Lie symmetry properties of nonlinear reaction-diffusion equations with gradient-dependent diffusivity

TL;DR

This paper classifies Lie symmetries of nonlinear reaction-diffusion equations with gradient-dependent diffusivity, revealing rich symmetry structures, conditions for linearizability, and exact solutions, especially in 1-D and 2-D cases.

Contribution

It provides a comprehensive Lie symmetry classification for these equations, identifying conditions for linearizability and wider invariance in specific cases.

Findings

Existence of an infinite-dimensional Lie algebra in 1-D case leading to linearization.

Wider Lie invariance for a specific power-law diffusivity in 2-D.

The diffusion equation without source term has limited Lie symmetries.

Abstract

Complete descriptions of the Lie symmetries of a class of nonlinear reaction-diffusion equations with gradient-dependent diffusivity in one and two space dimensions are obtained. A surprisingly rich set of Lie symmetry algebras depending on the form of diffusivity and source (sink) in the equations is derived. It is established that there exists a subclass in 1-D space admitting an infinite-dimensional Lie algebra of invariance so that it is linearisable. A special power-law diffusivity with a fixed exponent, which leads to wider Lie invariance of the equations in question in 2-D space, is also derived. However, it is shown that the diffusion equation without a source term (which often arises in applications and is sometimes called the Perona-Malik equation) possesses no rich variety of Lie symmetries depending on the form of gradient-dependent diffusivity. The results of the Lie symmetry classification for the reduction to lower dimensionality, and a search for exact solutions of the nonlinear 2-D equation with power-law diffusivity, also are included.