Scalar reaction-diffusion equations: Conditional symmetry, exact solutions and applications

Abstract

All the main results on Q-conditional symmetry (nonclassical symmetry) of the general class of nonlinear reaction-diffusion-convection equations are summarized. Although some of them were published about 25 years ago, and the others were derived in the 2000s, it is the first attempt to present an extensive review of this matter. It is shown that several well-known equations arising in applications and their direct generalizations possess conditional symmetry. Notably, the Murray, Fitzhugh–Nagumo, and Huxley equations and their natural generalizations are identified. Moreover, several exact solutions (including travelling fronts) are constructed using the conditional symmetries obtained in order to find exact solutions with a biological interpretation.