On perturbations retaining conservation laws of difftrential equations

TL;DR

This paper introduces a method to identify perturbations of differential equations that preserve specific conservation laws, with applications to equations like KdV-Burgers and magnetodynamics, revealing new solution properties.

Contribution

The paper presents a novel procedure for finding perturbations that retain particular conservation laws in differential equations.

Findings

Identified perturbations that preserve energy and momentum.

Demonstrated the method on KdV-Burgers and magnetodynamics equations.

Revealed unique properties of solutions under such perturbations.

Abstract

The paper deals with perturbations of the equation that have a number of conservation laws. When a small term is added to the equation its conserved quantities usually decay at individual rates, a phenomenon known as a selective decay. These rates are described by the simple law using the conservation laws' generating functions and the added term. Yet some perturbation may retain a specific quantity(s), such as energy, momentum and other physically important characteristics of solutions. We introduce a procedure for finding such perturbations and demonstrate it by examples including the KdV-Burgers equation and a system from magnetodynamics. Some interesting properties of solutions of such perturbed equations are revealed and discussed.