Nonlinear Waves and Solitons on Contours and Closed Surfaces

Nonlinear Waves and Solitons on Contours and Closed Surfaces

by Andrei Ludu

Hardcover(2nd ed. 2012)

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Product Details

ISBN-13: 9783642228940
Publisher: Springer Berlin Heidelberg
Publication date: 01/16/2012
Series: Springer Series in Synergetics
Edition description: 2nd ed. 2012
Pages: 490
Product dimensions: 6.10(w) x 9.25(h) x 0.04(d)

About the Author

Nonlinear and solitary waves are historically related to quasi one-dimensional systems where the spatial extent in one direction is much bigger than in the other direction, such as channels and fibres. The present book treats the case of more compact systems and their nonlinear ascillations, which have only recently come into forms. Such systems include liquid drop models, Bose-Einstein condensates and even living cells. A general formalism is developed, based on the differential geometry of curved manifolds, and various applications are considered in the physical sciences and beyond.

Table of Contents

Mathematical Prerequisites
Introduction     5
Introduction to Soliton Theory     5
Algebraic and Geometric Approaches     6
A List of Useful Derivatives     8
Mathematical Prerequisites     11
Elements of Topology     11
Separation Axioms     12
Compactness     15
Weierstrass-Stone Theorem     17
Connectedness, Connectivity, and Homotopy     18
Separability and Basis     20
Metric and Normed Spaces     20
Elements of Homology     21
The Importance of the Boundary     23
The Power of Compact Boundaries: Representation Formulas     23
Representation Formula for n = 1: Taylor Series     24
Representation Formula for n = 2: Cauchy Formula     24
Representation Formula for n = 3: Green Formula     25
Representation Formula in General: Stokes Theorem     26
Comments and Examples     28
Vector Fields, Differential Forms, and Derivatives     31
Manifolds and Maps     32
Differential and Vector Fields     35
Existence and Uniqueness Theorems: Differential Equation Approach     39
Existence and Uniqueness Theorems: Flow BoxApproach     45
Compact Supported Vector Fields     47
Lie Derivative and Differential Forms     47
Invariants     52
Fiber Bundles     54
Poincare Lemma     57
Tensor Analysis, Covariant Derivative, and Connections     58
The Mixed Covariant Derivative     60
Curvilinear Orthogonal Coordinates     62
Gradient     64
Divergence     64
Curl     65
Laplacian     65
Special Two-Dimensional Nonlinear Orthogonal Coordinates     66
Problems     67
Geometry of Curves     69
Elements of Differential Geometry of Curves     69
Closed Curves     76
Curves Lying on a Surface     78
Problems     79
Motion of Curves and Solitons     81
Nonlinear Kinematics of Two-Dimensional Curves and Solitons     82
The Time Evolution of Length and Area in General     94
Kinematics of Curve Motion: Three Dimension     101
Problems     102
Geometry of Surfaces     103
Elements of Differential Geometry of Surfaces     105
Covariant Derivative and Connections      112
Geometry of Parametrized Surfaces Embedded in R[subscript 3]     116
Christoffel Symbols and Covariant Differentiation for Hybrid Tensors     118
Compact Surfaces     120
Surface Differential Operators     122
Surface Gradient     123
Surface Divergence     125
Surface Laplacian     126
Surface Curl     127
Integral Relations for Surface Differential Operators     129
Applications     131
Problems     134
Theory of Motion of Surfaces     137
Coordinates and Velocities on a Fluid Surface     137
Geometry of Moving Surfaces     143
Dynamics of Moving Surfaces     145
Boundary Conditions for Moving Fluid Interfaces     148
Dynamics of the Fluid Interfaces     149
Problems     151
Solitons and Nonlinear Waves on Closed Curves and Surfaces
Kinematics of Hydrodynamics     157
Lagrangian vs. Eulerian Frames     157
Introduction     158
Geometrical Picture for Lagrangian vs. Eulerian     159
Fluid Fiber Bundle     161
Introduction     161
Motivation for a Geometrical Approach     164
The Fiber Bundle      167
Fixed Fluid Container     168
Free Surface Fiber Bundle     172
How Does the Time Derivative of Tensors Transform from Euler to Lagrange Frame?     174
Path Lines, Stream Lines, and Particle Contours     178
Eulerian-Lagrangian Description for Moving Curves     184
The Free Surface     184
Equation of Continuity     186
Introduction     186
Solutions of the Continuity Equation on Compact Intervals     192
Problems     198
Dynamics of Hydrodynamics     201
Momentum Conservation: Euler and Navier-Stokes Equations     201
Boundary Conditions     204
Circulation Theorem     206
Surface Tension     212
Physical Problem     212
Minimal Surfaces     214
Application     216
Isothermal Parametrization     219
Topological Properties of Minimal Surfaces     222
General Condition for Minimal Surfaces     224
Surface Tension for Almost Isothermal Parametrization     225
Special Fluids     228
Representation Theorems in Fluid Dynamics     228
Helmholtz Decomposition Theorem in R[subscript 3]      228
Decomposition Formula for Transversal Isotropic Vector Fields     231
Solenoidal-Toroidal Decomposition Formulas     234
Problems     234
Nonlinear Surface Waves in One Dimension     237
KdV Equation Deduction for Shallow Waters     237
Smooth Transitions Between Periodic and Aperiodic Solutions     242
Modified KdV Equation and Generalizations     246
Hydrodynamic Equations Involving Higher-Order Nonlinearities     249
A Compact Version for KdV     249
Small Amplitude Approximation     252
Dispersion Relations     254
The Full Equation     255
Reduction of GKdV to Other Equations and Solutions     257
The Finite Difference Form     261
Boussinesq Equations on a Circle     264
Nonlinear Surface Waves in Two Dimensions     267
Geometry of Two-Dimensional Flow     267
Two-Dimensional Nonlinear Equations     275
Two-Dimensional Fluid Systems with Boundary     278
Oscillations in Two-Dimensional Liquid Drops     281
Contours Described by Quartic Closed Curves     283
Surface Nonlinear Waves in Two-Dimensional Liquid Nitrogen Drops     284
Nonlinear Surface Waves in Three Dimensions      289
Oscillations of Inviscid Drops: The Linear Model     291
Drop Immersed in Another Fluid     293
Drop with Rigid Core     295
Moving Core     301
Drop Volume     305
Oscillations of Viscous Drops: The Linear Model     307
Model 1     308
Nonlinear Three-Dimensional Oscillations of Axisymmetric Drops     322
Nonlinear Resonances in Drop Oscillation     330
Other Nonlinear Effects in Drop Oscillations     340
Solitons on the Surface of Liquid Drops     344
Problems     353
Other Special Nonlinear Compact Systems     355
Nonlinear Compact Shapes and Collective Motion     355
The Hamiltonian Structure for Free Boundary Problems on Compact Surfaces     359
Physical Nonlinear Systems at Different Scales
Filaments, Chains, and Solitons     367
Vortex Filaments     367
Gas Dynamics Filament Model and Solitons     372
Special Solutions     375
Integration of Serret-Frenet Equations for Filaments     377
The Riccati Form of the Serret-Frenet Equations     380
Soliton Solutions on the Vortex Filament     381
Vortex Filaments and the Nonlinear Schrodinger Equation      384
Nonlinear Dynamics of Stiff Chains     387
Problems     390
Solitons on the Boundaries of Microscopic Systems     391
Field Theory Model on a Closed Contour and Instantons     392
Quantization: Excited States     394
Quantization: Instantons and Tunneling     394
Clusters as Solitary Waves on the Nuclear Surface     396
Solitons and Quasimolecular Structure     404
Soliton Model for Heavy Emitted Nuclear Clusters     406
Quintic Nonlinear Schrodinger Equation for Nuclear Cluster Decay     408
Contour Solitons in the Quantum Hall Liquid     411
Perturbative Approach     414
Geometric Approach     417
Nonlinear Contour Dynamics in Macroscopic Systems     423
Plasma Vortex     423
Effective Surface Tension in Magnetohydrochynamics and Plasma Systems     423
Trajectories in Magnetic Field Configurations     424
Magnetic Surfaces in Static Equilibrium     433
Elastic Spheres     440
Nonlinear Evolution of Oscillation Modes in Neutron Stars     441
Mathematical Annex     445
Differentiable Manifolds     445
Riccati Equation     446
Special Functions     446
One-Soliton Solutions for the KdV, MKdV, and Their Combination     448
Scaling and Nonlinear Dispersion Relations     450
References     453
Index     461

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