Multiphase Weakly Nonlinear Geometric Optics for Schrödinger Equations

Handle URI:
http://hdl.handle.net/10754/598908
Title:
Multiphase Weakly Nonlinear Geometric Optics for Schrödinger Equations
Authors:
Carles, Rémi; Dumas, Eric; Sparber, Christof
Abstract:
We describe and rigorously justify the nonlinear interaction of highly oscillatory waves in nonlinear Schrödinger equations, posed on Euclidean space or on the torus. Our scaling corresponds to a weakly nonlinear regime where the nonlinearity affects the leading order amplitude of the solution, but does not alter the rapid oscillations. We consider initial states which are superpositions of slowly modulated plane waves, and use the framework of Wiener algebras. A detailed analysis of the corresponding nonlinear wave mixing phenomena is given, including a geometric interpretation of the resonance structure for cubic nonlinearities. As an application, we recover and extend some instability results for the nonlinear Schrödinger equation on the torus in negative order Sobolev spaces. © 2010 Society for Industrial and Applied Mathematics.
Citation:
Carles R, Dumas E, Sparber C (2010) Multiphase Weakly Nonlinear Geometric Optics for Schrödinger Equations. SIAM J Math Anal 42: 489–518. Available: http://dx.doi.org/10.1137/090750871.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Mathematical Analysis
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
Jan-2010
DOI:
10.1137/090750871
Type:
Article
ISSN:
0036-1410; 1095-7154
Sponsors:
Received by the editors February 26, 2009; accepted for publication (in revised form) December 21, 2009; published electronically March 12, 2010. This work was supported by the French ANR project R.A.S. (ANR-08-JCJC-0124-01) and by award KUK-I1-007-43, provided by King Abdullah University of Science and Technology (KAUST).Department of Applied Mathematics and Theoretical Physics, CMS, Wilberforce Road, Cambridge CB3 0WA, England. This author's research was supported by a Royal Society University Research Fellowship (c.sparber@damtp.cam.ac.uk).
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Full metadata record

DC FieldValue Language
dc.contributor.authorCarles, Rémien
dc.contributor.authorDumas, Ericen
dc.contributor.authorSparber, Christofen
dc.date.accessioned2016-02-25T13:43:30Zen
dc.date.available2016-02-25T13:43:30Zen
dc.date.issued2010-01en
dc.identifier.citationCarles R, Dumas E, Sparber C (2010) Multiphase Weakly Nonlinear Geometric Optics for Schrödinger Equations. SIAM J Math Anal 42: 489–518. Available: http://dx.doi.org/10.1137/090750871.en
dc.identifier.issn0036-1410en
dc.identifier.issn1095-7154en
dc.identifier.doi10.1137/090750871en
dc.identifier.urihttp://hdl.handle.net/10754/598908en
dc.description.abstractWe describe and rigorously justify the nonlinear interaction of highly oscillatory waves in nonlinear Schrödinger equations, posed on Euclidean space or on the torus. Our scaling corresponds to a weakly nonlinear regime where the nonlinearity affects the leading order amplitude of the solution, but does not alter the rapid oscillations. We consider initial states which are superpositions of slowly modulated plane waves, and use the framework of Wiener algebras. A detailed analysis of the corresponding nonlinear wave mixing phenomena is given, including a geometric interpretation of the resonance structure for cubic nonlinearities. As an application, we recover and extend some instability results for the nonlinear Schrödinger equation on the torus in negative order Sobolev spaces. © 2010 Society for Industrial and Applied Mathematics.en
dc.description.sponsorshipReceived by the editors February 26, 2009; accepted for publication (in revised form) December 21, 2009; published electronically March 12, 2010. This work was supported by the French ANR project R.A.S. (ANR-08-JCJC-0124-01) and by award KUK-I1-007-43, provided by King Abdullah University of Science and Technology (KAUST).Department of Applied Mathematics and Theoretical Physics, CMS, Wilberforce Road, Cambridge CB3 0WA, England. This author's research was supported by a Royal Society University Research Fellowship (c.sparber@damtp.cam.ac.uk).en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectDinger equationen
dc.subjectHigh-frequency limiten
dc.subjectInstabilityen
dc.subjectNonlinear schröen
dc.subjectResonancesen
dc.titleMultiphase Weakly Nonlinear Geometric Optics for Schrödinger Equationsen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Mathematical Analysisen
dc.contributor.institutionUniversite Montpellier 2 Sciences et Techniques, Montpellier, Franceen
dc.contributor.institutionCNRS Centre National de la Recherche Scientifique, Paris, Franceen
dc.contributor.institutionInstitut Fourier Universite Joseph Fourier, Saint Martin d'Heres, Franceen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
kaust.grant.numberKUK-I1-007-43en
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