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

Abstract

We consider the space of solutions of the Bethe ansatz equations of the $\widehat{{\mathfrak{sl}}_N}$ XXX quantum integrable model, associated with the trivial representation of $\widehat{{\mathfrak{sl}}_N}$. 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.

Latex, 57 pages