Related papers: Dynamical r-matrix for the elliptic Ruijsenaars-Sc…
The purpose of this paper is to construct a generalized r-matrix structure of finite dimensional systems and an approach to obtain the algebro-geometric solutions of integrable nonlinear evolution equations (NLEEs). Our starting point is a…
We describe the most general ${\rm GL}_{NM}$ classical elliptic finite-dimensional integrable system, which Lax matrix has $n$ simple poles on elliptic curve. For $M=1$ it reproduces the classical inhomogeneous spin chain, for $N=1$ it is…
By applying the Hamiltonian reduction scheme we recover the R-matrix of the trigonometric and elliptic Calogero-Moser system.
A family of integrable $GL(NM)$ models is described. On the one hand it generalizes the classical spin Ruijsenaars--Schneider systems (the case $N=1$), and on the other hand it generalizes the relativistic integrable tops on $GL(N)$ Lie…
We present a direct construction of compact real forms of the trigonometric and elliptic $n$-particle Ruijsenaars-Schneider systems whose completed center-of-mass phase space is the complex projective space $\mathbb{CP}^{n-1}$ with the…
We discuss the classical elliptic Toda chain introduced by Krichever and the elliptic Ruijsenaars-Toda chain introduced by Adler, Shabat and Suris. It is shown that these models can be obtained as particular cases of the elliptic…
It is shown that the elliptic Ruijsenaars-Schneider model can be obtained from the cotangent bundle over the two-dimensional current group by means of the Hamiltonian reduction procedure.
We introduce an integrable time-discretized version of the classical Calogero-Moser model, which goes to the original model in a continuum limit. This discrete model is obtained from pole solutions of a discretized version of the…
We propose and prove a set of identities for ${\rm GL}_M$ elliptic $R$-matrix (in the fundamental representation). In the scalar case ($M=1$) these are elliptic function identities derived by S.N.M. Ruijsenaars as necessary and sufficient…
The Hamiltonian structure of spin generalization of the rational Ruijsenaars-Schneider model is found by using the Hamiltonian reduction technique. It is shown that the model possesses the current algebra symmetry. The possibility of…
In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum $R$-matrices. Here we study the simplest case -- the 11-vertex $R$-matrix and related ${\rm gl}_2$ rational…
The expression of the quantum Ruijsenaars-Schneider Hamiltonian is obtained in the framework of the dynamical $R$-matrix formalism. This generalizes to the case of $U_q(sl_n)$ the result obtained by O. Babelon, D. Bernard and E. Billey for…
The classical (dynamical) $R$-matrices for the 2- and 3-body Calogero-Moser models with elliptic potentials are given. The 3-body case has an interesting nontrivial structure that goes beyond the known ansatz for momentum independent…
We consider a hierarchy of the natural type Hamiltonian systems of $n$ degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax representation in terms of $2\times 2$…
A new method for the construction of classical integrable systems, that we call loop coproduct formulation, is presented. We show that the linear r-matrix formulation, the Sklyanin algebras and the reflection algebras can be obtained as…
We introduce an integrable time-discretized version of the classical Calogero-Moser model, which goes to the original model in a continuum limit. This discrete model is obtained from pole solutions of a semi-discretized version of the…
The new integrable systems associated to the space of elliptic branched coverings are constructed. The relationship of these systems with elliptic Schlesinger's system is described. For the standard two-fold elliptic coverings the…
A simple discrete planar dynamical model for the ideal (logical) R-S flip-flop circuit is developed with an eye toward mimicking the dynamical behavior observed for actual physical realizations of this circuit. It is shown that the model…
We briefly review the possible Poisson structures on the chiral WZNW phase space and discuss the associated Poisson-Lie groupoids. Many interesting dynamical r-matrices appear naturally in this framework. Particular attention is paid to the…
J. F. van Diejen and H. Puschmann have recently shown that the dynamics of zeros of the n-solitonic solutions to the Schrodinger equation with the reflectionless potential is governed by a rational Ruijsenaars-Schneider system. We use the…