On Darboux integrable semi-discrete systems of exponential type

TL;DR

This paper investigates semi-discrete hyperbolic systems of exponential type, establishing that Darboux integrability in 2x2 cases occurs only when associated with Cartan matrices of semi-simple Lie algebras.

Contribution

It proves that Darboux integrability for 2x2 semi-discrete exponential systems is characterized by Cartan matrices of semi-simple Lie algebras.

Findings

Darboux integrability occurs only for systems linked to semi-simple Lie algebra Cartan matrices.

The paper provides a classification criterion for integrability based on algebraic structures.

Results connect integrability of discretized systems with Lie algebra theory.

Abstract

In the present paper we consider a discretization of hyperbolic systems of exponential type. We proved that, in the case of $2\times 2$ systems, the resulting semi-discrete system is Darboux integrable only if it corresponds to a Cartan matrix of a semi-simple Lie algebra.