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

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 $\bar{\mathcal{L}}={\mathcal{L}}^{†}$ 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.

24 pages, no figures

References in corpus (1)

• BKP and CKP revisited: The odd KP system

Cited by in corpus (6)

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• Multi-component Toda lattice hierarchy
• Dispersionless version of the constrained Toda hierarchy and symmetric radial Löwner equation