Comptes Rendus
no. 5
Partial differential equations/Dynamical systems
Periodic solitons for the elliptic–elliptic focussing Davey–Stewartson equations
[Solitons périodiques du système de Davey–Stewartson elliptique–elliptique focalisant]

Présenté par : the Editorial Board

Mark D. Groves1, 2 ; Shu-Ming Sun3 ; Erik Wahlén4
1 FR 6.1 - Mathematik, Universität des Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany
2 Department of Mathematical Sciences, Loughborough University, Loughborough, Leics, LE11 3TU, UK
3 Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA
4 Department of Mathematics, Lund University, 22100 Lund, Sweden
Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 486-492.

Nous considérons les équations de Davey–Stewartson focalisantes dans le cas elliptique–elliptique, lorsqu'elles possèdent une solution unidimensionnelle de type soliton. En utilisant des méthodes de la théorie des systèmes dynamiques, nous montrons l'existence d'une famille de solutions bidimensionnelles qui ont le profil d'un soliton dans la direction spatiale longitudinale et sont périodiques dans la direction spatiale transverse. Nous montrons également que le soliton unidimensionnel est linéairement instable vis-à-vis des perturbations transverses.

We consider the elliptic–elliptic, focussing Davey–Stewartson equations, which have an explicit bright line soliton solution. The existence of a family of periodic solitons, which have the profile of the line soliton in the longitudinal spatial direction and are periodic in the transverse spatial direction, is established using dynamical systems arguments. We also show that the line soliton is linearly unstable with respect to perturbations in the transverse direction.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.02.005
Mark D. Groves 1, 2 ; Shu-Ming Sun 3 ; Erik Wahlén 4

1 FR 6.1 - Mathematik, Universität des Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany
2 Department of Mathematical Sciences, Loughborough University, Loughborough, Leics, LE11 3TU, UK
3 Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA
4 Department of Mathematics, Lund University, 22100 Lund, Sweden 
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     title = {Periodic solitons for the elliptic{\textendash}elliptic focussing {Davey{\textendash}Stewartson} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {486--492},
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Mark D. Groves; Shu-Ming Sun; Erik Wahlén. Periodic solitons for the elliptic–elliptic focussing Davey–Stewartson equations. Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 486-492. doi : 10.1016/j.crma.2016.02.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.02.005/

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[3] T. Arai; K. Takeuchi; M. Tajiri Note on periodic soliton resonance: solutions to the Davey–Stewartson II equation, J. Phys. Soc. Jpn., Volume 70 (2001), pp. 55-59

[4] P.G. Drazin Solitons, Lond. Math. Soc. Lect. Note Ser., vol. 85, Cambridge University Press, Cambridge, UK, 1983

[5] C. Godey A simple criterion for transverse linear instability of nonlinear waves, C. R. Acad. Sci. Paris, Ser. I, Volume 354 (2016), pp. 175-179

[6] M.D. Groves; S.M. Sun; E. Wahlén A dimension-breaking phenomenon for water waves with weak surface tension, Arch. Ration. Mech. Anal., Volume 220 (2016), pp. 747-807

[7] G. Iooss Gravity and capillary-gravity periodic travelling waves for two superposed fluid layers, one being of infinite depth, J. Math. Fluid Mech., Volume 1 (1999), pp. 24-63

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