Miura-type transformations for lattice equations and Lie group actions associated with Darboux-Lax representations

TL;DR

This paper introduces a geometric method to construct Miura-type transformations for lattice equations using Darboux-Lax representations and Lie group actions, enabling the derivation of new integrable models.

Contribution

The paper presents a novel geometric approach to generate Miura-type transformations for lattice equations from Darboux-Lax representations, expanding the toolkit for integrable systems.

Findings

Constructed multiple Miura-type transformations for known lattices.

Derived new modified lattice equations from Lie group invariants.

Demonstrated the method on several classical lattice models.

Abstract

Miura-type transformations (MTs) are an essential tool in the theory of integrable nonlinear partial differential and difference equations. We present a geometric method to construct MTs for differential-difference (lattice) equations from Darboux-Lax representations (DLRs) of such equations. The method is applicable to parameter-dependent DLRs satisfying certain conditions. We construct MTs and modified lattice equations from invariants of some Lie group actions on manifolds associated with such DLRs. Using this construction, from a given suitable DLR one can obtain many MTs of different orders. The main idea behind this method is closely related to the results of Drinfeld and Sokolov on MTs for the partial differential KdV equation. Considered examples include the Volterra, Narita-Itoh-Bogoyavlensky, Toda, and Adler-Postnikov lattices. Some of the constructed MTs and modified lattice equations seem to be new.