A simple criterion for transverse linear instability of nonlinear waves

Godey, Cyril

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

Godey, Cyril ¹

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

Comptes Rendus. Mathématique, Tome 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
Mot clés : Instabilité transverse, Dynamique spatiale, Équations de Davey–Stewartson, Rupture de dimension

Affiliations des auteurs :

Godey, Cyril ¹

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

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Godey, Cyril. A simple criterion for transverse linear instability of nonlinear waves. Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 175-179. doi : 10.1016/j.crma.2015.10.017. http://www.numdam.org/articles/10.1016/j.crma.2015.10.017/

Bibliographie
Cité par

[1] Ablowitz, M.J.; Segur, H. On the evolution of packets of water waves, J. Fluid Mech., Volume 92 (1979) no. 04, pp. 691-715

[2] Chow, S.N.; Hale, J.K. Methods of Bifurcation Theory, Springer, 1982

[3] Dias, F.; Hărăguş-Courcelle, M. On the transition from two-dimensional to three-dimensional water waves, Stud. Appl. Math., Volume 104 (2000) no. 2, pp. 91-127

[4] Groves, M.D.; Haragus, M.; Sun, S.M. Transverse instability of gravity-capillary line solitary water waves, C. R. Acad. Sci. Paris, Ser. I, Volume 333 (2001) no. 5, pp. 421-426

[5] Groves, M.D.; Haragus, M.; Sun, S.M. A dimension-breaking phenomenon in the theory of steady gravity-capillary water waves, Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci., Volume 360 (2002) no. 1799, pp. 2189-2243

[6] Groves, M.D.; Sun, S.M.; Wahlén, E. A dimension-breaking phenomenon for water waves with weak surface tension, Arch. Ration. Mech. Anal. (2015) (preprint in press) | DOI

[7] Haragus, M. Transverse spectral stability of small periodic traveling waves for the KP equation, Stud. Appl. Math., Volume 126 (2011) no. 2, pp. 157-185

[8] Haragus, M. Transverse dynamics of two-dimensional gravity-capillary periodic water waves, J. Dyn. Differ. Equ. (2015) (in press) | DOI

[9] Johnson, M.A.; Zumbrun, K. Transverse instability of periodic traveling waves in the generalized Kadomtsev–Petviashvili equation, SIAM J. Math. Anal., Volume 42 (2010) no. 6, pp. 2681-2702

[10] Rousset, F.; Tzvetkov, N. A simple criterion of transverse linear instability for solitary waves, Math. Res. Lett., Volume 17 (2010) no. 1, pp. 157-170

[11] Rousset, F.; Tzvetkov, N. Transverse instability of the line solitary water-waves, Invent. Math., Volume 184 (2011) no. 2, pp. 257-388

[12] Stefanov, A.; Stanislavova, M.; Hakkaev, Z. Transverse instability for periodic waves of KP-I and Schrödinger equations, Indiana Univ. Math. J., Volume 61 (2012) no. 2, pp. 461-492

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