[PDF] A Numerical Study Of The Radial Nonlinear Schrodinger Equation eBook

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Page : pages
File Size : 31,82 MB
Release : 2003
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(Uncorrected OCR) Abstract of the thesis entitled A NUMERICAL STUDY OF COUPLED NONLINEAR SCHRODINGER EQUATIONS ARISING IN HYDRODYNAMICS AND OPTICS submitted by Suk-Chong TSANG for the degree of Master of Philosophy at the University of Hong Kong in June 2003 This thesis reports the fmdings of numerical studies of coupled nonlinear Schrodinger equations (CNLS) in both hydrodynamics and optics applications, and its focus was a model for interaction between wavepackets in the framework of CNLS. Generally, group velocity dispersion, self-phase modulation and cross-phase modulation terms are present in these equations. However, intermodal dispersion and linear coupling terms may also exist when CNLS are applied in optics. The interplay between these effects plays a crucial role in pulse evolution. A numerical method, the Hopscotch method, was introduced to solve CNLS. This is a particularly simple method for solving CNLS, and its accuracy was verified by ascertaining the evolution of a single soliton solution of CNLS and comparing this numerical solution with the exact solution. Two applications of CNLS were studied, m hydrodynamics and optics respectively. In hydrodynamics, CNLS is used to govern the interaction of wavepackets in layered fluid. The long-time evolution of periodic solution of CNLS 1 was studied. The initial phase difference, amplitude ratio between perturbations and the ratio between the self-phase and cross-phase modulations were found to be important factors in long-time evolution. Different patterns of evolution may result from different combinations of these three effects mentioned above. In optics, CNLS can be used as the governing equation for wavepackets in directional couplers. Soliton interaction in directional couplers was studied, as their performance can be quite different from how they behave in single-mode fibers. Their behaviour was influenced by group-velocity dispersion, intermodal dispersion and linear coupling terms within the equation.

Author : Jialin Hong
Publisher : Springer Nature
Page : 220 pages
File Size : 46,22 MB
Release : 2019-08-22
Category : Mathematics
ISBN : 9813290692

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This book provides some recent advance in the study of stochastic nonlinear Schrödinger equations and their numerical approximations, including the well-posedness, ergodicity, symplecticity and multi-symplecticity. It gives an accessible overview of the existence and uniqueness of invariant measures for stochastic differential equations, introduces geometric structures including symplecticity and (conformal) multi-symplecticity for nonlinear Schrödinger equations and their numerical approximations, and studies the properties and convergence errors of numerical methods for stochastic nonlinear Schrödinger equations. This book will appeal to researchers who are interested in numerical analysis, stochastic analysis, ergodic theory, partial differential equation theory, etc.

Author : Panayotis G. Kevrekidis
Publisher : Springer Science & Business Media
Page : 417 pages
File Size : 37,53 MB
Release : 2009-07-07
Category : Science
ISBN : 3540891994

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This book constitutes the first effort to summarize a large volume of results obtained over the past 20 years in the context of the Discrete Nonlinear Schrödinger equation and the physical settings that it describes.

Author : Gadi Fibich
Publisher : Springer
Page : 870 pages
File Size : 21,25 MB
Release : 2015-03-06
Category : Mathematics
ISBN : 3319127489

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This book is an interdisciplinary introduction to optical collapse of laser beams, which is modelled by singular (blow-up) solutions of the nonlinear Schrödinger equation. With great care and detail, it develops the subject including the mathematical and physical background and the history of the subject. It combines rigorous analysis, asymptotic analysis, informal arguments, numerical simulations, physical modelling, and physical experiments. It repeatedly emphasizes the relations between these approaches, and the intuition behind the results. The Nonlinear Schrödinger Equation will be useful to graduate students and researchers in applied mathematics who are interested in singular solutions of partial differential equations, nonlinear optics and nonlinear waves, and to graduate students and researchers in physics and engineering who are interested in nonlinear optics and Bose-Einstein condensates. It can be used for courses on partial differential equations, nonlinear waves, and nonlinear optics. Gadi Fibich is a Professor of Applied Mathematics at Tel Aviv University. “This book provides a clear presentation of the nonlinear Schrodinger equation and its applications from various perspectives (rigorous analysis, informal analysis, and physics). It will be extremely useful for students and researchers who enter this field.” Frank Merle, Université de Cergy-Pontoise and Institut des Hautes Études Scientifiques, France

Author : R‚mi Carles
Publisher : World Scientific
Page : 256 pages
File Size : 33,12 MB
Release : 2008
Category : Mathematics
ISBN : 9812793127

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These lecture notes review recent results on the high-frequency analysis of nonlinear Schr”dinger equations in the presence of an external potential. The book consists of two relatively independent parts: WKB analysis, and caustic crossing. In the first part, the basic linear WKB theory is constructed and then extended to the nonlinear framework. The most difficult supercritical case is discussed in detail, together with some of its consequences concerning instability phenomena. Applications of WKB analysis to functional analysis, in particular to the Cauchy problem for nonlinear Schr”dinger equations, are also given. In the second part, caustic crossing is described, especially when the caustic is reduced to a point, and the link with nonlinear scattering operators is investigated.These notes are self-contained and combine selected articles written by the author over the past ten years in a coherent manner, with some simplified proofs. Examples and figures are provided to support the intuition, and comparisons with other equations such as the nonlinear wave equation are provided.