no. 7-8
Partial differential equations/Dynamical systems
Non-existence of small-amplitude doubly periodic waves for dispersive equations
[Non-existence d'onde de petites amplitudes doublement périodiques pour les équations dispersives]

Présenté par : Jean-Michel Bony

Ambrose, David M.1 ; Wright, J. Douglas1
1 Department of Mathematics, Drexel University, 3141 Chestnut St., Philadelphia, PA 19104, USA
Comptes Rendus. Mathématique, Tome 352 (2014) no. 7-8, pp. 597-602.

Nous exprimons le problème d'existence de solutions périodiques en temps et en espace d'opérateurs d'évolution sous forme de problèmes de points fixes, pour certaines périodes de temps. Nous prouvons que, si une certaine estimation pour l'integrale de Duhamel existe, alors les solutions périodiques en temps ne peuvent être arbitrairement petites. Cela donne des résultats analogues pour le cas de la diffusion d'ondes périodiques dans l'espace sur la droite réelle, puisque la diffusion implique la non-existence d'onde de petites amplitudes. De plus, nos résultats viennent compléter les méthodes des petits diviseurs (comme par exemple la méthode de Craig–Wayne–Bourgain) pour prouver l'existence de solutions périodiques en temps de petites amplitudes (pour des frequences dans un certain ensemble).

We formulate the question of the existence of spatially periodic, time-periodic solutions for evolution equations as a fixed point problem, for certain temporal periods. We prove that if a certain estimate applies for the Duhamel integral, then time-periodic solutions cannot be arbitrarily small. This provides a partial analogue in the spatially periodic case of scattering results for dispersive equations on the real line, as scattering implies the non-existence of small-amplitude traveling waves. Furthermore, it also complements small-divisor methods (e.g., the Craig–Wayne–Bourgain method) for proving the existence of small-amplitude time-periodic solutions (again, for frequencies in certain set).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.05.003
Ambrose, David M. 1 ; Wright, J. Douglas 1

1 Department of Mathematics, Drexel University, 3141 Chestnut St., Philadelphia, PA 19104, USA 
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Ambrose, David M.; Wright, J. Douglas. Non-existence of small-amplitude doubly periodic waves for dispersive equations. Comptes Rendus. Mathématique, Tome 352 (2014) no. 7-8, pp. 597-602. doi : 10.1016/j.crma.2014.05.003. http://www.numdam.org/articles/10.1016/j.crma.2014.05.003/

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