Periodic solitons for the elliptic–elliptic focussing Davey–Stewartson equations

Mark D. Groves; Shu-Ming Sun; Erik Wahlén

doi:10.1016/j.crma.2016.02.005

Partial differential equations/Dynamical systems

Periodic solitons for the elliptic–elliptic focussing Davey–Stewartson equations

Presented by: the Editorial Board

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

Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 486-492.

Abstract
Résumé

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.

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.

Received: 2015-03-30
Accepted: 2016-02-08
Published online: 2016-03-25

DOI: 10.1016/j.crma.2016.02.005

Author's affiliations:

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},
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     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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