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arXiv · 1907.06621

Toda lattice hierarchy and trigonometric Ruijsenaars-Schneider hierarchy

V. Prokofev·A. Zabrodin
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Abstract

We consider solutions of the 2D Toda lattice hierarchy which are trigonometric functions of the ``zeroth'' time $t_0=x$. It is known that their poles move as particles of the trigonometric Ruijsenaars-Schneider model. We extend this correspondence to the level of hierarchies: the dynamics of poles with respect to the $m$-th hierarchical time $t_m$ (respectively, $\bar t_m$) of the 2D Toda lattice hierarchy is shown to be governed by the Hamiltonian which is proportional to the $m$-th Hamiltonian $\mbox{tr}\, L^m$ (respectively, $\mbox{tr}\, L^{-m}$) of the Ruijsenaars-Schneider model, where $L$ is the Lax matrix.

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V. Prokofev, A. Zabrodin. 2019-07-15. Toda lattice hierarchy and trigonometric Ruijsenaars-Schneider hierarchy. https://doi.org/10.1088/1751-8121%2Fab520c

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