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Nonlinear dispersive partial differential equations of physical relevance with applications to vortex dynamics

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Date Issued:
2014
Abstract/Description:
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.
Title: Nonlinear dispersive partial differential equations of physical relevance with applications to vortex dynamics.
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Name(s): VanGorder, Robert, Author
Kaup, David, Committee Chair
Vajravelu, Kuppalapalle, Committee CoChair
Nevai, Andrew, Committee Member
Mohapatra, Ram, Committee Member
Kassab, Alain, Committee Member
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2014
Publisher: University of Central Florida
Language(s): English
Abstract/Description: 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.
Identifier: CFE0005272 (IID), ucf:50545 (fedora)
Note(s): 2014-05-01
Ph.D.
Sciences, Mathematics
Doctoral
This record was generated from author submitted information.
Subject(s): nonlinear partial differential equations -- vortex filament dynamics -- superfluid Helium -- nonlinear dynamics -- non-local equations
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0005272
Restrictions on Access: campus 2019-05-15
Host Institution: UCF

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