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arXiv · 2109.05240

Constrained Toda hierarchy and turning points of the Ruijsenaars-Schneider model

I. Krichever·A. Zabrodin
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Abstract

We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the constraint $\bar {\cal L}={\cal L}^{\dag}$ on the two Lax operators (in the symmetric gauge). We prove the existence of the tau-function of the C-Toda hierarchy and show that it is the square root of the 2D Toda lattice tau-function. In this and some other respects the C-Toda is a Toda analogue of the CKP hierarchy. It is also shown that zeros of the tau-function of elliptic solutions satisfy the dynamical equations of the Ruijsenaars-Schneider model restricted to turning points in the phase space. The spectral curve has holomorphic involution which interchange the marked points in which the Baker-Akhiezer function has essential singularities.

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I. Krichever, A. Zabrodin. 2021-09-11. Constrained Toda hierarchy and turning points of the Ruijsenaars-Schneider model. https://doi.org/10.1007/s11005-022-01519-0

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