Reaction-diffusion systems with constant diffusivities: conditional symmetries and form-preserving transformations

TL;DR

This paper classifies reaction-diffusion systems with constant diffusivities that admit Q-conditional symmetries, constructs form-preserving transformations, and derives exact solutions, including biologically relevant models.

Contribution

It provides an exhaustive classification of such systems with Q-conditional symmetries and constructs explicit exact solutions, advancing symmetry analysis in reaction-diffusion equations.

Findings

Classified all systems with Q-conditional symmetries of the first type.

Constructed form-preserving transformations for these systems.

Derived explicit exact solutions, including biologically motivated models.

Abstract

Q-conditional symmetries (nonclassical symmetries) for a general class of two-component reaction-diffusion systems with constant diffusivities are studied. Using the recently introduced notion of Q-conditional symmetries of the first type (R. Cherniha J. Phys. A: Math. Theor., 2010. vol. 43., 405207), an exhaustive list of reaction-diffusion systems admitting such symmetry is derived. The form-preserving transformations for this class of systems are constructed and it is shown that this list contains only non-equivalent systems. The obtained symmetries permit to reduce the reaction-diffusion systems under study to two-dimensional systems of ordinary differential equations and to find exact solutions. As a non-trivial example, multiparameter families of exact solutions are explicitly constructed for two nonlinear reaction-diffusion systems. A possible interpretation to a biologically motivated model is presented.