Lie discrete symmetries of lattice equations

TL;DR

This paper extends methods for finding discrete symmetries from differential equations to difference and differential-difference equations, demonstrating their application on the discrete Painlevé I and Toda lattice equations.

Contribution

It introduces an extension of existing symmetry-finding methods to discrete and differential-difference equations, with practical examples.

Findings

Discrete symmetries of the discrete Painlevé I equation identified.

Discrete symmetries of the Toda lattice equation constructed.

Methods successfully applied to difference equations.

Abstract

We extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential-difference equations. As an example of the application of the methods, we construct the discrete symmetries of the discrete Painlev\'e I equation and of the Toda lattice equation.