Comptes Rendus
Partial differential equations
A simple criterion for transverse linear instability of nonlinear waves
Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 175-179.

We prove a simple criterion for transverse linear instability of nonlinear waves for partial differential equations in a spatial domain Ω×RRn×R. For stationary solutions depending upon xΩ only, the question of transverse (in)stability is concerned with their (in)stability with respect to perturbations depending upon (x,y)Ω×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.

Nous montrons un critère simple d'instabilité transverse linéaire d'ondes non linéaires d'équations aux dérivées partielles posées dans un domaine spatial Ω×RRn×R. Pour des solutions stationnaires dépendant de xΩ, la question de l'(in)stabilité transverse concerne leur (in)stabilité par rapport à des perturbations dépendantes de (x,y)Ω×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.

Received:
Accepted:
Published online:
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

Cyril Godey 1

1 Laboratoire de mathématiques de Besançon, Université de Franche-Comté, 16, route de Gray, 25030 Besançon cedex, France
@article{CRMATH_2016__354_2_175_0,
     author = {Cyril Godey},
     title = {A simple criterion for transverse linear instability of nonlinear waves},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {175--179},
     publisher = {Elsevier},
     volume = {354},
     number = {2},
     year = {2016},
     doi = {10.1016/j.crma.2015.10.017},
     language = {en},
}
TY  - JOUR
AU  - Cyril Godey
TI  - A simple criterion for transverse linear instability of nonlinear waves
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 175
EP  - 179
VL  - 354
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2015.10.017
LA  - en
ID  - CRMATH_2016__354_2_175_0
ER  - 
%0 Journal Article
%A Cyril Godey
%T A simple criterion for transverse linear instability of nonlinear waves
%J Comptes Rendus. Mathématique
%D 2016
%P 175-179
%V 354
%N 2
%I Elsevier
%R 10.1016/j.crma.2015.10.017
%G en
%F CRMATH_2016__354_2_175_0
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/

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

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

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

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

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

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

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

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

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

Cited by Sources:

Comments - Policy