Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation

Grégoire Nadin; Benoît Perthame; Min Tang

doi:10.1016/j.crma.2011.03.008

Partial Differential Equations

Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation

Presented by: the Editorial Board

Grégoire Nadin ¹; Benoît Perthame ^1,²; Min Tang ^1,²

¹ UPMC, CNRS UMR 7598, laboratoire Jacques-Louis-Lions, 4, place Jussieu, 75005 Paris, France
² INRIA Paris-Rocquencourt, équipe BANG, domaine de Voluceau, Rocquencourt, B.P. 105, 78153 Le Chesnay, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 553-557.

Abstract
Résumé

This Note investigates the properties of the traveling waves solutions of the nonlocal Fisher equation. The existence of such solutions has been proved recently in Berestycki et al. (2009) [3] but their asymptotic behavior was still unclear. We use here a new numerical approximation of these traveling waves which shows that some traveling waves connect the two homogeneous steady states 0 and 1, which is a striking fact since 0 is dynamically unstable and 1 is unstable in the sense of Turing.

Nous étudions dans cette Note les propriétés des solutions de type ondes progressives pour lʼéquation de Fisher non-locale. Lʼexistence de telles solutions a été prouvée récemment dans Berestycki et al. (2009) [3] mais leur comportement asymptotique était encore mal compris. Nous développons ici une nouvelle méthode dʼapproximation numérique montrant que certaines ondes progressives connectent les deux états dʼéquilibre homogènes 0 et 1, ce qui est surprenant puisque 0 est dynamiquement instable et 1 est instable au sens de Turing.

Received: 2010-11-12
Accepted: 2011-03-09
Published online: 2011-04-01

DOI: 10.1016/j.crma.2011.03.008

Author's affiliations:

Grégoire Nadin ¹; Benoît Perthame ^{1, 2}; Min Tang ^{1, 2}

¹ UPMC, CNRS UMR 7598, laboratoire Jacques-Louis-Lions, 4, place Jussieu, 75005 Paris, France
² INRIA Paris-Rocquencourt, équipe BANG, domaine de Voluceau, Rocquencourt, B.P. 105, 78153 Le Chesnay, France

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     author = {Gr\'egoire Nadin and Beno{\^\i}t Perthame and Min Tang},
     title = {Can a traveling wave connect two unstable states? {The} case of the nonlocal {Fisher} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {553--557},
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     year = {2011},
     doi = {10.1016/j.crma.2011.03.008},
     language = {en},
}

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Grégoire Nadin; Benoît Perthame; Min Tang. Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation. Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 553-557. doi : 10.1016/j.crma.2011.03.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.03.008/

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