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Comptes Rendus Mathématique
Volume 344, n° 8
pages 483-486 (avril 2007)
Doi : 10.1016/j.crma.2007.03.006
Received : 28 February 2007 ;  accepted : 6 Mars 2007
On instability for the cubic nonlinear Schrödinger equation
Sur lʼinstabilité pour lʼéquation de Schrödinger avec non-linéarité cubique
 

Rémi Carles
CNRS and Université Montpellier 2, Mathématiques, CC 051, Place Eugène Bataillon, 34095 Montpellier cedex 5, France 

Abstract

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  , where  ,   the critical index, and perturbations in  , where   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. To cite this article: R. Carles, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

The full text of this article is available in PDF format.
Résumé

Nous étudions lʼéquation de Schrödinger cubique défocalisante en dimension dʼespace au moins trois. Pour des données initiales de taille quelconque dans  ,  , où   est lʼindice critique, nous considérons des perturbations dans  , avec   indépendant de s . On montre une instabilité dans des espaces de Sobolev dʼordre inférieur à s . La preuve repose sur une analyse de type optique géométrique en régime sur-critique. Pour citer cet article : R. Carles, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

The full text of this article is available in PDF format.


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