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more ch4 explanation

git-svn-id: svn+ssh://leto.net/usr/local/svn/thesis@135 c868c573-c6a3-dc11-90ff-0002b3153201
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1 parent ee68834 commit 9ec603925cc4100eb1e916afff29dc93bbdcf5e5 leto committed Apr 18, 2008
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  1. +7 −2 ucf_thesis/chapter_4.tex
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@@ -29,14 +29,19 @@ \chapter{CHAPTER FOUR: RESULTS} \label{chapter_4}
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 solitary waves are called delocalized because they have exponentially
+small oscillations as $|z|\rightarrow\infty$ and so are not localized in space.
+This is often referred to as a soliton in a "sea of radiation."
+
These curves are defined by the behavior of the four eigenvalues of the characteristic
equation $ \lambda^4 - q \lambda^2 - \epsilon = 0$. Specifically, the
multiplicity of the eigenvalues change as the parameters vary across these curves.
-Thus, these curves define seperatrices between vastly different dynamics.
+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
-$A(z)$ comprimises a homoclinic orbit, since it connects the fixed point $0$ to
+$A(z)$ 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}.

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