An exactly solvable travelling wave equation in the Fisher-KPP class

TL;DR

This paper derives an exact relation for a one-dimensional lattice Fisher-KPP type travelling wave equation, linking initial conditions to wave front positions over time, and recovers known properties and corrections through complex analysis.

Contribution

It introduces an exact inverse relation between initial conditions and wave front positions for a lattice Fisher-KPP equation, extending known results.

Findings

Derived an exact relation between initial profile and front position.

Recovered Bramson's shifts and Ebert-van Saarloos' corrections.

Demonstrated the use of complex analysis in solving travelling wave equations.

Abstract

For a simple one dimensional lattice version of a travelling wave equation, we obtain an exact relation between the initial condition and the position of the front at any later time. This exact relation takes the form of an inverse problem: given the times $t_n$ at which the travelling wave reaches the positions $n$, one can deduce the initial profile. We show, by means of complex analysis, that a number of known properties of travelling wave equations in the Fisher-KPP class can be recovered, in particular Bramson's shifts of the positions. We also recover and generalize Ebert-van Saarloos' corrections depending on the initial condition.