Self-dual form of Ruijsenaars-Schneider models and ILW equation with discrete Laplacian

We discuss a self-dual form or the B\"acklund transformations for the continuous (in time variable) ${\rm gl}_N$ Ruijsenaars-Schneider model. It is based on the first order equations in $N+M$ complex variables which include $N$ positions of particles and $M$ dual variables. The latter satisfy equations of motion of the ${\rm gl}_M$ Ruijsenaars-Schneider model. In the elliptic case it holds $M=N$ while for the rational and trigonometric models $M$ is not necessarily equal to $N$. Our consideration is similar to the previously obtained results for the Calogero-Moser models which are recovered in the non-relativistic limit. We also show that the self-dual description of the Ruijsenaars-Schneider models can be derived from complexified intermediate long wave equation with discrete Laplacian be means of the simple pole ansatz likewise the Calogero-Moser models arise from ordinary intermediate long wave and Benjamin-Ono equations.


Introduction
It was observed in [8] that N-body classical Calogero-Moser model [9] appears from N-soliton solution of the Benjamin-Ono equation [6] on the real line where − is the principal value integral. Namely, (1.1) is fulfilled by the pole ansatz , Im(q k ) < 0 (1. (1. 4) This trick was generalized by Wojciechowski in [29] to the Bäcklund transformations. Later, in [1,2], it was referred to as the self-dual form of the Calogero-Sutherland model including harmonic and other [15] external potentials. From the point of view of the Benjamin-Ono equation (1.1) these models are related to the intermediate long wave (ILW) equation [13,14,4] where δ and ν are constants and T is trigonometric or elliptic analogue of the Hilbert transformation H used in (1.1). (See Appendix B for a brief review.) For example, in the elliptic case [4,17] T f (x) = ı 2π − where E 1 (z) is the first Eisenstein function (A.3).
We are going to consider complexified integrable many-body systems mentioned above, i.e., in our setting the positions of particles are complex numbers. For this purpose we need the complexified version of the ILW equation (1.5). It is written in terms of a pair of complex functions f ± (z) as follows 3 [1,2]: provides N + M first order equations (the Bäcklund transformations or the self-dual form of the Calogero-Sutherland model) (1.9) In the elliptic case N = M; in the trigonometric and rational cases N and M can be not equal to each other. In the latter cases the E 1 (z) function in (1.9) should be replaced by coth(z) and 1/z respectively, see (A.17).
By differentiating (1.9) with respect to the time variable and some tedious calculations it can be shown that both sets of variables q and µ satisfy the Calogero-Moser equations of motion: . The derivation of (1.8)-(1.10) in the elliptic case was presented in [7].
Equations of type (1.9) are known to be embedded into discrete time dynamics [21,27]. Then the two sets of variables {q i } and {µ i } are related by a discrete time shift. The discrete equations of motion involve three sets of variables (related by two subsequent time shifts). We do not use this approach since we deal with only two sets of variables.
The purpose of the paper. First, we describe the self-dual form of the gl N Ruijsenaars-Schneider models [23], which have the following equations of motion: where q ij = q i − q j . Hyperbolic and rational analogues of E 1 (z) are given by coth(z) and 1/z respectively, see (A.17). We claim that the described above construction for self-dual representation of the Calogero-Moser model is generalized to the Ruijsenaars-Schneider one.
Theorem 1 Equations of motion (1.11) for the Ruijsenaars-Schneider model follow from the set of N + M equationsq where ϑ(z) should be replaced by sinh(z) and z in hyperbolic and rational cases respectively. The variables {µ α } satisfy gl M Ruijsenaars-Schneider equations of motion: In the elliptic case N = M, while in hyperbolic and rational cases N and M are arbitrary.
Note that equations (1.12) are well known in the theory of time discretization [22] (and/or Bäcklund transformations [16]) of the Ruijsenaars-Schneider model 4 . Here we give a direct proof (without usage of the discrete time dynamics) likewise it was presented in [29] for the Calogero-Moser models.
Let N ≥ M for definiteness. It was shown in [2] for the Calogero-Sutherland models that M integrals of motion coincide, and other N − M are equal to some constants. We give a proof of a similar result for the Ruijsenaars-Schneider models using determinant identities from [11,5].
Next, we show that equations (1.12) follow from some multi-pole anzats for a pair of complex functions satisfying (complexified version of) the ILW equation with discrete Laplacian. The latter was suggested in [24,25,26]: (1.14) It can be reduced to the following equation for a single real function: where f = f (x, t), x ∈ R and It should be mentioned that a relation between the Ruijsenaars-Schneider (and Calogero-Moser) models and ILW-Benjamin-Ono equations is known [3,10] from the collective field theory description of integrable many-body systems, which is adapted to the N → ∞ limit. Related algebraic structures and possible applications can be found in [7,18,19,20].
The paper is organized as follows. In the next section we prove Theorem 1 and coincidence of (a part of) action variables for the Ruijsenaars-Schneider models (1.11) and (1.13). In Section 3 we review the ILW equation with discrete Laplacian following [24] and describe its relation to the self-dual form (1.12).

Self-dual form of Ruijsenaars-Schneider models
We are going to prove Theorem 1. Let us start with elliptic case, i.e. M = N. Its hyperbolic and rational counterparts are discussed in the end of the section. It is convenient to deal with the Kronecker function (A.2). Using (A.7), we rewrite (1.11) in the form Also, rescale formally the time variable as This has no affect on equations (1.11) or (2.1) since they are homogeneous in t. At the same time (1.12) acquires the forṁ For the proof of the theorem we need the following identity: It is a particular case of (A.9). Indeed, consider n = 2N −1 in (A.9) or, what is more convenient, let the summation (and multiplication) index i in (A.9) to be i = 2, ..., 2N. Substitute Then, using the property (A.5) we get (2.4) from (A.9) in the formq 1 i . Consider derivative of the identity (2.5) with respect to q i . It reads as follows: Proof of Theorem 1. Computeq i by differentiating the upper line of (2.3) by the time variable. This yieldsq Equations of motion (2.1) appear by subtracting the l.h.s. of (2.7) from the r.h.s. of (2.8).
The hyperbolic and rational cases are obtained as follows. All the proof is the same since the functions in each line of (A.17) satisfy the same identities. So that in the M = N case the statement is proved. In the case M = N, say M < N we apply the argument called "dimensional reduction" in [2]: N − M coordinates µ α , α = M + 1...N may go to infinity in the rational or hyperbolic case, i.e. there is a limit |µ α | → ∞, α = M + 1...N in which (1.12) with N = M turns into the same system of equations with M < N.
where in the r.h.s. the non-relativistic velocities are implied. In order to make the non-relativistic limit in (2.3) one should first come back to (1.12) (through the inverse rescaling of the time variable (2.2)) and make an additional rescaling t → ct (this leads to multiplication of the r.h.s. of (1.12) by c). Then the limit (2.9), (2.10) of (1.12) reproduces the self-dual form of the elliptic Calogero-Moser model (1.9). For M = N in rational or trigonometric cases the "dimensional reduction" argument is used again. In this way we reproduce the self-dual representation of the Calogero-Moser models [8,1,2,7].
Let us discuss the relation between gl N and gl M Ruijsenaars-Schneider models in variables {q} (1.11) and {µ} (1.13). For this purpose introduce the following pair of matrices used in [11] for the direct proof of the quantum-classical correspondence between the classical (rational) Ruijsenaars-Schneider model and generalized (XXX) quantum spin chains: The following relation holds true for matrices (2.11) and (2.12): Here I is the unity matrix. Set g = 1 and substitute the products in (2.11)  In the trigonometric case the analogues of relations (2.11)-(2.13) were used in [5] for the proof of the quantum-classical duality between the classical (trigonometric) Ruijsenaars-Schneider model and generalized (XXZ) quantum spin chain.
and the pair of N × N and M × M matrices: (2.16) Then the following identity is valid for (2.14), (2.15), (2.16): Again, we substitute the products (2.15), (2.16) from (1.12) (in the hyperbolic case ϑ(z) is replaced by sinh(z), see (A.17)). Then L andL are the Lax matrices of gl N and gl M Ruijsenaars-Schneider models respectively. And again, this means that M action variables in both models coincide, and the other N −M of the first one take degenerated values given by diagonal elements of the matrix S. Notice that in N = M case similar result is easily obtained in the context of discrete time dynamics [22] (it comes immediately from the discrete Lax equation).

ILW equation and Ruijsenaars-Schneider model
A brief review of the Benjamin-Ono and ILW equations is given in Appendix B. Here we describe the construction of the (double periodic) ILW equation with discrete Laplacian [24] and its relation to the self-dual form of the Ruijsenaars-Schneider model.

ILW equation with discrete Laplacian
The periodic ILW equation with discrete Laplacian was proposed in [24]: The integral transformation (3.2) is defined for a periodic function f (x + 1, t) = f (x, t) on the real axis x ∈ R, and 0 < Im(η) < Im(τ ). The modular parameter of the elliptic curve is assumed to be purely imaginary: Re(τ ) = 0 (and Im(τ ) > 0).
The last term in (3.2) is proportional to T = (1/2ı)T with T (B.8), (B.9) and normalization of the real half-period L = 1/2: It was argued in [24,25] that the integral operator (3.2) can be written in the following form: In order to define F ± (x) let us denote byf (x) the zero mean part of f (x), i.e.f (

5)
This function is periodic, F (z) = F (z + 1), holomorphic in the strip domain 0 < Im(z) < Im(τ ) and continuous up to its boundary. It can be τ -periodically continued to piecewise 5 holomorphic function on C due to the properties (A.6) and zero mean off (x) (see [17]). Then, for we have due to the Sokhotski−Plemelj formulae: and The functions F ± (x ± η) are analytical continuations of F ± (x) (3.6). The limit to the ILW equation is achieved as follows. Consider T in the limit η → 0. From (3.4) and (3.7), (3.8) we have Choose also c = νη + aη 2 + O(η 3 ) with another constant a and rescale t →t = t/η 2 . Then, taking the limit η → 0, we obtain ut = au y − uu y + νT u yy .

Ruijsenaars-Schneider models and discrete ILW equation
Let us consider the complexified version of (3.9) with a complex variable x. In [1] the self-dual form of the Calogero-Sutherland models were obtained by passing from equation (3.12) to the following one 6 u t = −uu z − ν 2ũ zz , (3.13) written for a pair of independent complex functions u = u + − u − andũ = u + + u − . It was called bidirectional Benjamin-Ono equation. The relation of type (3.7) was treated as an additional reduction. In a similar manner, the self-dual form of the elliptic Calogero-Moser model was described through the complexified periodic ILW equation [7].
In order to get the self-dual form for the Ruijsenaars-Schneider model (2.3), it is reasonable to study equation (3.1) with T given by (3.4): (3.14) Equation (3.14) generalizes (3.9) in the same way as (3.13) generalizes (3.12) (with a = 0). In what follows we deal with (3.14). It is written for two independent complex functions F + (z), F − (z), z ∈ C (f 0 is a constant). We are going to show that a natural multi-pole ansatz provides the Ruijsenaars-Schneider model 7 in the form (2.3). Introduce Res and As a by-product, we have an alternative proof of (2.4): notice that f (z) is double-periodic in z due to (A.6) (here M = N is necessary). Therefore, the sum of residues equals zero.
Due to the structure of poles the double-periodic function f (z) (3.15) is represented also as with some constant f 0 . Denote and  (3.19), (3.20) satisfies (3.14) and provides equations of motion of the Ruijsenaars-Schneider models (2.3).
Proof: Indeed, differentiating f (z) (3.15) with respect to t and using (A.7) we obtain For the r.h.s. of (3.14) with F ± (z) given by (3.19), (3.20) we have By comparing (3.21) and (3.22) we get: Res Notice that the time variable in (3.14) is assumed to be rescaled as given in (2.2). Alternatively one could consider the ansatz (3.15) in a slightly different form: In this casef The ansatz (3.24) leads to equations (1.12), and the additional rescaling of time variable is not needed.

Appendix A: elliptic functions
We use the odd theta-function with the modular parameter τ , Im(τ ) > 0 [12,28]: the Kronecker function and the Eisenstein functions The Eisenstein functions are simply related to the Weierstrass functions: The parities are: The behavior on the lattice of periods Γ = Z + τ Z is The derivative of the Kronecker function with respect to the second argument is The Fay trisecant identity for genus one reads There are also higher (n-th order) identities: Proof is by induction in n. Suppose (A.9) is true. We then need to prove that Consider the r.h.s. of (A.10). Let us write the first n terms of the sum separately: The first terms in (A.11) and (A.12) are the same. Therefore, we need to prove that the second terms are equal as well, i.e.
After cancellation of the common factor φ x n+1 , The first Eisenstein function E 1 (z) possesses the following series representation: (here −Im(τ ) <Im(z) <Im(τ )). The first term in (A.13) regarded as a generalized function can be represented via sin(kx) . (A.14) Then (A.13) acquires the form We also need the following modular transformation: 16) Rational and hyperbolic analogues of the functions (A.1)-(A.3) are as follows: rational :

Appendix B: Benjamin-Ono and ILW equations
Here we review the Benjamin-Ono and ILW equations.
1. Rational case. Let Hf be the Hilbert transform of the function 8 f (x, t) in the variable x ∈ R: The Benjamin-Ono equation [6] is as follows 9 : 2. Trigonometric case. Similarly, the periodic Benjamin-Ono equation [14] reads with the integral transformation When L → ∞ we reproduce (B.2).