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This book concentrates upon the study of mathematical models of nonlinear solitary waves known as solitons. As the name indicates, a soliton is a localized wave that travels without changing its shape, such as an ocean wave. An important feature of solitons is that they keep their shapes after collisions with other solitons.
Interesting to anybody who wants to unearth the real sense and nature of solitary waves, and the relevant mathematical tools to use for effective investigation and analysis of these phenomena, the text focuses on numerical analysis of solitons. The integrability and multidimensionality of solitons is inextricably bound up with the approach of investigation and, as the more physical systems are not fully integrable, even in one dimension, numerical analysis is the main tool to investigate and understand the pertinent physical mechanisms.
The investigations
were conducted in the last 10 years in close collaboration with Professor C I
Christov (University of Louisiana at Lafayette, LA, USA), Professor V S
Gerdjikov (Institute of Nuclear Research and Nuclear Energy, BAS, Sofia,
Bulgaria), Professor I G Koprinkov (Technical University of Sofia, Bulgaria).
Part of them was supported by the Bulgarian Science Foundations under two
grants—
http://oldweb.tu-sofia.bg/fond-NI/klasirani-proekti/FPMI-DDBU02-71/project.htm
or http://www.math.bas.bg/nummeth/boussinesq/, and http://www.math.bas.bg/
~nummeth/nonlinear/index.htm.
Contents
Preface
Chapter 1. Two-dimensional Boussinesq Equation. Boussinesq Paradigm and Soliton Solutions • Bousinessq Equations. Generalized Wave Equation • Investigation of the Long-Time Evolution of Localized Solutions of a Dispersive Wave System • Numerical Implementation of Fourier-transform Method for Generalized Wave Equations • Perturbation Solution for the 2D Shallow-water Waves • Boussinesq Paradigm Equation and the Experimental Measurement • Development and Realization of Efficient Numerical Methods, Algorithms and Scientific Software for 2D Nonsteady Boussinesq Paradigm Equation. Comparative Analysis of Results
Chapter 2. Systems of Coupled Nonlinear Schr¨odinger Equations. Vector Schr¨odinger Equation • Conservative Scheme in Complex Arithmetic for Vector Nonlinear Schr¨odinger Equation • Finite-Difference Implementation of Conserved Properties of Vector NLSE • Collision Dynamics of Circularly Polarized Solitons in Nonintegrable Vector NLSE • Impact of the Large Cross-Modulation Parameter on the Collision Dynamics of Quasi- Particles Governed by Vector NLSE • Repelling Soliton Collisions in Vector NLSE with Negative Cross Modulation • On the Solution of the System of Ordinary Differential Equations Governing the Polarized Stationary Solutions of Vector NLSE • Collision Dynamics of Elliptically Polarized Solitons in Vector NLSE • Collision Dynamics of Polarized Solitons in Linearly Coupled Vector NLSE • Polarization Dynamics during Takeover Collisions of Solitons in Vector NLSE • The Effect of the Elliptic Polarization on the Quasi-Particle Dynamics of Linearly Coupled Vector NLSE • Vector NLSE with Different Cross-Modulation Rates • Asymptotic Behavior of Manakov Solitons • Manakov Solitons and Effects of External Potential Wells and Humps
Chapter 3. Ultrashort Optical Pulses. Envelope Dispersive Equations • On a Method for Solving of Multidimensional Equations of Mathematical Physics • Dynamics of High-Intensity Ultrashort Light Pulses at Some Basic Propagation Regimes • (3+1)D Nonlinear Schr¨odinger Equation • (3+1)D Nonlinear Envelope Equation • Summary of the Studies