On the construction of conservation laws: a mixed approach

TL;DR

This paper introduces a combined method for systematically deriving local conservation laws of PDEs, capable of recovering known laws and identifying symmetry sources, with new laws found for specific equations.

Contribution

It presents a novel mixed approach that integrates two existing methods to compute conservation laws and identify their symmetry origins.

Findings

Successfully recovers all known conservation laws using the new method.

Identifies new local conservation laws for the Short Pulse and Fornberg Whitham equations.

Provides examples demonstrating the effectiveness of the combined approach.

Abstract

A new approach, combining the Ibragimov method and the one by Anco and Bluman, with the aim of algorithmically computing local conservation laws of partial differential equations, is discussed. Some examples of the application of the procedure are given. The method, of course, is able to recover all the conservation laws found by using the direct method; at the same time we can characterize which symmetry, if any, is responsible for the existence of a given conservation law. Some new local conservation laws for the Short Pulse equation and for the Fornberg Whitham equation are also determined.