Lie Point Symmetries and Commuting Flows for Equations on Lattices

TL;DR

This paper reviews symmetry methods for difference equations on lattices, applying Lie point and generalized symmetries to reduce and analyze linear and nonlinear discrete equations like the heat equation and Toda lattice.

Contribution

It introduces a comprehensive approach to symmetry analysis for difference equations, including applications to integrable lattice models.

Findings

Symmetry reductions simplify discrete heat and Toda lattice equations.

Lie point and generalized symmetries are effectively applied to discrete systems.

The methods facilitate the analysis of integrability and solution structures.

Abstract

Different symmetry formalisms for difference equations on lattices are reviewed and applied to perform symmetry reduction for both linear and nonlinear partial difference equations. Both Lie point symmetries and generalized symmetries are considered and applied to the discrete heat equation and to the integrable discrete time Toda lattice.