A comparison of four approaches to the calculation of conservation laws

TL;DR

This paper compares four computational methods for calculating conservation laws of differential equations, highlighting their capabilities, restrictions, and providing examples including well-known equations.

Contribution

It introduces a comparative analysis of four approaches to compute conservation laws, including new examples with non-polynomial and variable-dependent laws.

Findings

Different methods have varying restrictions and capabilities.

New conservation laws with complex features are identified.

The approaches are demonstrated on classical differential equations.

Abstract

The paper compares computational aspects of four approaches to compute conservation laws of single differential equations (DEs) or systems of them, ODEs and PDEs. The only restriction, required by two of the four corresponding computer algebra programs, is that each DE has to be solvable for a leading derivative. Extra constraints for the conservation laws can be specified. Examples include new conservation laws that are non-polynomial in the functions, that have an explicit variable dependence and families of conservation laws involving arbitrary functions. The following equations are investigated in examples: Ito, Liouville, Burgers, Kadomtsev-Petviashvili, Karney-Sen-Chu-Verheest, Boussinesq, Tzetzeica, Benney.