Closed-form travelling-wave solutions for the Fisher-KPP boundary value problem

Determine exact, closed-form travelling-wave solutions, if any, for the Fisher–KPP travelling-wave boundary value problem U''(z) + c U'(z) + U(z)(1 − U(z)) = 0 on (−∞, ∞) with limits U(−∞) = 1 and U(∞) = 0, beyond the known special case c^2 = 25/6 where an explicit solution exists.

Background

The paper studies travelling-wave solutions of the Fisher–KPP equation by transforming the PDE to the ODE U'' + c U' + U(1−U) = 0 with boundary conditions U(−∞)=1 and U(∞)=0. This boundary value problem characterizes the travelling-wave profiles associated with the Fisher–KPP model.

Aside from a special speed where the equation has the Painlevé property (c^2 = 25/6), general closed-form solutions are not known. The authors highlight this limitation before proceeding with phase-plane and perturbation analyses.