Negative flows and non-autonomous reductions of the Volterra lattice

TL;DR

This paper investigates reductions of the Volterra lattice linked to stationary equations of noncommutative symmetries, leading to new Painlevé-type difference equations with associated Lax pairs and Bäcklund transformations.

Contribution

It introduces a novel class of Painlevé-type difference equations derived from non-autonomous reductions of the Volterra lattice, including their Lax pairs and Bäcklund transformations.

Findings

Derivation of $(m+1)$-component difference equations of Painlevé type.

Construction of isomonodromic Lax pairs for these equations.

Development of Bäcklund transformations forming a $ obreakbZ^m$ lattice.

Abstract

We study reductions of the Volterra lattice corresponding to stationary equations for the additional, noncommutative subalgebra of symmetries. It is shown that, in the case of general position, such a reduction is equivalent to the stationary equation for a sum of the scaling symmetry and the negative flows, and is written as $(m+1)$-component difference equations of the Painlev\'e type generalizing the dP$_1$ and dP$_{34}$ equations. For these reductions, we present the isomonodromic Lax pairs and derive the B\"acklund transformations which form the $\mathbb{Z}^m$ lattice.