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
5-15-2019
Length of Campus-only Access
5 years
Access Status
Doctoral Dissertation (Open Access)
Subjects
Dissertations, Academic -- Sciences; Sciences -- Dissertations, Academic
STARS Citation
VanGorder, Robert, "Nonlinear Dispersive Partial Differential Equations of Physical Relevance with Applications to Vortex Dynamics" (2014). Electronic Theses and Dissertations. 4835.
https://stars.library.ucf.edu/etd/4835