Symmetries and Integrability of Generalized Fisher Type Nonlinear Diffusion Equations

Abstract:
Generalized Fisher type nonlinear differentical equations occur in many physical and biological problems. Different propagating wave solutions and spatio-temporal patterns arise in them. In this paper we investigate this system both in (1+1) and (2+1) dimensions from singularity structure and Lie symmetry points of view. In particular we show that the Painleve property exists for a particular parametric value in the problem. A Backlund transformation is shown to give rise to the linearizing transformation to the linear heat equation for this case. A Lie symmetry analysis also picks out the same case as the only system among this class as having nontrivial infinite dimensional Lie algebra of symmetries and that the similarity variables and similarity reductions lead in a natural way to the linearizing tranformation and physically important classes of solutions, thereby giving a group theoretical understanding of the system. For nonintegrable cases in (2+1) dimensions, associated Lie symmetries and similarity reductions are discussed.