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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the role of quadratic oscillations in nonlinear Schrödinger equations II. The $ L^2$-critical case


Authors: Rémi Carles and Sahbi Keraani
Journal: Trans. Amer. Math. Soc. 359 (2007), 33-62
MSC (2000): Primary 35Q55; Secondary 35B40, 35B05
Published electronically: April 11, 2006
MathSciNet review: 2247881
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Abstract: We consider a nonlinear semi-classical Schrödinger equation for which quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. The relevance of the nonlinearity was discussed by R. Carles, C. Fermanian-Kammerer and I. Gallagher for $ L^2$-supercritical power-like nonlinearities and more general initial data. The present results concern the $ L^2$-critical case, in space dimensions $ 1$ and $ 2$; we describe the set of non-linearizable data, which is larger, due to the scaling. As an application, we make precise a result by F. Merle and L. Vega concerning finite time blow up for the critical Schrödinger equation. The proof relies on linear and nonlinear profile decompositions.


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Additional Information

Rémi Carles
Affiliation: MAB, UMR CNRS 5466, Université Bordeaux 1, 351 cours de la Libération, 33 405 Talence cedex, France
Email: Remi.Carles@math.cnrs.fr

Sahbi Keraani
Affiliation: IRMAR, Université de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex, France
Email: sahbi.keraani@univ-rennes1.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03955-9
PII: S 0002-9947(06)03955-9
Received by editor(s): September 13, 2004
Published electronically: April 11, 2006
Additional Notes: This work was done while the first author was a guest at IRMAR (University of Rennes), and he would like to thank this institution for its hospitality. This work was partially supported by the ACI grant “Équation des ondes: oscillations, dispersion et contrôle”, and by the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.