Miura transformations for Toda--type integrable systems, with applications to the problem of integrable discretizations

TL;DR

This paper investigates Miura transformations for various Toda-type integrable systems, exploring their properties and applications in discretization, providing new modifications and analyzing their Poisson and permutability features.

Contribution

It introduces multiple new modifications of Miura transformations for Toda and Volterra lattices, enhancing understanding of their role in integrable discretizations.

Findings

Three modifications for Toda lattice

Two modifications for Volterra lattice

Analysis of Poisson and permutability properties

Abstract

We study lattice Miura transformations for the Toda and Volterra lattices, relativistic Toda and Volterra lattices, and their modifications. In particular, we give three successive modifications for the Toda lattice, two for the Volterra lattice and for the relativistic Toda lattice, and one for the relativistic Volterra lattice. We discuss Poisson properties of the Miura transformations, their permutability properties, and their role as localizing changes of variables in the theory of integrable discretizations.