Related papers: Elliptic Ruijsenaars-Toda and elliptic Toda chains…

In this letter we discuss the classical integrable elliptic Toda chain proposed by I. Krichever. Our goal is to construct an open elliptic Toda chain with boundary terms. This is achieved using the factorized form of the Lax matrix and…

We suggest a field extension of the classical elliptic Ruijsenaars-Schneider model. The model is defined in two different ways which lead to the same result. The first one is via the trace of a chain product of $L$-matrices which allows one…

We demonstrate that in a certain gauge the elliptic Ruijsenaars-Shneider model with N=2 admits a nondynamical r-matrix structure and the corresponding classical r-matrix is the same as that of its non-relativistic counterpart…

The classical r-matrix structure for the generic elliptic Ruijsenaars-Schneider model is presented. It makes the integrability of this model as well as of its discrete-time version that was constructed in a recent paper manifest.

We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the…

We propose a relation between the elliptic SL(N,C) top and Toda systems and obtain a new class of integrable systems in a specific limit of the elliptic SL(N,C) top. The relation is based on the Inozemtsev limit (IL) and a symplectic map…

From the dynamical twisting of the classical r-matrix, we obtain a new Lax operator for the elliptic Ruijsenaars-Schneider model (cf. Ruijsenaars'). The corresponding r-matrix is shown to be the classical $Z_n$-symmetric elliptic r-matrix,…

We use the Hamiltonian reduction method to construct the Ruijsenaars dual systems to generalized Toda chains associated with the classical Lie algebras of types $B, C, D$. The dual systems turn out to be the $B, C$ and $D$ analogues of the…

We explore various aspects of the correspondence between dimer models and integrable systems recently introduced by Goncharov and Kenyon. Dimer models give rise to relativistic integrable systems that match those arising from 5d N=1 gauge…

We study elliptic solutions of the recently introduced Toda lattice with the constraint of type B and derive equations of motion for their poles. The dynamics of poles is given by the deformed Ruijsenaars-Schneider system. We find its…

It is shown that the classical L-operator algebra of the elliptic Ruijsenaars-Schneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions.…

We demonstrate that in a certain gauge the elliptic Ruijsenaars--Schneider models admit Lax representation governed by the same dynamical $r$--matrix as their non--relativistic counterparts (Calogero--Moser models). This phenomenon was…

For the integrable $N$-particle Calogero-Moser system with elliptic potential it is shown that the Lax operator found by Krichever possesses a classical $r$-matrix structure. The $r$-matrix is a natural generalisation of the matrix found…

The hierarchy of the classical nonlinear integrable equations associated with relativistic Toda chain model is considered. It is formulated for the N-th powers of the quantum operators of the corresponding quantum integrable models.…

We demonstrate that the generalization of the relativistic Toda chain (RTC) is a special reduction of two-dimensional Toda Lattice hierarchy (2DTL). We also show that the RTC is gauge equivalent to the discrete AKNS hierarchy and the…

We present a description of the classical elliptic ${\rm BC}_1$ Ruijsenaars-van Diejen model with 8 independent coupling constants through a pair of ${\rm BC}_1$ type classical Sklyanin algebras generated by the (classical) quadratic…

We construct new elliptic solutions of the restricted Toda chain. These solutions give rise to a new explicit class of orthogonal polynomials which can be considered as a generalization of the Stieltjes-Carlitz elliptic polynomials.…

We consider solutions of the 2D Toda lattice hierarchy which are elliptic functions of the zeroth time t_0=x. It is known that their poles as functions of t_1 move as particles of the elliptic Ruijsenaars-Schneider model. The goal of this…

We demonstrate that the generalization of the relativistic Toda chain (RTC) is a special reduction of two-dimensional Toda Lattice hierarchy (2DTL). This reduction implies that the RTC is gauge equivalent to the discrete AKNS hierarchy and,…

We present the Lax pair formalism for certain extension of the continuous limit of the classical Toda lattice hierarchy, provide a well defined notion of tau function for its solutions, and give an explicit formulation of the relationship…