Symmetries of differential-difference dynamical systems in a two-dimensional lattice

TL;DR

This paper classifies the Lie point symmetries of differential-difference equations on a two-dimensional triangular lattice, revealing upper bounds on symmetry group dimensions based on algebraic properties.

Contribution

It provides a systematic classification of symmetries for differential-difference equations on a 2D lattice, establishing maximum dimensions for abelian and nonsolvable symmetry algebras.

Findings

Maximum 12-dimensional symmetry group for abelian algebras

Maximum 13-dimensional symmetry group for nonsolvable algebras

Symmetry classification aids in understanding integrability and solution methods

Abstract

Classification of differential-difference equation of the form $\ddot{u}_{nm}=F_{nm}\big(t, \{u_{pq}\}|_{(p,q)\in \Gamma}\big)$ are considered according to their Lie point symmetry groups. The set $\Gamma$ represents the point $(n,m)$ and its six nearest neighbors in a two-dimensional triangular lattice. It is shown that the symmetry group can be at most 12-dimensional for abelian symmetry algebras and 13-dimensional for nonsolvable symmetry algebras.