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

TL;DR

This paper introduces a self-dual formulation of the Ruijsenaars-Schneider models using Bäcklund transformations, linking them to the ILW equation with a discrete Laplacian, and extends previous Calogero-Moser results.

Contribution

It presents a novel self-dual form of Ruijsenaars-Schneider models based on first order equations involving dual variables, connecting these models to ILW equations with discrete Laplacian.

Findings

Self-dual form derived from complexified ILW equation with pole ansatz

Connection established between Ruijsenaars-Schneider and ILW models

Extension of Calogero-Moser results to relativistic models

Abstract

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.