Rational Ruijsenaars-Schneider hierarchy and bispectral difference operators

TL;DR

This paper establishes a connection between tau-functions of the discrete KP hierarchy, the dynamics of zeros governed by Ruijsenaars-Schneider systems, and their role in the bispectral problem, revealing a new integrable structure.

Contribution

It demonstrates that tau-functions with zeros following Ruijsenaars-Schneider dynamics characterize solutions to a difference-differential bispectral problem.

Findings

Tau-functions correspond to zeros evolving via Ruijsenaars-Schneider systems.

The work links discrete KP hierarchy to bispectral difference operators.

Provides a new integrable hierarchy related to rational Ruijsenaars-Schneider models.

Abstract

We show that a monic polynomial in a discrete variable $n$, with coefficients depending on time variables $t_1, t_2,...$ is a $\tau$-function for the discrete Kadomtsev-Petviashvili hierarchy if and only if the motion of its zeros is governed by a hierarchy of Ruijsenaars-Schneider systems. These $\tau$-functions were considered in [12], where it was proved that they parametrize rank one solutions to a difference-differential version of the bispectral problem.