Integrable Time-Discretisation of the Ruijsenaars-Schneider Model

TL;DR

This paper introduces an exactly integrable discrete-time model related to the Ruijsenaars-Schneider system, providing a Lax pair, symplectic structure, and exact solutions, bridging quantum and classical integrable systems.

Contribution

It presents a novel integrable symplectic correspondence for the Ruijsenaars-Schneider model, including a Lax pair, involutivity proof, and connections to Bethe Ansatz equations and soliton solutions.

Findings

Derived an integrable symplectic correspondence for the model

Established a Lax pair and proved involutivity of invariants

Connected Bethe Ansatz equations with discrete soliton solutions

Abstract

An exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the Calogero-Moser system, that was introduced for the first time by Ruijsenaars and Schneider. For the discrete-time model the equations of motion take the form of Bethe Ansatz equations for the inhomogeneous spin-1/2 Heisenberg magnet. We present a Lax pair, the symplectic structure and prove the involutivity of the invariants. Exact solutions are investigated in the rational and hyperbolic (trigonometric) limits of the system that is given in terms of elliptic functions. These solutions are connected with discrete soliton equations. The results obtained allow us to consider the Bethe Ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding Bethe wavefunction.