Negative flows and non-autonomous reductions of the Volterra 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 Painlevé type generalizing the dP${}_1$ and dP${}_{34}$ equations. For these reductions, we present the isomonodromic Lax pairs and derive the Bäcklund transformations which form the ${\mathbb{Z}}^m$ lattice.

arXiv comment17 pages. This article is part of an OCNMP Special Issue in Memory of Professor Decio Levi

References in corpus (8)

• Darboux transformations and Recursion operators for differential--difference equations
• Backlund transformations for many-body systems related to KdV
• Painlevé type reductions for the non-Abelian Volterra lattices
• Some exact solutions of the Volterra lattice
• Symmetries of the Continuous and Discrete Krichever-Novikov Equation
• Nonautonomous symmetries of the KdV equation and step-like solutions
• Negative Volterra flows
• Non-autonomous reductions of the KdV equation and multi-component analogs of the Painlevé equations P${}_{34}$ and P${}_3$

Cited by in corpus (2)

• Negative flows for several integrable models
• 3D-consistency of negative flows