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

We discuss a self-dual form or the Backlund transformations for the continuous (in time variable) 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 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 by means of the simple pole ansatz likewise the Calogero-Moser models arise from ordinary intermediate long wave and Benjamin-Ono equations. (c) 2018 The Authors. Published by Elsevier B.V.