Symmetry properties of conservation laws

TL;DR

This paper explores how symmetry properties influence conservation laws in partial differential equations, providing conditions for invariance and homogeneity, and clarifying the relationship between different conservation law formulas.

Contribution

It introduces simple conditions to determine when conservation laws are invariant or homogeneous under symmetries, and relates Ibragimov's conservation law formula to standard symmetry actions.

Findings

Conditions for invariance and homogeneity of conservation laws

Equivalence of Ibragimov's formula to standard symmetry actions

Application of conservation law multipliers to symmetry analysis

Abstract

Symmetry properties of conservation laws of partial differential equations are developed by using the general method of conservation law multipliers. As main results, simple conditions are given for characterizing when a conservation law and its associated conserved quantity are invariant (and, more generally, homogeneous) under the action of a symmetry. These results are used to show that a recent conservation law formula (due to Ibragimov) is equivalent to a standard formula for the action of an infinitesimal symmetry on a conservation law multiplier.