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%%%
%%% File: utthesis.doc, version 2.0, January 1995
%%% =============================================
%%% Copyright (c) 1995 by Dinesh Das.  All rights reserved.
%%% This file is free and can be modified or distributed as long as
%%% you meet the following conditions:
%%%
%%% (1) This copyright notice is kept intact on all modified copies.
%%% (2) If you modify this file, you MUST NOT use the original file name.
%%%
%%% This file contains a template that can be used with the package
%%% utthesis.sty and LaTeX2e to produce a thesis that meets the requirements
%%% of the Graduate School of The University of Texas at Austin.
%%%
%%% All of the commands defined by utthesis.sty have default values (see
%%% the file utthesis.sty for these values).  Thus, theoretically, you
%%% don't need to define values for any of them; you can run this file
%%% through LaTeX2e and produce an acceptable thesis, without any text.
%%% However, you probably want to set at least some of the macros (like
%%% \thesisauthor).  In that case, replace "..." with appropriate values,
%%% and uncomment the line (by removing the leading %'s).
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\RE {I\kern-6pt R    }
\def\Z  {Z\kern-13pt Z   }
\def\be {\begin{equation}}
\def\ee {\end{equation}  }
\def\beq{\begin{eqnarray}}
\def\eeq{\end{eqnarray}  }
\def\lb{\left(}
\def\rb{\right)}
\def\bi {\begin{itemize} }
\def\i#1{\item{#1}       }
\def\ei {\end{itemize}   }
\def\kkbar {$KK^{^{^{_{\! \! \! \! \! \! \rule{2.4mm}{0.25mm}}}}}$ }
\def\kbar   {$K^{^{^{_{\! \! \! \! \! \! \rule{2.4mm}{0.25mm}}}}}$ }
\def\sech { {\rm sech} }
\newcommand{\scri}{\mathcal {J}}

\documentclass[11pt]{report}         %% LaTeX2e document.
\usepackage {utthesis}               %% Preamble.
\usepackage {latexsym}               %% for \Box and other symbols.
\input epsf
\input macros

% \mastersthesis                     %% Uncomment one of these; if you don't
 \phdthesis                         %% use either, the default is \phdthesis.

% \thesisdraft                       %% Uncomment this if you want a draft
                                     %% version; this will print a timestamp
                                     %% on each page of your thesis.

% \leftchapter                       %% Uncomment one of these if you want
% \centerchapter                     %% left-justified, centered or
% \rightchapter                      %% right-justified chapter headings.
                                     %% Chapter headings includes the
                                     %% Contents, Acknowledgments, Lists
                                     %% of Tables and Figures and the Vita.
                                     %% The default is \centerchapter.

% \singlespace                       %% Uncomment one of these if you want
% \oneandhalfspace                   %% single-spacing, space-and-a-half
 \doublespace                        %% or double-spacing; the default is
                                     %% \oneandhalfspace, which is the
                                     %% minimum spacing accepted by the
                                     %% Graduate School.

\renewcommand{\thesisauthor}{Ethan Philip Honda}    %% Your official UT name.

\renewcommand{\thesismonth}{August}   %% Your month of graduation.

\renewcommand{\thesisyear}{2000}      %% Your year of graduation.

\renewcommand{\thesistitle}{ Resonant Dynamics within the nonlinear\\ Klein-Gordon Equation: 
	\hbox{\it Much ado about Oscillons} }
				     %% The title of your thesis; use
                                     %% mixed-case.

\renewcommand{\thesisauthorpreviousdegrees}{B.S.}
                                     %% Your previous degrees, abbreviated;
                                     %% separate multiple degrees by commas.

\renewcommand{\thesissupervisor}{Matthew William Choptuik \\ \ \ \ \ \  Philip J. Morrison}
                                     %% Your thesis supervisor; use mixed-case
                                     %% and don't use any titles or degrees.

\renewcommand{\thesisauthoraddress}{7507 N. Lowell Ave. \\ Skokie, IL 60076\\ USA}
                                     %% Your permanent address; use "\\" for
                                     %% linebreaks.

\renewcommand{\thesisdedication}{To Trish...}
                                     %% Your dedication, if you have one; use
                                     %% "\\" for linebreaks.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%% The following commands are all optional, but useful if your requirements
%%% are different from the default values in utthesis.sty.  To use them,
%%% simply uncomment (remove the leading %) the line(s).

\renewcommand{\thesiscommitteesize}{6}
                                     %% Uncomment this only if your thesis
                                     %% committee does NOT have 5 members
                                     %% for \phdthesis or 2 for \mastersthesis.
                                     %% Replace the "..." with the correct
                                     %% number of members.

% \renewcommand{\thesisdegree}{...}  %% Uncomment this only if your thesis
                                     %% degree is NOT "DOCTOR OF PHILOSOPHY"
                                     %% for \phdthesis or "MASTER OF ARTS"
                                     %% for \mastersthesis.  Provide the
                                     %% correct FULL OFFICIAL name of
                                     %% the degree.

% \renewcommand{\thesisdegreeabbreviation}{...}
                                     %% Use this if you also use the above
                                     %% command; provide the OFFICIAL
                                     %% abbreviation of your thesis degree.

% \renewcommand{\thesistype}{...}    %% Use this ONLY if your thesis type
                                     %% is NOT "Dissertation" for \phdthesis
                                     %% or "Thesis" for \mastersthesis.
                                     %% Provide the OFFICIAL type of the
                                     %% thesis; use mixed-case.

% \renewcommand{\thesistypist}{...}  %% Use this to specify the name of
                                     %% the thesis typist if it is anything
                                     %% other than "the author".

%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-----------------------------------------------------------------------

\thesiscopyrightpage                 %% Generate the copyright page.

\thesistitlepage                     %% Generate the title page.

\thesissignaturepage                 %% Generate the signature page.

\thesisdedicationpage                %% Generate the dedication page.

\begin{thesisacknowledgments}        %% Use this to write your
                                     %% acknowledgments; it can be anything
\small

I sincerely thank my supervisor Matt Choptuik for all of his help, guidance, and 
support over the last four years.
I appreciate the  patience and encouragement he showed me and thank him for 
continuing to supervise me despite the logistical challenges.

I thank Philip J. Morrison for his help on the trial function methods 
and for becoming my advisor-of-record.

I am indebted to my friends (many in the Center for Relativity) 
for sharing their wisdom and knowledge, both about numerical relativity and life.  

Finally, I pay special respect and give all my thanks to my family for their love and 
support over the last twenty-five years.  Mom, Dad, Graham, Grandpa and Gary,  
I would not be here now if it weren't for the sacrifices you have made.
I also 
thank my fiance, Trish, for all her patience, support, and encouragement.



I also would like to acknowledge financial support from the 
National Science Foundation grant  and from a Texas Advanced Research
Projects grant. 
The bulk of the computations described here were carried out 
on the {\tt vn.physics.ubc.ca} Beowulf cluster, which was 
funded by the Canadian Foundation for Innovation, with 
operations support from the Natural Sciences and Engineering 
Research Council of Canada, and the Canadian Institute for 
Advanced Research.
Some computations were also carried out using the 
Texas Advanced Computing Center's
SGI Cray SV-1 {\tt aurora.hpc.utexas.edu} and SGI Cray T3E 
{\tt lonestar.hpc.utexas.edu}.


\normalsize
\end{thesisacknowledgments}          %% allowed in LaTeX2e par-mode.

\begin{thesisabstract}               %% Use this to write your thesis
                                     %% abstract; it can be anything


This dissertation discusses solutions to the nonlinear Klein-Gordon (nlKG) equation
with symmetric and asymmetric double-well potentials, focusing on the 
collapse and collision of bubbles and critical phenomena found therein.

A new method is presented that allows the solution of massive field equations
on a (relatively) small static grid. 
A coordinate transformation is used that transforms typical 
flatspace coordinates to coordinates that move outward 
(near the outer boundary) at nearly the speed of 
light.  The outgoing radiation is
compressed to nearly the Nyquist limit of the grid where it 
is quenched by dissipation.   
The method is implemented successfully in both spherically symmetric and
axisymmetric codes.

The new method is first used in a code to explore spherically 
symmetric bubble collapse.
New resonant oscillon solutions are found within the solution space of the
nlKG model with a symmetric double-well potential (SDWP).  A time-scaling relation 
is found to exist for the lifetime of each resonance.
The resonant solutions are also obtained independently using 
a set of ordinary differential equations derived from a non-radiative  
periodic ansatz. 
The method is also applied to the nlKG model with an asymmetric double-well potential 
(ADWP); the threshold of expanding bubble formation is 
investigated and a time-scaling law is shown to exist.

The method is then used in an axisymmetric code to simulate bubble collisions.
A technique for boosting arbitrary spherically symmetric finite difference 
solutions is presented and used to generate initial data for the collisions.
The 2D parameter space of bubble width versus collision velocity is
explored and the threshold of expanding bubble formation is again
considered.
On the threshold, there exists a time-scaling law with critical 
exponent similar to the spherically symmetric case.

Lastly, resonant oscillon solutions are constructed using trial function methods
and variational principles.  The solutions are found to be consistent
with the dynamical evolutions.


%total word count: 294 which is LESS THAN 350!




\end{thesisabstract}                 %% allowed in LaTeX2e par-mode.

\tableofcontents                     %% Generate table of contents.
\listoftables                      %% Uncomment this to generate list
                                     %% of tables.
\listoffigures                     %% Uncomment this to generate list
                                     %% of figures.

\input{intro.tex}
\input{bgdtheory.tex}
\input{nummeth.tex}
\input{chap4.tex}
\input{chap5.tex}
\input{chap6.tex}
\input{conc.tex}
\input{app1.tex}

\begin{thebibliography}{..}          %% Start your bibliography here; you can
%\bibitem{...} ...                    %% also use the \bibliography command
\input{bib.tex}

\end{thebibliography}                %% to generate your bibliography.

\begin{thesisauthorvita}             %% Write your vita here; it can be
                                     %% anything in LaTeX2e par-mode.
Ethan Philip Honda was born in Chicago, IL on December 16, 1974,
the son of Sheila and Edward Honda.
After completing high school at the Illinois Math and Science Academy,
Aurora, IL, in 1992 he entered Loyola University Chicago as a freshman.
%During the summer of 1993, he completed the PLC I commissioning program 
%at MCCDC, Quantico, VA. 
%Although an extremely motivating experience, he decided to focus on a career 
%in physics and transfered to MIT during the fall of 1993 where he 
%received his S.B. in physics in 1996.
He transfered to MIT during the fall of 1993 and received a S.B. in 
physics in 1996.
In the fall of 1996, he entered the graduate school at the University of
Texas at Austin, where he began studying numerical relativity under
the supervision of Matthew Choptuik.
As an adjunct faculty member from 1997 to 2000, he lectured introductory physics classes 
at Austin Community College.
While in graduate school he was engaged to Patricia Eileen Stafford, who he
will be marrying in the spring of 2001.
After graduating he will be working as a postdoctoral fellow
with the Signal Physics Group
at the Applied Research Laboratories in Austin. 
\end{thesisauthorvita}               %%
\end{document}                       %% Done.








