Comptes Rendus
no. 19-20
Partial Differential Equations
Existence of bound states for the coupled Schrödinger–KdV system with cubic nonlinearity
[Existence d'ondes solitaires pour le système couplé de Schrödinger–KdV avec non linearité cubique]

Présenté par : Gilles Lebeau

João-Paulo Dias1 ; Mário Figueira1 ; Filipe Oliveira2
1 CMAF/UL and FCUL, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal
2 Dep. Matemática, FCT/UNL, Monte da Caparica, Portugal
Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1079-1082.

Nous prouvons dans cette Note l'existence d'une famille infinie d'ondes solitaires régulières pour le système couplé de Schrödinger–Korteweg–de Vries, qui décroissent exponentiellement a l'infini.

We prove in this Note the existence of an infinite family of smooth positive bound states for the coupled Schrödinger–Korteweg–de Vries system, which decays exponentially at infinity.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.09.018
João-Paulo Dias 1 ; Mário Figueira 1 ; Filipe Oliveira 2

1 CMAF/UL and FCUL, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal
2 Dep. Matemática, FCT/UNL, Monte da Caparica, Portugal 
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     title = {Existence of bound states for the coupled {Schr\"odinger{\textendash}KdV} system with cubic nonlinearity},
     journal = {Comptes Rendus. Math\'ematique},
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João-Paulo Dias; Mário Figueira; Filipe Oliveira. Existence of bound states for the coupled Schrödinger–KdV system with cubic nonlinearity. Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1079-1082. doi : 10.1016/j.crma.2010.09.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.09.018/

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[2] A. Ambrosetti; E. Colorado Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris, Ser. I, Volume 342 (2006), pp. 453-458

[3] J. Angulo Pava; J.F. Montenegro Existence and evenness of solitary-wave solutions for an equation of short and long dispersive waves, Nonlinearity, Volume 13 (2000), pp. 1595-1611

[4] T. Cazenave An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matemáticos, vol. 22, Instituto de Matemática, UFRJ, Rio de Janeiro, 1989

[5] A.J. Corcho; F. Linares Well-posedness for the Schrödinger–Korteweg–de Vries system, Trans. Amer. Math. Soc., Volume 359 (2007), pp. 4089-4106

[6] P.L. Lions The concentration-compactness principle in the calculus of variations, Part 1, Ann. Inst. H. Poincaré, Volume 1 (1984), pp. 109-145

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