Lie symmetries of nonlinear multidimensional reaction-diffusion systems: I

Abstract

We present a complete description of the classical (Lie) symmetries of a coupled system of partial differential equations comprising a pair of semilinear reaction–diffusion equations with constant diffusivities and arbitrary nonlinearities in the reaction terms, in any number of spatial dimensions. Part I (Cherniha R M and King J R 2000 J. Phys. A: Math. Gen. 33 267–82, J. Phys. A: Math. Gen. 33 7839–41) addressed the case of unequal diffusivities; here we complete the analysis by treating the case of equal diffusivities in which the symmetry structure is richer still. Such models arise in the description of numerous physical, chemical and biological systems and we also indicate the possible application in such contexts of some of the specific cases arising from the group classification. Specifically, a variety of Lie's ansatze and exact solutions of the so-called λ − ω reaction–diffusion systems, of a type that arises in mathematical biology, are constructed.