Exact smooth and sharp-fronted travelling waves of reaction–diffusion equations with Weak Allee effects

Abstract

We provide new exact forms of smooth and sharp-fronted travelling wave solutions of the reaction–diffusion equation, ∂tu=R(u)+∂xD(u)∂xu, where the reaction term, R(u), employs a Weak Allee effect. The resulting ordinary differential equation system is solved by means of constructing a power series solution of the heteroclinic trajectory in phase plane space. For specific choices of wavespeeds and standard Weak Allee reaction terms, extending the celebrated exact travelling wave solution of the FKPP equation with wavespeed 5/6, we determine a family of exact travelling wave solutions that are smooth or sharp-fronted.