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## 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 be means of the simple pole ansatz likewise the Calogero-Moser models arise from ordinary intermediate long wave and Benjamin-Ono equations.