# Toda lattice hierarchy and trigonometric Ruijsenaars-Schneider hierarchy

## TL;DR

This paper establishes a connection between the dynamics of poles in solutions of the 2D Toda lattice hierarchy and the Ruijsenaars-Schneider model, extending known particle correspondence to the entire hierarchy through Hamiltonian flows.

## Contribution

It extends the pole-particle correspondence from the 2D Toda lattice to its hierarchy, linking pole dynamics to Ruijsenaars-Schneider Hamiltonians at all hierarchical times.

## Key findings

- Pole dynamics governed by hierarchy times match Ruijsenaars-Schneider Hamiltonians.
- Hierarchy flows correspond to traces of powers of the Lax matrix.
- Extension of particle-model correspondence to entire Toda hierarchy.

## 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.