Symmetry operators and generation of symmetry transformations of partial differential equations

TL;DR

This paper introduces an algebraic method using abstract operators to analyze symmetries of both scalar and matrix-valued partial differential equations, overcoming geometric approach limitations.

Contribution

It presents a novel algebraic framework for symmetry analysis of PDEs, including matrix-valued cases where traditional geometric methods struggle.

Findings

Algebraic approach effectively handles matrix-valued PDE symmetries.

Abstract operators enable standard differential-operator representations for scalar PDEs.

Several examples demonstrate the method's applicability.

Abstract

The study of symmetries of partial differential equations (PDEs) has been traditionally treated as a geometrical problem. Although geometrical methods have been proven effective with regard to finding infinitesimal symmetry transformations, they present certain conceptual difficulties in the case of matrix-valued PDEs; for example, the usual differential-operator representation of the symmetry-generating vector fields is not possible in this case. In this article an algebraic approach to the symmetry problem of PDEs - both scalar and matrix-valued - is described, based on abstract operators (characteristic derivatives) that admit a standard differential-operator representation in the case of scalar-valued PDEs. A number of examples are given.