Nonlinear reaction-diffusion systems with a non-constant diffusivity: conditional symmetries in no-go case

TL;DR

This paper explores Q-conditional symmetries in nonlinear reaction-diffusion systems with variable diffusivity, extending previous work to classify systems with such symmetries and derive explicit solutions.

Contribution

It provides an exhaustive classification of reaction-diffusion systems with Q-conditional symmetries of the first type and constructs explicit solutions for systems with power-law diffusivity.

Findings

Classified reaction-diffusion systems admitting Q-conditional symmetries.

Derived explicit multiparameter families of solutions.

Compared new results with previous classifications.

Abstract

Q-conditional symmetries (nonclassical symmetries) for a general class of two-component reaction-diffusion systems with non-constant diffusivities are studied. The work is a natural continuation of our paper (Cherniha and Davydovych, 2012) in order to extend the results on so-called no-go case. Using the notion of Q-conditional symmetries of the first type, an exhaustive list of reaction-diffusion systems admitting such symmetry is derived. The results obtained are compared with those derived earlier. The symmetries for reducing reaction-diffusion systems to two-dimensional dynamical systems (ODE systems) and finding exact solutions are applied. As result, multiparameter families of exact solutions in the explicit form for nonlinear reaction-diffusion systems with an arbitrary power-law diffusivity are constructed and their properties for possible applicability are established.