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better info in the intro

git-svn-id: svn+ssh://leto.net/usr/local/svn/thesis@136 c868c573-c6a3-dc11-90ff-0002b3153201
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1 parent 9ec6039 commit 460f9bc7c9a602bcf0807d3568f771b0874b71ac leto committed Apr 18, 2008
Showing with 18 additions and 16 deletions.
  1. +17 −13 ucf_thesis/chapter_1.tex
  2. +1 −3 ucf_thesis/chapter_4.tex
@@ -53,10 +53,10 @@ \chapter{CHAPTER ONE: INTRODUCTION} \label{chapter_1}
\end{equation}
\end{itemize}
-The phase space of the traveling-wave equation will be studied, specifically
-the homoclinic orbits, which correspond to solitary wave solutions in the original
-PDE. A homoclinic orbit is defined as any orbit which connects a fixed point to itself
-\cite{Strogatz}.
+The phase space of the traveling-wave equation will be studied, specifically the
+homoclinic orbits, which correspond to solitary wave solutions in the original
+PDE. A homoclinic orbit is defined as any orbit which connects a fixed point to
+itself \cite{Strogatz}.
Limited analytic results exist for the occurrence of one family
of solitary wave solutions for each of these equations. Since, as
@@ -67,15 +67,19 @@ \chapter{CHAPTER ONE: INTRODUCTION} \label{chapter_1}
alternative approach using a Hamiltonian formulation has also
been used to analyze the traveling wave ODE \cite{LiZhang}.
-Besides confirming
-the existence of the known family of solitary waves for each model,
-we find a continuum of delocalized solitary waves
-(or homoclinics to small-amplitude periodic orbits).
-On isolated curves in the relevant parameter region, the delocalized
-waves reduce to genuine embedded solitons.
-For the generalized Microstructure equation, the new family of solutions occur in regions of
-parameter space distinct from the known solitary wave solutions and
-are thus entirely new.
+Besides confirming the existence of the known family of solitary waves for each
+model, we find a continuum of delocalized solitary waves (or homoclinics to
+small-amplitude periodic orbits). On isolated curves in the relevant parameter
+region, the delocalized waves reduce to genuine embedded solitons. These curves
+are determined from the regions of different eigenvalue configurations in the
+characteristic equation of the traveling wave ODE. An example of this
+would be an eigenvalue of multiplicity two splitting into two simple eigenvalues,
+or two simple eigenvalues coalescing into an eigenvalue of multiplicity two
+as a parameter is varied.
+
+For the generalized Microstructure equation, the new family of solutions occur
+in regions of parameter space distinct from the known solitary wave solutions
+and are thus entirely new.
Directions for future work, including the dynamics of each family of
solitary waves using exponential asymptotics techniques, are also mentioned.
@@ -39,15 +39,13 @@ \chapter{CHAPTER FOUR: RESULTS} \label{chapter_4}
Thus, these curves define separatrices between vastly different dynamics in
the traveling wave ODE as well as the original PDE.
-
One may easily verify that $\lim_{z\rightarrow\pm\infty} A(z) = 0$, therefore
$A(z)$ compromises a homoclinic orbit, since it connects the fixed point $0$ to
itself. The importance of homoclinic orbits in the traveling wave ODE is that
they correspond to soliton pulse solutions of the original PDE \cite{IA}.
For the Microstructure equation , the new family of solutions occur in regions
of parameter space distinct from the known solitary wave solutions and are thus
-entirely new. Directions for future work, including the dynamics of each family
-of solitary waves using exponential asymptotics techniques, are also mentioned.
+entirely new.
%42

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