Characteristics of conservation laws for difference equations

TL;DR

This paper extends the concept of conservation laws and characteristics from differential equations to difference equations, addressing unique challenges and establishing key theoretical results including a converse of Noether's Theorem.

Contribution

It introduces methods to transfer conservation law concepts to difference equations, proving the converse of Noether's Theorem and analyzing specific equations like the potential Lotka-Volterra.

Findings

Established the converse of Noether's Theorem for difference equations

Demonstrated the distinctness of conservation laws generated by the Gardner transformation

Derived all five-point conservation laws for the potential Lotka-Volterra equation

Abstract

Each conservation law of a given partial differential equation is determined (up to equivalence) by a function known as the characteristic. This function is used to find conservation laws, to prove equivalence between conservation laws, and to prove the converse of Noether's Theorem. Transferring these results to difference equations is nontrivial, largely because difference operators are not derivations and do not obey the chain rule for derivatives. We show how these problems may be resolved and illustrate various uses of the characteristic. In particular, we establish the converse of Noether's Theorem for difference equations, we show that the infinite family of conservation laws generated by Rasin and Schiff using the Gardner transformation are distinct (without taking a continuum limit), and we obtain all five-point conservation laws for the potential Lotka-Volterra equation.