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speling and ch4 expansion

git-svn-id: svn+ssh://leto.net/usr/local/svn/thesis@132 c868c573-c6a3-dc11-90ff-0002b3153201
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1 parent 5706dbf commit 2f229c4ba7b1397465199e3f16bd8cd5a6d8b6d7 leto committed Apr 17, 2008
Showing with 25 additions and 29 deletions.
  1. +2 −2 ucf_thesis/chapter_2.tex
  2. +23 −27 ucf_thesis/chapter_4.tex
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@@ -129,11 +129,11 @@ \section{NORMAL FORM NEAR $C_0$: SOLITARY WAVE SOLUTIONS}
\end{subequations}
We now derive a linear operator $L_{pq}$ which is equivalent to $A=L_0+L_1$ and in reversible form.
-Let $L_{pq} = L_0 + M $ where $M$ must satifisfy the following properties:
+Let $L_{pq} = L_0 + M $ where $M$ must satisfy the following properties:
\begin{itemize}
\item $ M L_0^* = L_0^* M $ or $ \left[M, L_0^*\right]=0$: $M$ commutes with $L_0^*$
-\item $ S M = -M S $ : $S$ and $M$ are antisymmetric with repect to each other
+\item $ S M = -M S $ : $S$ and $M$ are antisymmetric with resect to each other
\end{itemize}
where $S = \left(\begin{array}{cccc}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{array}\right) $
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@@ -13,32 +13,28 @@ \chapter{CHAPTER FOUR: RESULTS} \label{chapter_4}
\item a generalized microstructure PDE.
\end{itemize}
-Limited analytic results exist for the occurrence of one family
-of solitary wave solutions for the Microstructure equation and
-results using a Hamiltonian formulation have recently been found in the Pochammer-Chree equations. Since, as
-mentioned above, solitary wave solutions often play a central role in
-the long-time evolution of an initial disturbance, we consider
-such solutions of both models here (via the normal form approach)
-within the framework of reversible systems theory.
-
-Besides confirming
-the existence of the known family of solitary waves for each model,
-of the form
-\begin{equation} \label{eq:ms_soliton1}
+Limited analytic results exist for the occurrence of one family of solitary
+wave solutions for the Microstructure equation and results using a Hamiltonian
+formulation have recently been found in the Pochhammer-Chree equations. Since, as
+mentioned above, solitary wave solutions often play a central role in the
+long-time evolution of an initial disturbance, we consider such solutions of
+both models here (via the normal form approach) within the framework of
+reversible systems theory.
+
+Besides confirming the existence of the known family of solitary waves for each
+model, of the form
+\begin{equation}
A\left(z\right) = \ell \space \mathrm{sech}^2\left(k z\right)
\end{equation}
-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 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.
-
-%\newpage
+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. 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.
+
+%42

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