Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations

TL;DR

This paper introduces a method to find explicit upper and lower bounds for traveling wave solutions of reaction-diffusion equations, enabling precise analytical localization of solutions, exemplified on the Fisher-Kolmogorov equation.

Contribution

The paper presents a novel approach for deriving explicit bounds on heteroclinic orbits associated with traveling wave solutions in reaction-diffusion PDEs.

Findings

Rational bounds for Fisher-Kolmogorov solutions are obtained

Bounds enable high-accuracy analytical localization

Method applicable to other reaction-diffusion equations

Abstract

It is well-known that the existence of traveling wave solutions for reaction-diffusion partial differential equations can be proved by showing the existence of certain heteroclinic orbits for related autonomous planar differential equations. We introduce a method for finding explicit upper and lower bounds of these heteroclinic orbits. In particular, for the classical Fisher-Kolmogorov equation we give rational upper and lower bounds which allow to locate these solutions analytically and with very high accuracy.