Q-Conditional Symmetries of Reaction-Diffusion Systems

Abstract

A recently developed theoretical background for searching Q-conditional (nonclassical) symmetries of systems of evolution partial differential equations is presented. We generalize the standard definition of Q-conditional symmetry by introducing the notion of Q-conditional symmetry of the p-th type and show that different types of symmetry of a given system generate a hierarchy of conditional symmetry operators. It is shown that Q-conditional symmetry of the p-th type possesses some special properties, which distinguish them from the standard conditional symmetry. The general class of two-component nonlinear reaction-diffusion systems is examined in order to find the Q-conditional symmetry operators. The relevant systems of so-called determining equations are solved under additional restrictions. As a result, several reaction-diffusion systems possessing conditional symmetry are constructed. In particular, it is shown that the diffusive Lotka–Volterra system, the Belousov–Zhabotinskii system (with the correctly specified coefficients) and some of their generalizations admit Q-conditional symmetry.