Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



High frequency solutions of the nonlinear Schrödinger equation on surfaces

Authors: Nicolas Burq, Patrick Gérard and Nikolay Tzvetkov
Journal: Quart. Appl. Math. 68 (2010), 61-71
MSC (2000): Primary 35Q55; Secondary 35B30
DOI: https://doi.org/10.1090/S0033-569X-09-01178-1
Published electronically: October 20, 2009
MathSciNet review: 2598880
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Abstract: We address the problem of describing solutions of the nonlinear Schrödin- ger equation on a compact surface in the high frequency regime. In this context, we introduce a nonnegative threshold, depending on the geometry of the surface, which can be seen as a measurement of the nonlinear character of the equation, and we compute this number for the torus and for the sphere, as a consequence of earlier arguments. The last part is devoted to the study, on the sphere, of the critical regime associated to this threshold. We prove that the effective dynamics are described by a new evolution equation, the Resonant Hermite-Schrödinger equation.

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Nicolas Burq
Affiliation: Université Paris-Sud, Laboratoire de Mathématiques d’Orsay, CNRS, UMR 8628 et Institut Universitaire de France, Bâtiment 425, 91405 Orsay Cedex, France
Email: nicolas.burq@math.u-psud.fr

Patrick Gérard
Affiliation: Université Paris-Sud, Laboratoire de Mathématiques d’Orsay, CNRS, UMR 8628, Bâtiment 425, 91405 Orsay Cedex, France
Email: patrick.gerard@math.u-psud.fr

Nikolay Tzvetkov
Affiliation: Université de Lille 1, Laboratoire Paul Painlevé, CNRS, UMR 8524, 59655 Villeneuve d’Asq Cedex, France
Email: nikolay.tzvetkov@math.univ-lille1.fr

DOI: https://doi.org/10.1090/S0033-569X-09-01178-1
Keywords: Nonlinear Schr\"odinger equations, Strichartz estimates, Propagation of oscillations, Spherical harmonics, Nonhomogeneous media
Received by editor(s): December 31, 2008
Published electronically: October 20, 2009
Dedicated: Dedicated to Walter Strauss for his 70th birthday, with our friendship and admiration
Article copyright: © Copyright 2009 Brown University

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