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

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 zero in far-field, is exceptional sense that it written simply terms an exponential function. This has property phase-plane trajectory a heteroclinic orbit beginning at saddle point and ends origin. For $c=-5/\sqrt{6}$, there also begins point, but this normally disregarded as being unphysical blows up finite $z$. We reinterpret special exact sharp-fronted \textit{Fisher-Stefan} type moving boundary problem, where population receding from, instead advancing into, empty space. By simulating full problem numerically, we demonstrate how time-dependent evolve large time. relevance such waves mathematical models cell migration proliferation discussed.