Construction of potential systems for systems of PDEs with multi-dimensional spaces of conservation laws

TL;DR

This paper generalizes the construction of potential systems for PDEs with multi-dimensional conservation laws by considering linear combinations of conservation laws, revealing new inequivalent potential systems and avoiding redundancies.

Contribution

It introduces a method to construct potential systems using arbitrary linear combinations of conservation laws, expanding the scope beyond basis conservation laws.

Findings

Using linear combinations reveals more inequivalent potential systems.

Considering symmetry groups prevents redundant analysis of equivalent systems.

Examples demonstrate the effectiveness of the generalized approach.

Abstract

In this paper we consider generalization of procedure of construction of potential systems for systems of partial differential equations with multidimensional spaces of conservation laws. More precisely, for construction of potential systems in cases when dimension of the space of local conservation laws is greater than one, instead of using only basis conservation laws we use their arbitrary linear combinations being inequivalent with respect to equivalence group of the class of systems or symmetry group of the fixed system. It appears that the basis conservation laws can be equivalent with respect to groups of symmetry or equivalence transformations, or vice versa, the number of independent in this sense linear combinations of conservation laws can be grater than the dimension of the space of conservation laws. The first possibility leads to an unnecessary, often cumbersome, investigation of equivalent systems, the second one makes possible missing a great number of inequivalent potential systems. Examples of all these possibilities are given.