Q-Conditional Symmetries and Exact Solutions of Nonlinear Reaction–Diffusion Systems

Oleksii Pliukhin

doi:10.3390/sym7041841

Abstract

A wide range of reaction–diffusion systems with constant diffusivities that are invariant under Q-conditional operators is found. Using the symmetries obtained, the reductions of the corresponding systems to the systems of ODEs are conducted in order to find exact solutions. In particular, the solutions of some reaction–diffusion systems of the Lotka–Volterra type in an explicit form and satisfying Dirichlet boundary conditions are obtained. An biological interpretation is presented in order to show that two different types of interaction between biological species can be described.

Highlights

IntroductionIn this paper he proposed the Turing hypothesis of pattern formation
In 1952, Alan Turing published his prominent paper [1]
There are many papers devoted to the construction of Q-conditional symmetries for this equation [14,15,16,17,18,19,20,21], starting from the pioneering work in [22]

Summary

Introduction

In this paper he proposed the Turing hypothesis of pattern formation He used reaction–diffusion equations of the form λ1 ut = (D1 (u)ux )x + F (u, v), λ2 vt = (D2 (v)vx )x + G(u, v). There are many papers devoted to the construction of Q-conditional symmetries for this equation [14,15,16,17,18,19,20,21], starting from the pioneering work in [22]. There is a non-trivial generalization of these results for the case of the reaction–diffusion–convection equation ([21] and papers cited therein).

Main Result

Ansätze and Exact Solutions of the Reaction–Diffusion System

Conclusions