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 Painleve property exists only for a special choice of the parameter (m = 2). A Backlund 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 having a 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.