New conditional symmetries and exact solutions of nonlinear reaction-diffusion-convection equations. II

In the first part of this paper [1], a complete description of Q-conditional symmetries for two classes of reaction-diffusion-convection equations with power diffusivities is derived. It was shown that all the known results for reaction-diffusion equations with power diffusivities follow as particular cases from those obtained in [1] but not vise versa. In the second part the symmetries obtained in are successfully applied for constructing exact solutions of the relevant equations. In the particular case, new exact solutions of nonlinear reaction-diffusion-convection (RDC) equations arising in application and their natural generalizations are found. 1.Introduction. This paper is a natural continuation of [1]. We apply step by step the Q-conditional symmetries obtained to construct exact solutions of the relevant nonlinear RDC equations, including the Murray equation with the fast and slow diffusions and the Fitzhugh-Nagumo equation with the fast diffusion and convection. It is well-known (see e.g. examples in [2, 3]) that new non-Lie ansätze don’t guarantee construction of new exact solutions. It turns out the relevant exact solutions may be also obtainable by the standard Lie machinery if the given equation admits a non-trivial Lie symmetry. Here we construct exact solutions using the Q-conditional symmetry operators and show that they are so called non-Lie solutions, i.e. cannot be obtained using Lie symmetry operators. As it follows from the proofs presented in section 3 [1], the Q-conditional symmetry operators have essentially simpler structure if one uses the substitution V = { U, m 6= −1, lnU, m = −1. (1) So we will firstly find exact solutions of equations of the form Vxx = V Vt − λVx + F (V ), Vxx = exp(V )Vt − λVx + F (V ), Vxx = V Vt − λV Vx + F (V ), Vxx = exp(V )Vt − λ exp(V )Vx + F (V ). and afterwards use (1) to obtain those of the RDC equations Ut = [U Ux]x + λU Ux + C(U), (2) Ut = [U Ux]x + λU Ux + C(U). (3)