The Generalized Symmetry Method for Discrete Equations

TL;DR

This paper extends the generalized symmetry method to classify and test integrability of a broad class of discrete equations, including well-known hyperbolic equations, by deriving specific integrability conditions.

Contribution

It introduces new integrability conditions based on generalized symmetries for discrete equations, aiding their classification and testing.

Findings

Derived integrability conditions for discrete equations.

Applied conditions to test discretizations of hyperbolic equations.

Included equations from the Adler-Bobenko-Suris list.

Abstract

The generalized symmetry method is applied to a class of completely discrete equations including the Adler-Bobenko-Suris list. Assuming the existence of a generalized symmetry, we derive a few integrability conditions suitable for testing and classifying equations of this class. Those conditions are used at the end to test for integrability discretizations of some well-known hyperbolic equations.