better introduction and more refs

git-svn-id: svn+ssh://leto.net/usr/local/svn/thesis@127 c868c573-c6a3-dc11-90ff-0002b3153201

ucf_thesis/chapter_1.tex

@@ -4,12 +4,26 @@ \chapter{CHAPTER ONE: INTRODUCTION} \label{chapter_1}
 important, both as possible information carriers,
 as well as organizing centers for the solution dynamics in regimes
 where the initial conditions naturally break into stable pulses or
-pulse-trains. Standard techniques for investigating solitary waves of
-integrable nonlinear PDEs,
-such as the Inverse Scattering Transform, do not carry over to the
+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}.
+
+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
+variational ones, and exponential asymptotics methods, each yielding
 results in certain regimes of the systems parameters.
 
 In this thesis, we apply a recently developed technique to
@@ -44,18 +58,21 @@ \chapter{CHAPTER ONE: INTRODUCTION} \label{chapter_1}
 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.
+within the framework of reversible systems theory. Recently, an
+alternative approach using a Hamiltonian formulation has also
+been used to analyze the traveling wave ODE \cite{LiZhang}.
 
 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
+For the generalized Microstructure equation, 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 entioned
+solitary waves using exponential asymptotics techniques, are also mentioned.
+
 

ucf_thesis/references.bib

@@ -1,3 +1,27 @@
+@ARTICLE{X,
+ AUTHOR = "{}",
+ TITLE = "{}",
+ JOURNAL = {},
+ VOLUME={},
+ YEAR={},
+ PAGES={}
+ }
+@ARTICLE{Lax,
+ AUTHOR = "{Peter D. Lax}",
+ TITLE = "{Integrals of Nonlinear Equations of Evolution and Solitary Waves}",
+ JOURNAL = {Comm. Pure Appl. Math.},
+ VOLUME={21},
+ YEAR={1968},
+ PAGES={467-490}
+ }
+@ARTICLE{ZK,
+ AUTHOR = "{Zabusky, N.J., \& Kruskal, M.D.}",
+ TITLE = "{Interactions of 'solitons' in a collisionless plasma and the recurrence of initial states}",
+ JOURNAL = {Phys. Rev. Lett.},
+ VOLUME={15},
+ YEAR={1965},
+ PAGES={240-243}
+ }
 @ARTICLE{Liu,
  AUTHOR = "{Y. Liu}",
  TITLE = "{Existence and blowup solutions of a nonlinear Pochhammer-Chree equation}",
@@ -6,6 +30,14 @@ @ARTICLE{Liu
  YEAR={1996},
  PAGES={797816}
  }
+@ARTICLE{AKNS,
+ AUTHOR = "{Ablowitz, M.J., Kaup, D.J., Newell, A.C \& Segur, H}",
+ TITLE = "{The Inverse Scattering Transform - Fourier analysis for nonlinear problem}",
+ JOURNAL = {Stud. Appl. Math.},
+ VOLUME={53},
+ YEAR={1974},
+ PAGES={2490315}
+ }
 @ARTICLE{LiZhang,
  AUTHOR = "{J. Li, L. Zhang}",
  TITLE = "{Bifurcations of traveling wave solutions in generalized Pochhammer-Chree Equations}",
@@ -32,6 +64,13 @@ @ARTICLE{EBS
  PAGES={}
  }
 
+@BOOK{DJ,
+ AUTHOR = "{P.G. Drazin \& R.S. Johnson}",
+ TITLE = "{Solitons: an introduction}",
+ PUBLISHER = {Cambridge University Press},
+ ADDRESS = {Cambridge},
+ YEAR={1989},
+ }
 @BOOK{Strogatz,
  AUTHOR = "{Steven H. Strogatz}",
  TITLE = "{Nonlinear Dynamics and Chaos}",