Conditional symmetries for systems of PDEs: new definitions and their application for reaction-diffusion systems

TL;DR

This paper introduces generalized definitions of Q-conditional symmetry for systems of PDEs, demonstrating their hierarchy and applicability to reaction-diffusion systems, and clarifying when different definitions coincide.

Contribution

It proposes new, generalized definitions of Q-conditional symmetry for PDE systems and explores their hierarchy and equivalence in reaction-diffusion models.

Findings

Different types of Q-conditional symmetry generate a hierarchy of operators.

The new definitions unify various existing notions of conditional symmetry.

Application to reaction-diffusion systems shows when definitions coincide.

Abstract

New definitions of $Q$-conditional symmetry for systems of PDEs are presented, which generalize the standard notation of non-classical (conditional) symmetry. It is shown that different types of $Q$-conditional symmetry of a system generate a hierarchy of conditional symmetry operators. A class of two-component nonlinear RD systems is examined to demonstrate applicability of the definitions proposed and it is shown when different definitions of $Q$-conditional symmetry lead to the same operators.