Equivalence of conservation laws and equivalence of potential systems

TL;DR

This paper investigates the relationship between conservation laws and potential symmetries of differential equations, using equivalence transformations and examples like the Fokker-Planck and Burgers equations to classify their conservation laws and symmetries.

Contribution

It introduces a framework linking conservation laws and potential systems through equivalence relations, providing complete hierarchies and interpreting existing results.

Findings

Constructed hierarchies of conservation laws for Fokker-Planck and Burgers equations

Described potential symmetries using equivalence transformations

Unified interpretation of known conservation law results

Abstract

We study conservation laws and potential symmetries of (systems of) differential equations applying equivalence relations generated by point transformations between the equations. A Fokker-Planck equation and the Burgers equation are considered as examples. Using reducibility of them to the one-dimensional linear heat equation, we construct complete hierarchies of local and potential conservation laws for them and describe, in some sense, all their potential symmetries. Known results on the subject are interpreted in the proposed framework. This paper is an extended comment on the paper of J.-q. Mei and H.-q. Zhang [Internat. J. Theoret. Phys., 2006, in press].