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 [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.

v2: correction of some typos, a couple more references and some clarifications

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• Phenomenological picture of fluctuations in branching random walks
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• Free boundary problems in PDEs and particle systems
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