2026-09-20T05:01:11Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1221732026-06-17T14:45:06Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Gigliola Staffilani.
Kurianski, Kristin Marie-Dettmers.
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2019-09-16T22:34:16Z
2019-09-16T22:34:16Z
2019
2019
https://hdl.handle.net/1721.1/122173
1117775272
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 119-124).
In 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.
Financial support from National Science Foundation and the MIT School of Science
by Kristin Marie-Dettmers Kurianski.
Ph.D.
Ph.D. Massachusetts Institute of Technology, Department of Mathematics
124 pages
eng
Massachusetts Institute of Technology
MIT 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.
http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Estimates for solutions to the Dysthe equation and numerical simulations of walking droplets in harmonic potentials
Thesis
application/pdf
2019-09-16T22:34:13Z
Publication
Doctoral
Math
unknown
unknown
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