Comptes Rendus
no. 8
Partial Differential Equations
On instability for the cubic nonlinear Schrödinger equation
Gilles Lebeau
Rémi Carles1
1CNRS 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 
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
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