New Q-conditional symmetries and exact solutions of some reaction–diffusion–convection equations arising in mathematical biology

Abstract

A theorem giving a complete description of Q-conditional symmetries of a class of nonlinear reaction–diffusion–convection equations is proved. Furthermore the Q-conditional symmetries obtained and the method of additional generating conditions are applied for finding exact solutions of the generalized Fisher, Fitzhugh–Nagumo and Kolmogorov–Petrovskii–Piskunov equations. The symmetries and solutions constructed are compared with those obtained by other authors. In particular, it was established that the known travelling wave solutions of these equations are particular cases of more general (non-Lie) solutions. The relation between Q-conditional symmetries and generalized conditional symmetries is also shown.