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dc.contributor.advisorGigliola Staffilani.en_US
dc.contributor.authorKurianski, Kristin Marie-Dettmers.en_US
dc.contributor.otherMassachusetts Institute of Technology. Department of Mathematics.en_US
dc.date.accessioned2019-09-16T22:34:16Z
dc.date.available2019-09-16T22:34:16Z
dc.date.copyright2019en_US
dc.date.issued2019en_US
dc.identifier.urihttps://hdl.handle.net/1721.1/122173
dc.descriptionThesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019en_US
dc.descriptionCataloged from PDF version of thesis.en_US
dc.descriptionIncludes bibliographical references (pages 119-124).en_US
dc.description.abstractIn this thesis, we study wave-type phenomena both from a numerical point of view and a theoretical one. We first present the results of a numerical investigation of droplets walking in a harmonic potential on a vibrating fluid bath. The droplet's trajectory is described by an integro-differential equation, which is simulated numerically in various parameter regimes. We produce a regime diagram that summarizes the dependence of the walker's behavior on the system parameters for a droplet of fixed size. At relatively low vibrational forcing, a number of periodic and quasiperiodic trajectories emerge. In the limit of large vibrational forcing, the walker's trajectory becomes chaotic, but the resulting trajectories can be decomposed into portions of unstable quasiperiodic states. We then recast the integro-differential equation as a coupled system of ordinary differential equations in time. This method is used to simulate droplet lattices in various configurations and in the presence of a harmonic potential, creating structures reminiscent of Wigner molecules. The development of this approach is presented in detail along with its future applications. We then switch focus to a fluid system described by a modified nonlinear Schrödinger equation. The surface of an incompressible, inviscid, irrotational fluid of infinite depth can be described in two dimensions by the Dysthe equation. Recently, this equation has been used to model extraordinarily large waves occurring on the ocean's surface called rogue waves. In this thesis, we prove dispersive estimates and Strichartz estimates for the Dysthe equation. We then prove a Kato-type smoothing effect in which we are able to bound uniformly in space the L² norm in time of a fractional derivative of the linear solution by the L² norm in space of the initial data. This section of the thesis lays the groundwork for further developments in proving well-posedness via a contraction argument.en_US
dc.description.sponsorshipFinancial support from National Science Foundation and the MIT School of Scienceen_US
dc.description.statementofresponsibilityby Kristin Marie-Dettmers Kurianski.en_US
dc.format.extent124 pagesen_US
dc.language.isoengen_US
dc.publisherMassachusetts Institute of Technologyen_US
dc.rightsMIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission.en_US
dc.rights.urihttp://dspace.mit.edu/handle/1721.1/7582en_US
dc.subjectMathematics.en_US
dc.titleEstimates for solutions to the Dysthe equation and numerical simulations of walking droplets in harmonic potentialsen_US
dc.typeThesisen_US
dc.description.degreePh. D.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Mathematicsen_US
dc.identifier.oclc1117775272en_US
dc.description.collectionPh.D. Massachusetts Institute of Technology, Department of Mathematicsen_US
dspace.imported2019-09-16T22:34:13Zen_US
mit.thesis.degreeDoctoralen_US
mit.thesis.departmentMathen_US


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