Semiclassical Asymptotics of the Focusing Nonlinear Schrodinger Equation for Square Barrier Initial Data

Persistent Link:
http://hdl.handle.net/10150/193553
Title:
Semiclassical Asymptotics of the Focusing Nonlinear Schrodinger Equation for Square Barrier Initial Data
Author:
Jenkins, Robert M.
Issue Date:
2009
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
The small dispersion limit of the focusing nonlinear Schroödinger equation (fNLS) exhibits a rich structure with rapid oscillations at microscopic scales. Due to the non self-adjoint scattering problem associated to fNLS, very few rigorous results exist in the semiclassical limit. The asymptotics for reectionless WKB-like initial data was worked out in [KMM03] and for the family q(x, 0) = sech^(1+(i/∈)μ in [TVZ04]. In both studies the authors observed sharp breaking curves in the space-time separating regions with disparate asymptotic behaviors. In this paper we consider another exactly solvable family of initial data, specifically the family of centered square pulses, q(x; 0) = qx[-L,L] for real amplitudes q. Using Riemann- Hilbert techniques we obtain rigorous pointwise asymptotics for the semiclassical limit of fNLS globally in space and up to an O(1) maximal time. In particular, we find breaking curves emerging in accord with the previous studies. Finally, we show that the discontinuities in our initial data regularize by the immediate generation of genus one oscillations emitted into the support of the initial data. This is the first case in which the genus structure of the semiclassical asymptotics for fNLS have been calculated for non-analytic initial data.
Type:
text; Electronic Dissertation
Keywords:
NLS; nonlinear waves; Schrodinger; square barrier
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Applied Mathematics; Graduate College
Degree Grantor:
University of Arizona
Advisor:
McLaughlin, Ken
Committee Chair:
McLaughlin, Ken

Full metadata record

DC FieldValue Language
dc.language.isoENen_US
dc.titleSemiclassical Asymptotics of the Focusing Nonlinear Schrodinger Equation for Square Barrier Initial Dataen_US
dc.creatorJenkins, Robert M.en_US
dc.contributor.authorJenkins, Robert M.en_US
dc.date.issued2009en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractThe small dispersion limit of the focusing nonlinear Schroödinger equation (fNLS) exhibits a rich structure with rapid oscillations at microscopic scales. Due to the non self-adjoint scattering problem associated to fNLS, very few rigorous results exist in the semiclassical limit. The asymptotics for reectionless WKB-like initial data was worked out in [KMM03] and for the family q(x, 0) = sech^(1+(i/∈)μ in [TVZ04]. In both studies the authors observed sharp breaking curves in the space-time separating regions with disparate asymptotic behaviors. In this paper we consider another exactly solvable family of initial data, specifically the family of centered square pulses, q(x; 0) = qx[-L,L] for real amplitudes q. Using Riemann- Hilbert techniques we obtain rigorous pointwise asymptotics for the semiclassical limit of fNLS globally in space and up to an O(1) maximal time. In particular, we find breaking curves emerging in accord with the previous studies. Finally, we show that the discontinuities in our initial data regularize by the immediate generation of genus one oscillations emitted into the support of the initial data. This is the first case in which the genus structure of the semiclassical asymptotics for fNLS have been calculated for non-analytic initial data.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectNLSen_US
dc.subjectnonlinear wavesen_US
dc.subjectSchrodingeren_US
dc.subjectsquare barrieren_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorMcLaughlin, Kenen_US
dc.contributor.chairMcLaughlin, Kenen_US
dc.contributor.committeememberErcolani, Nicken_US
dc.contributor.committeememberFlaschka, Hermannen_US
dc.identifier.proquest10562en_US
dc.identifier.oclc659752301en_US
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