Conditional symmetries and exact solutions of nonlinear reaction-diffusion systems with non-constant diffusivities

TL;DR

This paper investigates Q-conditional symmetries in two-component reaction-diffusion systems with variable diffusivities, deriving systems with such symmetries, and constructs explicit exact solutions applicable to biological and physical models.

Contribution

It provides a comprehensive classification of reaction-diffusion systems admitting Q-conditional symmetries of the first type and constructs explicit solutions for these systems.

Findings

Derived reaction-diffusion systems with Q-conditional symmetries.

Constructed explicit exact solutions for nonlinear systems.

Applied solutions to biological and physical models.

Abstract

Q-conditional symmetries (nonclassical symmetries) for the general class of two-component reaction-diffusion systems with non-constant diffusivities are studied. Using the recently introduced notion of Q-conditional symmetries of the first type, an exhausted list of reaction-diffusion systems admitting such symmetry is derived. The results obtained for the reaction-diffusion systems are compared with those for the scalar reaction-diffusion equations. The symmetries found for reducing reaction-diffusion systems to two-dimensional dynamical systems, i.e., ODE systems, and finding exact solutions are applied. As result, multiparameter families of exact solutions in the explicit form for a nonlinear reaction-diffusion system with an arbitrary diffusivity are constructed. Finally, the application of the exact solutions for solving a biologically and physically motivated system is presented.