# 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.
Type de document :
Article dans une revue
Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2018, 51 (3), 〈10.1088/1751-8121/aa899f〉

https://hal.archives-ouvertes.fr/hal-01723857
Contributeur : Bernard Derrida <>
Soumis le : lundi 5 mars 2018 - 18:38:29
Dernière modification le : vendredi 4 janvier 2019 - 17:33:39

### Citation

Julien Berestycki, Éric Brunet, Bernard Derrida. Exact solution and precise asymptotics of a Fisher–KPP type front. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2018, 51 (3), 〈10.1088/1751-8121/aa899f〉. 〈hal-01723857〉

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