Enhanced group analysis and conservation laws of variable coefficient reaction-diffusion equations with power nonlinearities

TL;DR

This paper performs a comprehensive symmetry and conservation law analysis of variable coefficient reaction-diffusion equations with power nonlinearities, providing classifications, transformations, and exact solutions.

Contribution

It introduces a complete group classification for these equations using various equivalence groups and constructs exact solutions via Lie symmetry methods.

Findings

Complete group classification achieved with respect to multiple equivalence groups

Exhaustive description of admissible transformations

Construction of exact solutions using Lie symmetry methods

Abstract

A class of variable coefficient (1+1)-dimensional nonlinear reaction-diffusion equations of the general form $f(x)u_t=(g(x)u^nu_x)_x+h(x)u^m$ is investigated. Different kinds of equivalence groups are constructed including ones with transformations which are nonlocal with respect to arbitrary elements. For the class under consideration the complete group classification is performed with respect to convenient equivalence groups (generalized extended and conditional ones) and with respect to the set of all point transformations. Usage of different equivalences and coefficient gauges plays the major role for simple and clear formulation of the final results. The corresponding set of admissible transformations is described exhaustively. Then, using the most direct method, we classify local conservation laws. Some exact solutions are constructed by the classical Lie method.