Q-conditional symmetries of the first type and exact solutions of nonlinear reaction-diffusion systems

Abstract

Two classes of two-component nonlinear reaction-diffusion systems are studied in order to find Q-conditional symmetries of the first type (a special subset of nonclassical symmetries), to construct exact solutions, and to show their applicability. The first class involves systems with constant coefficient of diffusivity, while the second contains systems with variable diffusivities only. The main theoretical results are given in the form of two theorems presenting exhaustive lists (up to the given sets of point transformations) of the reaction-diffusion systems belonging to the above classes and admitting Q-conditional symmetries of the first type. The reaction-diffusion systems obtained allow one to extract specific systems occurring in real-world models. A few examples are presented, including a modification of the classical prey–predator system with diffusivity and a system modelling the gravity-driven flow of thin films of viscous fluid. Exact solutions with attractive properties are found for these nonlinear systems and their possible biological and physical interpretations are presented.