Elliptic solutions to Toda lattice hierarchy and elliptic Ruijsenaars-Schneider model

TL;DR

This paper extends the known correspondence between elliptic solutions of the 2D Toda lattice hierarchy and the elliptic Ruijsenaars-Schneider model to the entire hierarchy, linking Hamiltonians to spectral curve expansions.

Contribution

It demonstrates that the Hamiltonians governing pole dynamics in the hierarchy are derived from spectral curve expansions of the Ruijsenaars-Schneider Lax matrix.

Findings

Hamiltonians for pole dynamics are obtained from spectral curve expansions.

The correspondence between Toda hierarchy solutions and Ruijsenaars-Schneider model is extended to all hierarchical times.

Spectral curve analysis reveals the structure of Hamiltonians in the hierarchy.

Abstract

We consider solutions of the 2D Toda lattice hierarchy which are elliptic functions of the zeroth time t_0=x. It is known that their poles as functions of t_1 move as particles of the elliptic Ruijsenaars-Schneider model. The goal of this paper is to extend this correspondence to the level of hierarchies. We show that the Hamiltonians which govern the dynamics of poles with respect to the m-th hierarchical times t_m and \bar t_m of the 2D Toda lattice hierarchy are obtained from expansion of the spectral curve for the Lax matrix of the Ruijsenaars-Schneider model at the marked points.