A simple criterion for transverse linear instability of nonlinear waves

Cyril Godey

doi:10.1016/j.crma.2015.10.017

Partial differential equations

A simple criterion for transverse linear instability of nonlinear waves
[Un critère simple d'instabilité transverse d'ondes non linéaires]

Présenté par : Gérard Iooss

Cyril Godey ¹

¹ Laboratoire de mathématiques de Besançon, Université de Franche-Comté, 16, route de Gray, 25030 Besançon cedex, France

Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 175-179.

Résumé
Abstract

Nous montrons un critère simple d'instabilité transverse linéaire d'ondes non linéaires d'équations aux dérivées partielles posées dans un domaine spatial $Ω \times R \subset R^{n} \times R$ . Pour des solutions stationnaires dépendant de $x \in Ω$ , la question de l'(in)stabilité transverse concerne leur (in)stabilité par rapport à des perturbations dépendantes de $(x, y) \in Ω \times R$ . En utilisant une formulation de l'équation comme système dynamique par rapport à la direction transverse y, nous donnons des conditions suffisantes d'instabilité transverse linéaire. Nous appliquons ce résultat aux équations de Davey–Stewartson, qui apparaissent comme équations de modulation dans le problème des vagues en trois dimensions.

We prove a simple criterion for transverse linear instability of nonlinear waves for partial differential equations in a spatial domain $Ω \times R \subset R^{n} \times R$ . For stationary solutions depending upon $x \in Ω$ only, the question of transverse (in)stability is concerned with their (in)stability with respect to perturbations depending upon $(x, y) \in Ω \times R$ . Starting with a formulation of the PDE as a dynamical system in the transverse direction y, we give sufficient conditions for transverse linear instability. We apply the general result to the Davey–Stewartson equations, which arise as modulation equations for three-dimensional water waves.

Reçu le : 2014-12-09
Accepté le : 2015-10-21
Publié le : 2015-12-07

DOI : 10.1016/j.crma.2015.10.017

Keywords: Transverse instability, Spatial dynamics, Davey–Stewartson equations, Dimension-breaking
Mots-clés : Instabilité transverse, Dynamique spatiale, Équations de Davey–Stewartson, Rupture de dimension

Affiliations des auteurs :

Cyril Godey ¹

¹ Laboratoire de mathématiques de Besançon, Université de Franche-Comté, 16, route de Gray, 25030 Besançon cedex, France

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     title = {A simple criterion for transverse linear instability of nonlinear waves},
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     pages = {175--179},
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     number = {2},
     year = {2016},
     doi = {10.1016/j.crma.2015.10.017},
     language = {en},
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Cyril Godey. A simple criterion for transverse linear instability of nonlinear waves. Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 175-179. doi : 10.1016/j.crma.2015.10.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.10.017/

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