On the relationship between the invariance and conservation laws of differential equations

TL;DR

This paper explores the connection between invariance and conservation laws in differential equations, unifying various approaches and extending previous results to broader symmetry contexts with illustrative examples.

Contribution

It demonstrates the relationship between symmetries and conservation laws, extending prior work to higher-order symmetries and unifying different methods for constructing conservation laws.

Findings

Unified the approaches of Anco & Bluman and Ibragimov.

Extended symmetry-conservation law relations to higher-order symmetries.

Provided multiple examples illustrating the theoretical results.

Abstract

In this paper, we highlight the complimentary nature of the results of Anco & Bluman and Ibragimov in the construction of conservation laws; that whilst the former establishes the role of multipliers, the latter presents a formal procedure to determine the flows. Secondly, we show that there is an underlying relationship between the symmetries and conservation laws in a general setting - extending the results of Kara & Mahomed. The results take apparently differently forms for point symmetry generators and higher-order symmetries. Similarities exist, to some extent, with a previously established result relating symmetries and multipliers of a differential equation. A number of examples are presented.