An algebraic method of classification of S-integrable discrete models

TL;DR

This paper introduces an algebraic classification method for S-integrable discrete models on quad-graphs, using Lie rings and the growth rate of commutator-generated spaces to distinguish integrable from non-integrable equations.

Contribution

It proposes a novel algebraic classification scheme for integrable discrete models based on Lie ring structures and growth rate analysis.

Findings

Integrable equations exhibit slower growth in Lie ring commutator spaces.

The classification scheme effectively distinguishes integrable from non-integrable models.

Examples demonstrate the applicability of the algebraic growth criterion.

Abstract

A method of classification of integrable equations on quad-graphs is discussed based on algebraic ideas. We assign a Lie ring to the equation and study the function describing the dimensions of linear spaces spanned by multiple commutators of the ring generators. For the generic case this function grows exponentially. Examples show that for integrable equations it grows slower. We propose a classification scheme based on this observation.