Simple Lax Description of the ILW Hierarchy

In this note we present a simple Lax description of the hierarchy of the intermediate long wave equation (ILW hierarchy). Although the linear inverse scattering problem for the ILW equation itself was well known, here we give an explicit expression for all higher flows and their Hamiltonian structure in terms of a single Lax difference-differential operator.


Introduction
The intermediate long wave (ILW) equation was introduced in [7] to describe the propagation of waves in a two-layer fluid of finite depth. The model represents the natural interpolation between the Benjamin-Ono deep water and Kortweg-de Vries shallow water theories. The ILW equation for the time s evolution of a function w = w(x) defined on the real line has the following form: where δ is a parameter and p.v. · dξ represents the principal value integral. This equation can be rewritten in the formal loop space formalism (see for instance [3], from which we borrow notations for the rest of the paper) as where B 2g are the Bernoulli numbers. Indeed the action of the operator T on a function f can be written in terms of the derivatives of f as T (f ) = n≥1 δ 2n−1 2 2n |B 2n | (2n)! ∂ 2n−1 x f , and, by setting √ µ t and δ = ε √ µ 2 , we see how the above equations are equivalent. In [10] the integrability of the ILW equation was established, finding an infinite number of conserved densities and giving a corresponding inverse scattering problem. From this inverse scattering problem a Lax representation can of course be deduced. Also, through a simple Hamiltonian structure, the conserved densities generate an infinite number of commuting flows. However the explicit relation between the Lax representation of the ILW equation, its higher flows and the Hamiltonian structure was never, to the best of our knowledge, clarified in the literature.
In this paper we give an explicit and remarkably simple Lax description of the full hierarchy of commuting flows of the ILW equation (the ILW hierarchy) and their Hamiltonian structure in terms of a single Lax mixed difference-differential operator.
As it turns out, this Lax description corresponds to the equivariant bi-graded Toda hierarchy of [9,Section B.2] in the somewhat degenerate case of bi-degree (1, 0), hence a reduction of the 2D-Toda hierarchy. In fact, in [9], because of the geometric origin of their problem, the authors always work with strictly positive bi-degree, but this is an unnecessary restriction. What we do here is prove that the bi-degree (1, 0) case corresponds to the ILW hierarchy in its Hamiltonian formulation. A similar question is raised in [5,Remark 31], where the author notices that the definition of all but the extended flows of the extended bi-graded Toda hierarchy survives when one of the bi-degrees vanishes. Since in the equivariant case there is no need of extended flows, this problem does not arise in the Lax representation of the equivariant bi-graded Toda hierarchy. Of course this means that equivariant bi-graded Toda hierarchies of any bidegree (N, 0), with N ≥ 1, are well defined (see also [1] for their relation with the rational reductions of the 2D-Toda hierarchy). We will study the relation of such hierarchies with the geometry of equivariant orbifold Gromov-Witten theory in an upcoming publication.

ILW hierarchy
Here and in what follows we will use the formal loop space formalism in the notations of [3]. The Hamiltonian structure of the ILW equation (1.1) is given by the Hamiltonian u , d ≥ 2, of the higher flows of the ILW hierarchy are uniquely determined by the properties For example, We refer the reader to the paper [2, Section 8], which explains a relation between the Hamiltonians h ILW d and the conserved quantities of the ILW equation, constructed in [10]. It is convenient to introduce an additional Hamiltonian h ILW 0 := u 2 2 dx, which generates spatial translations. So the flows of the ILW hierarchy are given by where we identify the times t 1 and t.

Lax description of the ILW hierarchy
Our Lax description of the ILW hierarchy is presented in Section 3.2, see Theorem 1. Before that, in Section 3.1, we recall necessary definitions from the theory of shift operators.

Shift operators
Let Λ := e iε∂x . We will consider formal series of the form A = n≤m a n Λ n , a n ∈ A u , m ∈ Z.
Via the operation of composition •, the vector space of such formal operators is endowed with the structure of a non-commutative associative algebra. The positive part A + , the negative part A − and the residue res A of the operator A are defined by Let z be a formal variable. The symbol A of the operator A is defined by A := n≤m a n e nz .
For an operator L of the form L = Λ + n≥0 a n Λ −n , a n ∈ A u , one can define the dressing operator P , by the identity Note that the coefficients p n of the dressing operator do not belong to the ring A u , but to a certain extension of it (see, e.g., [6, Section 2]). The dressing operator P is defined up to the multiplication from the right by an operator of the form 1 + n≥1 p n Λ −n , where p n are some constants.
The logarithm log L is defined by The ambiguity in the choice of dressing operator is cancelled in the definition of log L and, moreover, the coefficients of log L do belong to A u (see the proof of Theorem 2.1 in [6]). To be more precise, one has the commutation relations log L, L m = 0, m ≥ 1, which imply that These relations allow to compute recursively all the coefficients of the operator iεP x • P −1 . As a result, if we write then the coefficient f n can be expressed as a differential polynomial in the coefficients a 0 , a 1 , . . ., a n−1 of the operator L, f n = f n (a 0 , . . . , a n−1 ) ∈ A [0] a 0 ,...,a n−1 . For example,

Lax description
Let τ be a formal variable and From the discussion of the construction of the logarithm log L in the previous section it is easy to see that there exists a unique operator L of the form Since L d+1 , L = 0, d ≥ 0, the commutator L d+1 + , L doesn't contain terms with non-zero powers of Λ. Consider the following system of PDEs: The following theorem is the main result of our paper.

The system of Lax equations (3.4) possesses a Hamiltonian structure given by the Hamiltonians
and the Poisson bracket associated to the operator ∂ x .
3. Let y be a formal variable and define polynomials P d (y) ∈ Q[y, τ ], d ≥ 1, by The ILW hierarchy is related to the hierarchy (3.4) by the following triangular transformation: Proof . 1. Hereafter, for simplicity, we will use L m + to denote (L m ) + . Let Let us first check that Equations (3.1) and (3.3) imply that When we know formulas (3.6), the commutativity of the flows ∂ ∂T d is proved by a standard computation: 2. Note that the flows ∂ ∂T d can be written as Let us compute the flow ∂ ∂T 1 . For the coefficients of the operator L, one can immediately see that a 0 = u and then, using formula (3.2), we get This allows to compute Lax d are conserved quantities for the flow ∂ ∂T 1 . Indeed, which is zero because Therefore, the local functionals h Lax d together with the Poisson bracket {·, ·} ∂x generate the flows which commute with the flow ∂ ∂T 1 . Then these flows are uniquely determined by their dispersionless parts (see [8,Lemma 3.3] or [4,Lemma 4.14]). Hence, it is sufficient to check the equation at the dispersionless level. Denote L 0 := L ε=0 . We see that it is sufficient to check that For this we compute This implies equation (3.9). 3. We see that Using again the result of [8,Lemma 3.3] (see also [4,Lemma 4.14]), we conclude that it is sufficient to prove equation (