# Exact solution and precise asymptotics of a Fisher-KPP type front

## TL;DR

This paper analyzes a modified Fisher-KPP equation with a saturation mechanism, deriving exact solutions and detailed asymptotics of the propagating front, including corrections and conditions on initial data.

## Contribution

It provides the first precise asymptotic analysis of a Fisher-KPP type front with a saturation mechanism, including correction terms and initial condition conditions.

## Key findings

- Recovered the Ebert and van Saarloos correction
- Derived an additional $rac{ ext{log } t}{t}$ term in the front asymptotics
- Established conditions on initial data for correction terms

## Abstract

The present work concerns a version of the Fisher-KPP equation where the nonlinear term is replaced by a saturation mechanism, yielding a free boundary problem with mixed conditions. Following an idea proposed in [BrunetDerrida.2015], we show that the Laplace transform of the initial condition is directly related to some functional of the front position $\mu_t$. We then obtain precise asymptotics of the front position by means of singularity analysis. In particular, we recover the so-called Ebert and van Saarloos correction [EbertvanSaarloos.2000], we obtain an additional term of order $\log t /t$ in this expansion, and we give precise conditions on the initial condition for those terms to be present.