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Chapter
Homoclinic Orbits in a Four Dimensional Model of a Perturbed NLS Equation: A Geometric Singular Perturbation Study
This paper contains a detailed and explicit study of an important mechanism leading to chaotic dynamics in a perturbation of the integrable nonlinear Schrödinger equation, a perturbation which contains dam**...
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Chapter and Conference Paper
Whiskered Tori and Chaotic Behavior in Nonlinear Waves
Chaotic behavior in deterministic dynamical systems is an important phenomenon with many physical ramifications. In finite-dimensional settings this behavior is well understood, because of several fundamental ...
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Chapter
Whiskered Tori for Integrable Pde’s: Chaotic Behavior in Near Integrable Pde’s
This article is a summary of our numerical and theoretical studies (which were done in various collaborations with Alan Bishop, Nick Ercolani, Greg Forest, and Steve Wiggins) of near integrable nonlinear wave ...
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Chapter
Semiclassical Behavior in the NLS Equation: Optical Shocks - Focusing Instabilities
We consider the semiclassical limit of the non-linear Schroedinger equation in both the defocusing and the focusing cases. We review the theory of the defocusing case, and verify it with numerical experiments....
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Chapter
The Behavior of Solutions of the NLS Equation in the Semiclassical Limit
We report a numerical and theoretical study of the generation and propogation of oscillations in the semiclassical limit (h → 0) of the Nonlinear Schrödinger equation. In a general setting of both dimension and n...
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Chapter and Conference Paper
Toward a Topological Classification of Integrable PDE’s
We model Fomenko’s topological classification of 2 degree of freedom integrable stratifications in an infinite dimensional soliton system. Specifically, the analyticity of the Floquet discriminant Δ(q, ») in both...
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Chapter and Conference Paper
On the Construction of a Modulating Multiphase Wavetrain for a Perturbed Kdv Equation
This paper summarizes the status of a direct construction of an asymptotic representation of a modulating multiphase wavetrain for a class of perturbed kdV equations. This class includes the kdV-Burgers equati...
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Chapter
Solitary Waves as Fixed Points of Infinite Dimensional Maps
Coherent structures which are localized in space arise and play important roles in turbulent fields. Examples include tornados, large vortices in a turbulent wake, and isolated vortices in the ocean. These coh...
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Chapter and Conference Paper
An Overview of Soliton Mathematics
I have been asked to provide some background about “soliton mathematics”. I will describe an overview of the subject which indicates how some of the different mathematical developments fit together and complim...
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Chapter and Conference Paper
Some comments on Bäcklund transformations, canonical transformations, and the inverse scattering method