An algebraic criterion of the Darboux integrability of differential-difference equations and systems

TL;DR

This paper establishes an algebraic criterion for Darboux integrability of hyperbolic differential-difference systems, linking integrability to the finite-dimensionality of characteristic Lie-Rinehart algebras.

Contribution

It introduces a new algebraic condition based on characteristic algebras that characterizes Darboux integrability of differential-difference equations.

Findings

Darboux integrability is equivalent to finite-dimensional characteristic algebras in both directions.

The concept of a complete set of characteristic integrals is clarified.

A connection between integrals and Lie-Rinehart algebras is established.

Abstract

The article investigates systems of differential-difference equations of hyperbolic type, integrable in sense of Darboux. The concept of a complete set of independent characteristic integrals underlying Darboux integrability is discussed. A close connection is found between integrals and characteristic Lie-Rinehart algebras of the system. It is proved that a system of equations is Darboux integrable if and only if its characteristic algebras in both directions are finite-dimensional.