Symmetry invariance of conservation laws of partial differential equations

TL;DR

This paper investigates how symmetries affect conservation laws in partial differential equations, introducing concepts of symmetry-invariant and symmetry-homogeneous laws, with applications to several important physical equations.

Contribution

It provides a simple characterization of symmetry actions on conservation laws using conservation law multipliers, enhancing understanding of symmetry invariance in PDEs.

Findings

Defined symmetry-invariant and symmetry-homogeneous conservation laws.

Applied the theory to equations like Korteveg-de Vries and Navier-Stokes.

Illustrated the approach with physically relevant examples.

Abstract

A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and symmetry-homogeneous conservation laws. The main results are applied to several examples of physically interest, including the generalized Korteveg-de Vries equation, a non-Newtonian generalization of Burger's equation, the b-family of peakon equations, and the Navier-Stokes equations for compressible, viscous fluids in two dimensions.