Enhanced group classification of nonlinear diffusion-reaction equations with gradient-dependent diffusion

TL;DR

This paper advances the classification of nonlinear diffusion-reaction equations with gradient-dependent diffusion by developing an optimized method for identifying equivalence groups, leading to new exact solutions.

Contribution

It introduces an optimized two-step method for group classification and constructs the first example of a finite-dimensional effective generalized equivalence group.

Findings

Developed an optimized method for classifying equations with gradient-dependent diffusivity.

Constructed the first example of a finite-dimensional effective generalized equivalence group.

Obtained exact solutions using Lie reduction and separation of variables.

Abstract

We carry out the enhanced group classification of a class of (1+1)-dimensional nonlinear diffusion-reaction equations with gradient-dependent diffusivity using the two-step version of the method of furcate splitting. For simultaneously finding the equivalence groups of an unnormalized class of differential equations and a collection of its subclasses, we suggest an optimized version of the direct method. The optimization includes the preliminary study of admissible transformations within the entire class and the successive splitting of the corresponding determining equations with respect to arbitrary elements and their derivatives depending on auxiliary constraints associated with each of required subclasses. In the course of applying the suggested technique to subclasses of the class under consideration, we construct, for the first time, a nontrivial example of finite-dimensional effective generalized equivalence group. Using the method of Lie reduction and the generalized separation of variables, exact solutions of some equations under consideration are found.