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.

arXiv comment24 pages, no figures

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Cited by in corpus (6)

• Toda lattice with constraint of type B
• Elliptic families of solutions to constrained Toda hierarchy
• Elliptic solutions of the Toda lattice with constraint of type B and deformed Ruijsenaars-Schneider system
• Monodromy free linear equations and many-body systems
• Multi-component Toda lattice hierarchy
• Dispersionless version of the constrained Toda hierarchy and symmetric radial Löwner equation