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add stuff the results!

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1 parent ed57634 commit ee68834f645cccf70f730d2431456c6373a872d2 leto committed Apr 18, 2008
Showing with 11 additions and 3 deletions.
  1. +11 −3 ucf_thesis/chapter_4.tex
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@@ -28,9 +28,17 @@ \chapter{CHAPTER FOUR: RESULTS} \label{chapter_4}
\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. The importance
-of homoclinic orbits in the traveling wave ODE is that they correspond to soliton
-pulse solutions of the original PDE \cite{IA}.
+region, the delocalized waves reduce to genuine embedded solitons.
+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.
+
+
+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
+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

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