Characteristic Lie Algebras of Integrable Differential-Difference Equations in 3D

TL;DR

This paper develops an algebraic framework using characteristic Lie algebras to classify integrable differential-difference equations with mixed continuous and discrete variables, proposing a new integrability criterion based on Darboux reductions.

Contribution

It introduces a novel algebraic approach to classify such equations by linking Darboux integrability of reductions to finite-dimensional Lie algebras, providing a new criterion for integrability.

Findings

All known integrable equations in the class satisfy the proposed criterion.

The approach offers effective conditions for testing integrability.

The hypothesis is supported by existing examples.

Abstract

The purpose of this article is to develop an algebraic approach to the problem of integrable classification of differential-difference equations with one continuous and two discrete variables. As a classification criterion, we put forward the following hypothesis. Any integrable equation of the type under consideration admits an infinite sequence of finite-field Darboux-integrable reductions. The property of Darboux integrability of a finite-field system is formalized as finite-dimensionality condition of its characteristic Lie-Rinehart algebras. That allows one to derive effective integrability conditions in the form of differential equations on the right hand side of the equation under study. To test the hypothesis, we use known integrable equations from the class under consideration. In this article, we show that all known examples do have this property.