Nous étudions l'équation de Schrödinger cubique défocalisante en dimension d'espace au moins trois. Pour des données initiales de taille quelconque dans , , où est l'indice critique, nous considérons des perturbations dans , avec 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.
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.
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@article{CRMATH_2007__344_8_483_0, author = {Carles, R\'emi}, title = {On instability for the cubic nonlinear {Schr\"odinger} equation}, journal = {Comptes Rendus. Math\'ematique}, pages = {483--486}, publisher = {Elsevier}, volume = {344}, number = {8}, year = {2007}, doi = {10.1016/j.crma.2007.03.006}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2007.03.006/} }
TY - JOUR AU - Carles, Rémi TI - On instability for the cubic nonlinear Schrödinger equation JO - Comptes Rendus. Mathématique PY - 2007 SP - 483 EP - 486 VL - 344 IS - 8 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2007.03.006/ DO - 10.1016/j.crma.2007.03.006 LA - en ID - CRMATH_2007__344_8_483_0 ER -
%0 Journal Article %A Carles, Rémi %T On instability for the cubic nonlinear Schrödinger equation %J Comptes Rendus. Mathématique %D 2007 %P 483-486 %V 344 %N 8 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2007.03.006/ %R 10.1016/j.crma.2007.03.006 %G en %F CRMATH_2007__344_8_483_0
Carles, Rémi. On instability for the cubic nonlinear Schrödinger equation. Comptes Rendus. Mathématique, Tome 344 (2007) no. 8, pp. 483-486. doi : 10.1016/j.crma.2007.03.006. http://archive.numdam.org/articles/10.1016/j.crma.2007.03.006/
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