Keywords

Nonlinear partial differential equations, vortex filament dynamics, superfluid helium, nonlinear dynamics, non local equations

Abstract

Nonlinear dispersive partial differential equations occur in a variety of areas within mathematical physics and engineering. We study several classes of such equations, including scalar complex partial differential equations, vector partial differential equations, and finally non-local integro-differential equations. For physically interesting families of these equations, we demonstrate the existence (and, when possible, stability) of specific solutions which are relevant for applications. While multiple application areas are considered, the primary application that runs through the work would be the nonlinear dynamics of vortex filaments under a variety of physical models. For instance, we are able to determine the structure and time evolution of several physical solutions, including the planar, helical, self-similar and soliton vortex filament solutions in a quantum fluid. Properties of such solutions are determined analytically and numerically through a variety of approaches. Starting with complex scalar equations (often useful for studying two-dimensional motion), we progress through more complicated models involving vector partial differential equations and non-local equations (which permit motion in three dimensions). In many of the examples considered, the qualitative analytical results are used to verify behaviors previously observed only numerically or experimentally.

Notes

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Graduation Date

2014

Semester

Spring

Advisor

Kaup, David

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Mathematics

Degree Program

Mathematics

Format

application/pdf

Identifier

CFE0005272

URL

http://purl.fcla.edu/fcla/etd/CFE0005272

Language

English

Release Date

May 2019

Length of Campus-only Access

5 years

Access Status

Doctoral Dissertation (Campus-only Access)

Subjects

Dissertations, Academic -- Sciences; Sciences -- Dissertations, Academic

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