Lie symmetries of multidimensional difference equations

TL;DR

This paper introduces a method to compute Lie point symmetries of scalar difference equations on 2D lattices, enabling symmetry-based reduction and generalization to multi-variable systems.

Contribution

It generalizes existing methods for ordinary difference equations to multidimensional cases, allowing symmetry analysis of complex difference systems.

Findings

Method calculates Lie symmetries for 2D lattice difference equations.

Symmetries can be used for solution reduction.

Approach extends to systems with multiple variables.

Abstract

A method is presented for calculating the Lie point symmetries of a scalar difference equation on a two-dimensional lattice. The symmetry transformations act on the equations and on the lattice. They take solutions into solutions and can be used to perform symmetry reduction. The method generalizes one presented in a recent publication for the case of ordinary difference equations. In turn, it can easily be generalized to difference systems involving an arbitrary number of dependent and independent variables.