Exact sharp-fronted travelling wave solutions of the Fisher-KPP equation

TL;DR

This paper derives exact sharp-fronted travelling wave solutions for the Fisher-KPP equation using Weierstrass elliptic functions, revealing new insights into wave behavior and their applications in biological models.

Contribution

It presents the first exact elliptic function solutions for the Fisher-KPP equation, including a novel interpretation of a normally disregarded solution as a sharp-fronted wave in a Fisher-Stefan model.

Findings

Exact solutions expressed via Weierstrass elliptic functions.

Reinterpretation of a blow-up solution as a receding wave.

Numerical simulations show solutions evolve to these exact waves.

Abstract

A family of travelling wave solutions to the Fisher-KPP equation with speeds $c=\pm 5/\sqrt{6}$ can be expressed exactly using Weierstrass elliptic functions. The well-known solution for $c=5/\sqrt{6}$, which decays to zero in the far-field, is exceptional in the sense that it can be written simply in terms of an exponential function. This solution has the property that the phase-plane trajectory is a heteroclinic orbit beginning at a saddle point and ends at the origin. For $c=-5/\sqrt{6}$, there is also a trajectory that begins at the saddle point, but this solution is normally disregarded as being unphysical as it blows up for finite $z$. We reinterpret this special trajectory as an exact sharp-fronted travelling solution to a \textit{Fisher-Stefan} type moving boundary problem, where the population is receding from, instead of advancing into, an empty space. By simulating the full moving boundary problem numerically, we demonstrate how time-dependent solutions evolve to this exact travelling solution for large time. The relevance of such receding travelling waves to mathematical models for cell migration and cell proliferation is also discussed.