Incarnations of XXX $\widehat{\frak{sl}_N}$ Bethe ansatz equations and integrable hierarchies

TL;DR

This paper explores the solution space of Bethe ansatz equations for the $\

Contribution

It introduces a family of commuting flows on the Bethe ansatz solution space and links them to integrable Ruijesenaars-Schneider systems.

Findings

Constructed a spectral transform for the rational Ruijesenaars-Schneider system.

Identified the flows of Bethe ansatz solutions with integrable hierarchies.

Established a connection between quantum integrable models and classical integrable systems.

Abstract

We consider the space of solutions of the Bethe ansatz equations of the $\widehat{\frak{sl}_N}$ XXX quantum integrable model, associated with the trivial representation of $\widehat{\frak{sl}_N}$. We construct a family of commuting flows on this space and identify the flows with the flows of coherent rational Ruijesenaars-Schneider systems. For that we develop in full generality the spectral transform for the rational Ruijesenaars-Schneider system.