The Invariance and Conservation Laws of fourth-order Difference Equations

TL;DR

This paper investigates the symmetry properties and conservation laws of fourth-order difference equations, providing methods to find their invariants and first integrals, which are crucial for understanding their solutions.

Contribution

It introduces a novel technique for identifying first integrals of fourth-order difference equations and explores the relationship between symmetries and conservation laws.

Findings

Identified the Lie symmetries of fourth-order difference equations.

Developed a method to find first integrals using symmetries.

Clarified the connection between symmetries, first integrals, and multipliers.

Abstract

We consider difference equations of order four and determine the one parameter Lie group of transformations (Lie symmetries) that leave them invariant. We introduce a technique for finding their first integrals and discuss the association between the symmetries and first integrals as well as the notion of multipliers (related to conservation laws) for difference equations.