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
[Solitons périodiques du système de Davey–Stewartson elliptique–elliptique focalisant]

Présenté par : 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 (VO)
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.

Reçu le : 2015-03-30
Accepté le : 2016-02-08
Publié le : 2016-03-25

DOI : 10.1016/j.crma.2016.02.005

Affiliations des auteurs :

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

@article{CRMATH_2016__354_5_486_0,
     author = {Mark D. Groves and Shu-Ming Sun and Erik Wahl\'en},
     title = {Periodic solitons for the elliptic{\textendash}elliptic focussing {Davey{\textendash}Stewartson} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {486--492},
     publisher = {Elsevier},
     volume = {354},
     number = {5},
     year = {2016},
     doi = {10.1016/j.crma.2016.02.005},
     language = {en},
}

TY  - JOUR
AU  - Mark D. Groves
AU  - Shu-Ming Sun
AU  - Erik Wahlén
TI  - Periodic solitons for the elliptic–elliptic focussing Davey–Stewartson equations
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 486
EP  - 492
VL  - 354
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crma.2016.02.005
LA  - en
ID  - CRMATH_2016__354_5_486_0
ER  -

%0 Journal Article
%A Mark D. Groves
%A Shu-Ming Sun
%A Erik Wahlén
%T Periodic solitons for the elliptic–elliptic focussing Davey–Stewartson equations
%J Comptes Rendus. Mathématique
%D 2016
%P 486-492
%V 354
%N 5
%I Elsevier
%R 10.1016/j.crma.2016.02.005
%G en
%F CRMATH_2016__354_5_486_0

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/

Bibliographie
Cité par

[1] M.J. Ablowitz; P.A. Clarkson Solitons, Nonlinear Evolution Equations and Inverse Scattering, Lond. Math. Soc. Lect. Note Ser., vol. 149, Cambridge University Press, Cambridge, UK, 1991

[2] M.J. Ablowitz; H. Segur On the evolution of packets of water waves, J. Fluid Mech., Volume 92 (1979), pp. 691-715

[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

[8] T. Kato Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1976

[9] Y. Watanabe; M. Tajiri Periodic soliton resonance: solutions to the Davey–Stewartson I equation, J. Phys. Soc. Jpn., Volume 67 (1998), pp. 705-708

Karim K. Ahmed; Hamdy Ahmed; Niveen M. Badra; Mohammad Mirzazadeh; Wafaa B. Rabie; Mostafa Eslami Diverse exact solutions to Davey-Stewartson model using modified extended mapping method, Nonlinear Analysis. Modelling and Control, Volume 29 (2024) no. 5, pp. 983-1002 | DOI:10.15388/namc.2024.29.36103 | Zbl:1547.35612
Dag Nilsson; Douglas Svensson Seth; Yuexun Wang Periodically modulated solitary waves of the CH-KP-I equation, ZAMP. Zeitschrift für angewandte Mathematik und Physik, Volume 75 (2024) no. 6, p. 11 (Id/No 201) | DOI:10.1007/s00033-024-02338-0 | Zbl:7930186
Kuldeep Singh; Michael McKerr; Ioannis Kourakis Dust ion-acoustic dromions in Saturn’s magnetosphere, Monthly Notices of the Royal Astronomical Society, Volume 521 (2023) no. 2, p. 2119 | DOI:10.1093/mnras/stad518
Kuldeep Singh; Michael McKerr; Ioannis Kourakis On the stability of 2D modulated electrostatic wavepackets in non-Maxwellian dusty plasma – application in Saturn’s magnetosphere, Monthly Notices of the Royal Astronomical Society, Volume 514 (2022) no. 1, p. 569 | DOI:10.1093/mnras/stac1315
Handan Borluk; Gabriele Bruell; Dag Nilsson Traveling waves and transverse instability for the fractional Kadomtsev-Petviashvili equation, Studies in Applied Mathematics, Volume 149 (2022) no. 1, pp. 95-123 | DOI:10.1111/sapm.12494 | Zbl:1529.35119
Mariana Haragus Transverse linear stability of line periodic traveling waves for water-wave models, Séminaire Laurent Schwartz. EDP et Applications, Volume 2018-2019 (2019), p. ex | DOI:10.5802/slsedp.133 | Zbl:1479.35076

Cité par 6 documents. Sources : Crossref, zbMATH

Commentaires - Politique