[Une onde progressive peut-elle connecter deux équilibres instables ? Le cas de lʼéquation de Fisher non-locale]
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.
Accepté le :
Publié le :
Nadin, Grégoire 1 ; Perthame, Benoît 1, 2 ; Tang, Min 1, 2
@article{CRMATH_2011__349_9-10_553_0,
author = {Nadin, Gr\'egoire and Perthame, Beno{\^\i}t and Tang, Min},
title = {Can a traveling wave connect two unstable states? {The} case of the nonlocal {Fisher} equation},
journal = {Comptes Rendus. Math\'ematique},
pages = {553--557},
year = {2011},
publisher = {Elsevier},
volume = {349},
number = {9-10},
doi = {10.1016/j.crma.2011.03.008},
language = {en},
url = {https://www.numdam.org/articles/10.1016/j.crma.2011.03.008/}
}
TY - JOUR AU - Nadin, Grégoire AU - Perthame, Benoît AU - Tang, Min TI - Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation JO - Comptes Rendus. Mathématique PY - 2011 SP - 553 EP - 557 VL - 349 IS - 9-10 PB - Elsevier UR - https://www.numdam.org/articles/10.1016/j.crma.2011.03.008/ DO - 10.1016/j.crma.2011.03.008 LA - en ID - CRMATH_2011__349_9-10_553_0 ER -
%0 Journal Article %A Nadin, Grégoire %A Perthame, Benoît %A Tang, Min %T Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation %J Comptes Rendus. Mathématique %D 2011 %P 553-557 %V 349 %N 9-10 %I Elsevier %U https://www.numdam.org/articles/10.1016/j.crma.2011.03.008/ %R 10.1016/j.crma.2011.03.008 %G en %F CRMATH_2011__349_9-10_553_0
Nadin, Grégoire; Perthame, Benoît; Tang, Min. Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation. Comptes Rendus. Mathématique, Tome 349 (2011) no. 9-10, pp. 553-557. doi: 10.1016/j.crma.2011.03.008
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