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

Abstract

We discuss a self-dual form or the Bäcklund transformations for the continuous (in time variable) ${\mathrm{gl}}_N$ 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 ${\mathrm{gl}}_M$ 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.

16 pages, references added

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Cited by in corpus (6)

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• Non-chiral Intermediate Long Wave equation and inter-edge effects in narrow quantum Hall systems
• Dualities in quantum integrable many-body systems and integrable probabilities -- I
• On Dimensional Transmutation in 1+1D Quantum Hydrodynamics
• Toda lattice hierarchy and trigonometric Ruijsenaars-Schneider hierarchy
• On R-matrix identities related to elliptic anisotropic spin Ruijsenaars-Macdonald operators