Comptes Rendus
no. 8
Partial Differential Equations
On instability for the cubic nonlinear Schrödinger equation

Presented by: Gilles Lebeau

Rémi Carles1
1 CNRS and Université Montpellier 2, Mathématiques, CC 051, Place Eugène Bataillon, 34095 Montpellier cedex 5, France
Comptes Rendus. Mathématique, Volume 344 (2007) no. 8, pp. 483-486.

We study the flow map associated to the cubic, defocusing, Schrödinger equation in space dimension at least three. We consider initial data of arbitrary size in Hs, where 0<s<sc, sc the critical index, and perturbations in Hσ, where σ<sc is independent of s. We show an instability mechanism in some Sobolev spaces of order smaller than s. The analysis relies on two features of super-critical geometric optics: the creation of oscillation, and the ghost effect.

Nous étudions l'équation de Schrödinger cubique défocalisante en dimension d'espace au moins trois. Pour des données initiales de taille quelconque dans Hs, 0<s<sc, où sc est l'indice critique, nous considérons des perturbations dans Hσ, avec σ<sc indépendant de s. On montre une instabilité dans des espaces de Sobolev d'ordre inférieur à s. La preuve repose sur une analyse de type optique géométrique en régime sur-critique.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.03.006

Rémi Carles 1

1 CNRS and Université Montpellier 2, Mathématiques, CC 051, Place Eugène Bataillon, 34095 Montpellier cedex 5, France 
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Rémi Carles. On instability for the cubic nonlinear Schrödinger equation. Comptes Rendus. Mathématique, Volume 344 (2007) no. 8, pp. 483-486. doi : 10.1016/j.crma.2007.03.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.03.006/

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