumptions.
The conditions (22) have a very simple meaning. The first of these conditions ensures that L Y ρ (J) = 0, i.e., Y ρ are Hamiltonian with respect to J. The second condition means that the action of the operator N = JP on δψ α /δu yields a variational derivative of another Hamiltonian density K α . Moreover, if the operator N † =P J is hereditary, the said second condition guarantees [23] that N k (δψ α /δu) are variational derivatives (of possibly nonlocal Hamiltonian densities) for all k = 2, 3, . . . .
Combining Theorem 3 and Corollary 2 we readily obtain the following results.
Corollary 3. Let P be a nondegenerate Hamiltonian operator such that J = P −1 is weakly nonlocal and can be written in the form (19) for suitable p, q, b m and ψ α . Then under the assumptions of Theorem 3 any formally skew-symmetric operatorP : V * → V such that [P,P ]=0 can be written asP = L τ (P ), where τ = −P γ and γ is given by (24).
Then τ = −P γ is weakly nonlocal. Moreover, if P is a differential operator then τ = −P γ has the form where τ 0 ∈ V is local.
Proof . Using (25) and Corollary 2 we readily see that under the assumptions made τ = −P γ is indeed weakly nonlocal. If P is a differential operator then we have P δψ α δu = 0 by Lemma 2, and a straightforward computation yields (26).
For instance, let n = 2, and u = (u, v) T . Consider the symplectic structure and the Hamiltonian structure for the nonlinear Schrödinger equation, see e.g. [32] and references therein. We can rewriteP as We have The conditions of Theorem 2 and Corollary 3 are readily seen to hold, and therefore we have Given a Hamiltonian operator P , it is natural to ask under which conditionsP = L τ (P ) also is a Hamiltonian operator. A straightforward but tedious computation yields the following JP (δK α /δu) = δM α /δu, α = 1, . . . , q.
ThenP = L τ (P ) is a Hamiltonian operator if and only if there exists a local γ 0 ∈ V * such that Proof . By Proposition 1 of [26] the operatorP = L τ (P ) is Hamiltonian if and only if If P is nondegenerate then by Lemma 1 the condition (29) is equivalent to the requirement that the operator JL 2 τ (P )J be symplectic. It is readily seen that In turn, as JP J is symplectic, we have L τ (JP J) = (JP Jτ ) ′ − (JP Jτ ) ′ † , and, as τ = −P γ = −J −1 γ, where γ is given by (24), we obtain so the operator L τ (JP J) is symplectic. Hence the operator JL 2 τ (P )J is symplectic if and only if so is (JP ) 2 J. By virtue of (27) the operator (JP ) 2 J is weakly nonlocal, so we can verify its symplecticity using Theorem 2, and the result follows.
Combining Theorem 4 and Corollary 2 we obtain the following Corollary 5. Under the assumptions of Theorem 4 suppose that P is normal, weakly nonlocal and has the form (16). Further assume that we have . . ,q, σ = 1, . . . ,q, ThenP is a Hamiltonian operator if and only if there exists a weakly nonlocalτ ∈Ṽ such that L 2 τ (P ) = Lτ (P ).
Proof . We readily find thatτ = −P (−JP γ + 2γ) + Q =P γ − 2Pγ + Q, where Q ∈ V because P is normal. In complete analogy with the proof of Corollary 2 we find that the conditions (30) ensure that the coefficients at the nonlocal variables inτ are local, and thereforeτ is weakly nonlocal.

Local Hamiltonian operators of zero order
Now assume that J has the form where b 0 is an n × n matrix with entries from A.
A complete description of all symplectic operators of this form can be found in [20]. Namely, if J (31) is symplectic then we have [20] b i.e., b 0 depends only on x, u, u 1 and, moreover, is linear in u 1 . Of course, for J (31) to be symplectic the quantities b (1,s) 0 and b (0) 0 must satisfy certain further conditions, see [20] for details.
Corollary 6. Let P be a nondegenerate Hamiltonian operator such that J = P −1 has the form (31). Then any formally skew-symmetric differential operatorP : V * → V such that [P,P ] = 0 can be written asP = L τ (P ) for a local τ ∈ V.
Proof . Indeed, by Corollary 3 we can take τ = −P γ and γ given by (24) is now local.
Theorem 5. Let P be a nondegenerate Hamiltonian operator such that J = P −1 has the form (31). Then a formally skew-symmetric differential operatorP : V * → V is a Hamiltonian differential operatorP : V * → V compatible with P if and only if there exist a local τ ∈ V and a localτ ∈ V such thatP = L τ (P ) and L 2 τ (P ) = Lτ (P ).
Proof . The existence of a local τ ∈ V such thatP = L τ (P ) is immediate from Corollary 6. By Proposition 1 of [26] the operatorP = L τ (P ) is Hamiltonian if and only if [L 2 τ (P ), P ] = 0. But by Corollary 6 the latter equality holds if and only if there exists a localτ ∈ V such that L 2 τ (P ) = Lτ (P ), and the result follows.

Local Hamiltonian operators of Dubrovin-Novikov type
Assume now that P is a Hamiltonian operator of Dubrovin-Novikov type [6,7], cf. also [8,9,10], i.e., it is a matrix differential operator with the entries and det g ij = 0, i.e., P , considered as formal series, is nondegenerate. An operator P (33) with det g ij = 0 is [6,7] a Hamiltonian operator if and only if g ij is a contravariant flat (pseudo-)Riemannian metric on an n-dimensional manifold M with local coordinates u i and b ij k = − n m=1 g im Γ j mk , where Γ j mk is the Levi-Civita connection associated with g ij : Γ k ij = (1/2) n s=1 g ks (∂g sj /∂x i + ∂g is /∂x j − ∂g ij /∂x s ). Here g ij is determined from the conditions n s=1 g ks g sm = δ k m , k, m = 1, . . . , n. Let us pass to the flat coordinates ψ α (u), α = 1, . . . , n, of g ij . In these coordinates g ij becomes a constant matrix η ij , where η ij = 0 for i = j and η ii satisfy (η ii ) 2 = 1, i, j = 1, . . . , n, and the Hamiltonian operator P of Dubrovin-Novikov type associated with g ij takes the form Theorem 6 ([4]). Let P be a nondegenerate Hamiltonian operator of Dubrovin-Novikov type andP : V * → V be a purely differential formally skew-symmetric operator such that ψ α dx, ψ β dx P = 0, α, β = 1, . . . , n, where ψ α = ψ α (u) are flat coordinates for the metric g ij associated with P . Then [P,P ] = 0 if and only if there exist a local τ ∈ V such thatP = L τ (P ).

Corollary 7.
Under the assumptions of Theorem 6 suppose that ThenP is a Hamiltonian operator compatible with P if and only if there exists a localτ ∈ V such that L 2 τ (P ) = Lτ (P ).
Proof . If there exist local τ ,τ ∈ V such thatP = L τ (P ) and L 2 τ (P ) = Lτ (P ) then by Proposition 3 of [26] the operatorP indeed is a Hamiltonian operator compatible with P . On the other hand, the existence of τ such thatP = L τ (P ) is guaranteed by Theorem 6. Thus we only have to show that if the operatorP is Hamiltonian then there exists a localτ ∈ V such that L 2 τ (P ) = Lτ (P ). By Proposition 1 of [26] the operatorP = L τ (P ) is Hamiltonian if and only if [L 2 τ (P ), P ] = 0. As (36) holds by assumption, by Theorem 6 we have [L 2 τ (P ), P ] = 0 if and only if there exists a localτ ∈ V such that L 2 τ (P ) = Lτ (P ), and the result follows.
For a simple example, let n = 1, u ≡ u, and let P = D andP = D 3 + 2uD + u 1 be the first and the second Hamiltonian structure of the KdV equation. We have [4]P = L τ (P ) for τ = −(u 2 + u 2 )/2, and it is readily seen that the conditions of Corollary 7 are satisfied, so there exists a localτ such that L 2 τ (P ) = Lτ (P ). An easy computation shows that the latter equality holds e.g. forτ = −u 4 /2 − u 2 1 /2 + 5u 3 /6.

Conclusions
In the present paper we extended the homotopy formula (7) to a large class of weakly nonlocal symplectic structures, see Theorem 2 above. Besides the potential applications to the construction of nonlocal extensions for the variational complex, this result enabled us to provide a complete description for a large class of weakly nonlocal Hamiltonian operators compatible with a given nondegenerate weakly nonlocal Hamiltonian operator P that possesses a weakly nonlocal inverse (Corollaries 2, 3, 4, and 5) or, more broadly, with a given weakly nonlocal symplectic operator J (Theorems 3 and 4). These results admit useful simplifications for the case of zero-and first-order differential Hamiltonian operators, as presented in Sections 4 and 5. In particular, in Section 5 we provide a simple description for a very large class of local higher-order Hamiltonian operators compatible with a given local Hamiltonian operator of Dubrovin-Novikov type. Note that finding an efficient complete description of the nondegenerate weakly nonlocal Hamiltonian operators with a weakly nonlocal inverse is an interesting open problem, because such operators would naturally generalize the Hamiltonian operators (33) of Dubrovin-Novikov type from Section 5 and the zero-order local Hamiltonian operators from Section 4. Thus, we extended the Lie derivative approach to the study of Hamiltonian operators compatible with a given Hamiltonian operator P from finite-dimensional Poisson structures [26,28] and Hamiltonian operators of Dubrovin-Novikov type [21,26] to the weakly nonlocal Hamiltonian operators of more general form. An important advantage of this approach is that the vector fields τ andτ in general involve a considerably smaller number of unknown functions than a generic formally skew-symmetric operator being a "candidate" for a Hamiltonian operator compatible with P , and the search for such vector fields is often much easier than calculating directly the Schouten brackets involved, cf. also the discussion in [26,28]. This could be very helpful in solving the classification problems like the following one: to describe all weakly nonlocal Hamiltonian operators compatible with a given Hamiltonian operator P and having a certain prescribed form.