On the role of quadratic oscillations in nonlinear Schrödinger equations II. The $L^2$-critical case
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- by Rémi Carles and Sahbi Keraani PDF
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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.References
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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
- ORCID: 0000-0002-8866-587X
- 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
- 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.
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 33-62
- MSC (2000): Primary 35Q55; Secondary 35B40, 35B05
- DOI: https://doi.org/10.1090/S0002-9947-06-03955-9
- MathSciNet review: 2247881