lots of fixes. VERY CLOSE

git-svn-id: svn+ssh://leto.net/usr/local/svn/thesis@88 c868c573-c6a3-dc11-90ff-0002b3153201

ucf_thesis/chapter_2.tex

@@ -329,21 +329,16 @@ \section{Normal form near $C_1$: possible solitary wave solutions}
 
 Hence
 \begin{equation}
-\lambda_{1,2} = \pm i \sqrt{-q}, q < 0
+\lambda = \pm i \sqrt{-q}, q < 0
 \end{equation}
 
-Matching this to the linear part of \eqref{eq:c1nfc} ( which corresponds to the imaginary eigenvalues), $\lambda = i d_0 = i \sqrt{-q}$ or 
+Matching this to the linear part of \eqref{eq:c1nfc} 
+ (which corresponds to the imaginary eigenvalues), $\lambda = i d_0 = i \sqrt{-q}$ or 
 \begin{equation}
 d_0 = \sqrt{-q}
 \end{equation}
 
-
-With a  dominant balance argument 
-%after the change of variable $\epsilon = \sqrt{-3 \alpha}$ 
-on the characteristic equation 
-\eqref{eq:charlinear} as $\lambda \rightarrow \lambda_{1,2} $ 
-and comparing to the linear eigenvalue of \eqref{eq:c1nf} 
-we find
+With a dominant balance argument on the characteristic equation \eqref{eq:charlinear} as  $\lambda \rightarrow \pm i \sqrt{-q}$ we find
 \begin{equation}
 d_1 = \frac{\sqrt{-q}}{2 q^2} 
 \end{equation}

ucf_thesis/chapter_3.tex

@@ -30,12 +30,10 @@ \section{Solitary waves; local bifurcations}
 \end{equation}
 
 where 
-% the notation $\mathcal{N[\phi]}$ means that the operator $\mathcal{N}$ operates on $\phi$ and all of it's derivatives, and 
 \begin{equation}
 \mathcal{N}\left[\phi\right] = -\Delta_1 \phi_z^2 - b \Delta_1 \phi \phi_{zz}
 \end{equation}
-
-%where 
+and
 \begin{subequations}
 \begin{eqnarray}
 z &\equiv& x - c t\\
@@ -73,7 +71,7 @@ \section{Solitary waves; local bifurcations}
 four-dimensional normal form in Section 4 that there exists a $\mathrm{sech}^2$ homoclinic orbit near $C_1$.
 \end{description}
 
-Having outlined the possible families of orbits homoclinic to the fixed point \eqref{eq:fp} of \eqref{eq:linode2},
+Having outlined the possible families of orbits homoclinic to the fixed point \eqref{eq:fp2} of \eqref{eq:linode2},
 corresponding to pulse solitary waves of \eqref{eq:MS}, we now derive normal forms near the transition curves $C_0$ and $C_1$
 to confirm the existence of regular or delocalized solitary waves in the corresponding regions of $\left(p,q\right)$ parameter space.
 
@@ -176,7 +174,6 @@ \subsection{ Near $C_0$ }
 terms.  To this end, using the standard 'suspension' trick of treating the
 perturbation parameter $\epsilon$ as a variable, we expand the function $\Psi$
 in \eqref{eq:c0cm2} as 
-
 \begin{equation}\label{eq:psiexp2}
 \Psi(\epsilon,A,B) = \epsilon A \Psi_{10}^1 + \epsilon B \Psi_{01}^1 + A^2 \Psi_{20}^0 + A B \Psi_{11}^0 + B^2 \Psi_{02}^0 + \cdots
 \end{equation}
@@ -185,12 +182,12 @@ \subsection{ Near $C_0$ }
 In the first way of computing $dY/dz$, we take
 the $z$ derivative of \eqref{eq:c0cm2} (using \eqref{eq:c0nf2} and \eqref{eq:psiexp2}). 
 The coefficient of $A^2$ in the resulting expression is $\tilde{c} \zeta_1 $. In the second way of computing $dY/dz$, we use \eqref{eq:c0cm2} and \eqref{eq:psiexp2} in \eqref{eq:bilinear2}. The coefficient of $A^2$ in the resulting expression is 
-$ L_{0,q} \Psi_{20}^0 - F_2\left(\zeta_0,\zeta_0\right)$.  Hence
+$ L_{0,q} \Psi_{20}^0 - F_2\left(\zeta_0,\zeta_0\right)$,
+which leads to 
 \begin{equation}\label{eq:A2coef2}
  \tilde{c} \zeta_1 = L_{0q} \Psi_{20}^0 - F_2(\zeta_0,\zeta_0) \end{equation}
 
 Using \eqref{eq:lineareigs2} and \eqref{eq:nonlinear2} and denoting $\Psi_{20}^0 = \left<x_1,x_2,x_3,x_4\right>$ in \eqref{eq:A2coef2} yields the equations
-
 \begin{subequations}
 \begin{eqnarray}
 0 &=& x_2 \\
@@ -263,7 +260,7 @@ \section{Normal form near $C_1$: possible solitary wave solutions}
 \end{equation}
 
 with  a corresponding four-dimensional normal form
-\begin{subequations}\label{eq:c1nf}
+\begin{subequations}\label{eq:c1nf2}
 \begin{eqnarray}
 \frac{dA}{dz} &=& B \label{eq:aq2} \\
 \frac{dB}{dz} &=& \bar{\nu} A + b_* A^2 + c_* \left|C\right|^2  \label{eq:bq2} \\
@@ -293,21 +290,19 @@ \section{Normal form near $C_1$: possible solitary wave solutions}
 \lambda = \pm i \sqrt{-q}, q < 0
 \end{equation}
 
-Matching this to the linear part of \eqref{eq:cq2} ( which corresponds to the
+Matching this to the linear part of \eqref{eq:cq2} 
+(which corresponds to the
 imaginary eigenvalues), $\lambda = i d_0 = i \sqrt{-q}$ or 
 \begin{equation}
 d_0 = \sqrt{-q}
 \end{equation}
 
-
-With a dominant balance argument after the change of variable $\epsilon =
-\sqrt{-3 \alpha}$ on the characteristic equation \eqref{eq:charlinear2} as $\lambda \rightarrow 0 $ we
-find $d_1 = \frac{ \sqrt{-3 \alpha} }{18 \alpha^2 } $. Using $\alpha=q/3$
-implies 
+With a dominant balance argument on the characteristic equation \eqref{eq:charlinear2} as  $\lambda \rightarrow \pm i \sqrt{-q}$ we find
 \begin{equation}
-d_1 = \frac{\sqrt{-q}}{2 q^2}
+d_1 = \frac{\sqrt{-q}}{2 q^2} 
 \end{equation}
 
+
 The remaining undetermined coefficients  in the normal form are the 
 coefficients $b_*,c_*$ and $d_2$ 
 which correspond to the $A^2, |C|^2$ and $AC$ terms respectively. In 
@@ -328,7 +323,6 @@ \section{Normal form near $C_1$: possible solitary wave solutions}
 $\bar{C}$ are once again read off.  Equating the coefficients of the
 corresponding terms in the two separate expressions for $dY/dz$ yields the
 following equations:
-
 \begin{subequations}
 \begin{eqnarray}
 \mathcal{O}(A^2): &		b_* \zeta_1 &= L_{0q} \Psi_{2000}^0 - F_2(\zeta_0,\zeta_0) \\
@@ -337,11 +331,10 @@ \section{Normal form near $C_1$: possible solitary wave solutions}
 \mathcal{O}(A C): 	&i d_2 \zeta_+ + i d_0 \Psi_{1010}^0 &= L_{0q} \Psi_{1010}^0 - 2 F_2(\zeta_0,\zeta_+) \label{eq:AC2}
 \end{eqnarray}
 \end{subequations}
-where we have used the fact that $F_2$ is a symmetric bilinear form. Equation \eqref{eq:cstar} is decoupled and yields 
+where we have used the fact that $F_2$ is a symmetric bilinear form. Equation \eqref{eq:cstar2} is decoupled and yields 
 $ c_* = 2 \Delta_1 \left( \frac{2 b}{3}  - 1\right)$. The only coefficient left to determine is $d_2$ which we shall compute now. 
 
 Using $\Psi_{1010}^0 = \left<x_1,x_2,x_3,x_4\right>^T$ in \eqref{eq:AC2} implies 
-
 \begin{subequations}
 \begin{eqnarray}
 i d_2 + i d_0 x_1 &=& x_2 \label{eq:one2} \\

ucf_thesis/chapter_4.tex

@@ -1,12 +1,36 @@
 \chapter{CHAPTER FOUR: RESULTS} \label{chapter_4}
 
+In this thesis, we apply a recently developed technique to
+comprehensively categorize all possible families of solitary wave
+solutions in two models of topical interest.
 
-\begin{figure}[!ht] \singlespacing
-\begin{center}
-\includegraphics[width=.8\textwidth]{figures/homoclinic}  %% good idea to keep your figures in a separate folder
-\end{center}
-\caption{Homoclinic Orbit}\label{tbl:my_first_figure}
-\end{figure}
+The models considered are:
+\begin{itemize}
+\item the Generalized Pochammer-Chree Equations, which  govern the propagation of longitudinal waves in elastic rods,
 
-%% Best way to generate pictures, especially graphs, is to save them (say from MATLB) as .eps (Encapsulated PostScript) and than
-%% convert to PDF using (attached) eps2pdf utility. Note that pdfTeX will not accept .eps files.
+and
+
+\item generalized microstructure PDE.
+\end{itemize}
+
+Limited analytic results exist for the occurrence of one family
+of  solitary wave solutions for each of these 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,
+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 both models, 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