The exact travelling wave solutions of a KPP equation

TL;DR

The paper addresses exact traveling-wave fronts for the KPP reaction–diffusion equation with a nonlinear term $F(u)$ that includes the generalized Fisher form. It introduces a variable-exchange method by taking $p = u_\xi$ and treating $u$ as the independent variable, which transforms the problem into an Abel equation and yields closed-form front solutions. For $F(u)=u(1-u^n)(u^n+a)$, it obtains $u(\xi)=\big[\exp(n P \xi)+1\big]^{-1/n}$ with $P=1/\sqrt{n+1}$ and wave speed $c_0 = P + a/P$, recovering the Fisher case $u(\xi)=1/[\exp(\xi/\sqrt{6})+1]^2$ for $(n, a)=(1/2,1)$; the approach also extends to the generalized Burgers–Huxley equation and clarifies the proper asymptotic boundary conditions. The work provides a versatile, exact-solution framework for a broad class of nonlinear reaction–diffusion fronts and highlights how boundary-condition analysis interacts with the Abel-type reduction to yield physically meaningful fronts with translation-invariant positions.

Abstract

We obtain the exact analytical traveling wave solutions of the Kolmogorov-Petrovskii-Piskunov equation with the reaction term belonging to the class of functions, which includes that of the (generalized) Fisher equation, for the particular values of the waves speed. Additionally we obtain the the exact analytical traveling wave solutions of the generalized Burgers-Huxley equation.