Nonlinear Schrödinger systems on the three-dimensional torus

dc.contributor.advisorChen, Thomas (Ph. D. in mechanical engineering and Ph. D. in mathematical physics)
dc.contributor.committeeMemberPavlović, Nataša
dc.contributor.committeeMemberGualdani, Maria P
dc.contributor.committeeMemberCzubak, Magdalena
dc.creatorUrban, Amie Bowles
dc.creator.orcid0000-0003-1681-4132
dc.date.accessioned2022-07-19T23:18:34Z
dc.date.available2022-07-19T23:18:34Z
dc.date.created2021-08
dc.date.issued2021-08-06
dc.date.submittedAugust 2021
dc.date.updated2022-07-19T23:18:35Z
dc.description.abstractIn this dissertation, we study cubic and quintic nonlinear Schrödinger systems on 3-dimensional tori, with initial data in an adapted Hilbert space H [superscript s over lambda underscore], and all of our results hold on rational and irrational rectangular, flat tori. In the cubic and quintic case, we prove local well-posedness for both focusing and defocusing systems. We show that local solutions of the defocusing cubic system with initial data in H [superscript 1 over lambda underscore] can be extended for all time. Additionally, we prove that global well-posedness holds in the quintic system, focusing or defocusing, for initial data with sufficiently small H [superscript 1 over lambda underscore] norm. Finally, we use the energy-Casimir method to prove the existence and uniqueness, and nonlinear stability of a class of stationary states of the defocusing cubic and quintic nonlinear Schrödinger systems.
dc.description.departmentMathematics
dc.format.mimetypeapplication/pdf
dc.identifier.urihttps://hdl.handle.net/2152/114977
dc.identifier.urihttp://dx.doi.org/10.26153/tsw/41880
dc.language.isoen
dc.subjectNLS systems
dc.subjectNonlinear Schrodinger systems
dc.subjectTorus
dc.subjectStationary states
dc.subjectNonlinear stability
dc.subjectEnergy-Casimir method
dc.titleNonlinear Schrödinger systems on the three-dimensional torus
dc.typeThesis
dc.type.materialtext
thesis.degree.departmentMathematics
thesis.degree.disciplineMathematics
thesis.degree.grantorThe University of Texas at Austin
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy
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