Exact solution and precise asymptotics of a Fisher–KPP type front

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 Brunet and Derrida (2015 J. Stat. Phys. 161 801), we show that the Laplace transform of the initial condition is directly related to some functional of the front position . 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 (Ebert and van Saarloos 2000 Physica D 146 1), we obtain an additional term of order in this expansion, and we give precise conditions on the initial condition for those terms to be present.