Singularity Structure, Symmetries and Integrability of Generalized Fisher Type Nonlinear Diffusion Equation

Abstract

In this letter, the integrability aspects of a generalized Fisher type equation with modified diffusion in (1+1) and (2+1) dimensions are studied by carrying out a singularity structure and symmetry analysis. It is shown that the Painlev\'e property exists only for a special choice of the parameter ($m=2$). A B\"acklund transformation is shown to give rise to the linearizing transformation to the linear heat equation for this case ($m=2$). A Lie symmetry analysis also picks out the same case ($m=2$) 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 transformation and physically important classes of solutions (including known ones in the literature), thereby giving a group theoretical understanding of the system. For nonintegrable cases in (2+1) dimensions, associated Lie symmetries and similarity reductions are indicated.