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better intro

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  1. +18 −13 ucf_thesis/chapter_1.tex
  2. BIN ucf_thesis/thesis.pdf
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@@ -6,24 +6,29 @@ \chapter{CHAPTER ONE: INTRODUCTION} \label{chapter_1}
where the initial conditions naturally break into stable pulses or
pulse-trains.
-Since the numerical discovery of solitons in the Korteweg \& de Vries
-equation $ u_t + u u_x + \delta^2 u_{xxx} = 0$ \cite{ZK} in 1965 there
-has been intense research in equations that admit soliton solutions.
-The applications of the KdV equation are ubiquitous because it is a
-canonical equation which describes weakly nonlinear long waves.
-Similarly, the nonlinear Schr\"odinger equations $ i \Psi_t + \Psi_{xx} \pm \Psi\|\Psi\|^2 = 0 $
-arise in diverse areas because they are canonical equations governing the
-modulation of the amplitude $\Psi$ of a weakly nonlinear wave packets \cite{DJ}.
-A general principle for associating nonlinear evolution equations with the eigenvalues
-of linear operators was discovered in 1968 \cite{Lax}. This led to a comprehensive
-theory, an extension of Fourier analysis for nonlinear systems, called
-the Inverse Scattering Transform \cite{AKNS}.
+Since the numerical "re-discovery" of solitons in the Korteweg \& de Vries
+equation $ u_t + u u_x + \delta^2 u_{xxx} = 0$ \cite{ZK} in 1965 there has been
+intense research in equations that admit soliton solutions. An analytic soliton
+solution to the KdV equation was found in 1967 \cite{GGKM} by quite unique means
+and at the time it was not clear whether the method was generally applicable.
+%The applications of the KdV equation are ubiquitous because it is a canonical
+%equation which describes weakly nonlinear long waves.
+
+A general principle for associating nonlinear evolution equations with the
+eigenvalues of linear operators was discovered in 1968 \cite{Lax}. Soon after,
+solitons were found in an even more fundamental and canoncial system, the
+nonlinear Schr\"odinger equations $ i \Psi_t + \Psi_{xx} \pm \Psi\|\Psi\|^2 = 0
+$ \cite{ZS}. These equations arise in diverse areas because they are canonical
+equations governing the modulation of the amplitude $\Psi$ of a weakly nonlinear
+wave packets \cite{DJ}. This led to a comprehensive theory, an extension of
+Fourier analysis for nonlinear systems, called the Inverse Scattering Transform
+\cite{AKNS}.
These standard techniques for investigating solitary waves of
integrable nonlinear PDEs do not carry over to the
non-integrable models which are of increasing relevance in modern
applications. Other techniques which have been devised, such as
-variational ones, and exponential asymptotics methods, each yielding
+variational ones, and exponential asymptotics methods, each yield
results in certain regimes of the systems parameters.
In this thesis, we apply a recently developed technique to
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