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better wording on intro

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  1. +14 −8 ucf_thesis/chapter_1.tex
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22 ucf_thesis/chapter_1.tex
@@ -6,8 +6,12 @@ \chapter{CHAPTER ONE: INTRODUCTION} \label{chapter_1}
where the initial conditions naturally break into stable pulses or
pulse-trains.
-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
+The Korteweg \& de Vries (KdV) equation $ u_t + u u_x + \delta^2 u_{xxx} = 0$
+was the first nonlinear equation found to admit solitons, first derived in 1895
+to describe weakly nonlinear long water waves. Some particular solutions were
+known at this time, but general solution method was known. It was not until
+seventy years later until any further progress was made. Since the numerical
+"re-discovery" of solitons in the KdV equation \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.
@@ -24,12 +28,14 @@ \chapter{CHAPTER ONE: INTRODUCTION} \label{chapter_1}
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 yield
-results in certain regimes of the systems parameters.
+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. By non-integrable we mean equations
+for which an Inverse Scattering Transform does not exist.
+
+Other techniques which have been devised, such as 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
comprehensively categorize all possible families of solitary wave

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