Nonlinear Waves and Solitons on Contours and Closed Surfaces

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001Download PDF (187.3 KB)front-matter Nonlinear Waves and Solitons on Contours and Closed Surfaces Preface to the Second Edition Contents Symbols 002Download PDF (43.6 KB)front-matter Part I Mathematical Prerequisites 003Download PDF (129.0 KB)fulltext Chapter 1: Introduction 1.1 Introduction to Soliton Theory 1.2 Algebraic and Geometric Approaches 1.3 A List of Useful Derivatives in Finite Dimensional Spaces 004Download PDF (362.4 KB)fulltext Chapter 2: Mathematical Prerequisites 2.1 Elements of Topology 2.1.1 Separation Axioms 2.1.2 Compactness 2.1.3 Weierstrass–Stone Theorem 2.1.4 Connectedness, Connectivity, and Homotopy 2.1.5 Separability and Basis 2.1.6 Metric and Normed Spaces 2.2 Elements of Homology 2.3 Group Action 005Download PDF (183.8 KB)fulltext Chapter 3: The Importance of the Boundary 3.1 The Power of Compact Boundaries: Representation Formulas 3.1.1 Representation Formula for n = 1: Taylor Series 3.1.2 Representation Formula for n = 2: Cauchy Formula 3.1.3 Representation Formula for n = 3: Green Formula 3.1.4 Representation Formula in General: Stokes Theorem 3.2 Comments and Examples 006Download PDF (768.1 KB)fulltext Chapter 4: Vector Fields, Differential Forms, and Derivatives 4.1 Manifolds and Maps 4.2 Differential and Vector Fields 4.3 Existence and Uniqueness Theorems: Differential Equation Approach 4.4 Existence and Uniqueness Theorems: Flow Box Approach 4.5 Compact Supported Vector Fields 4.6 Differential Forms and the Lie Derivative 4.7 Differential Systems, Integrability and Invariants 4.8 Poincaré Lemma 4.9 Fiber Bundles and Covariant Derivative 4.9.1 Principal Bundle and Frames 4.9.2 Connection Form and Covariant Derivative 4.10 Tensor Analysis 4.11 The Mixed Covariant Derivative 4.12 Curvilinear Orthogonal Coordinates 4.13 Special Two-Dimensional Nonlinear Orthogonal Coordinates 4.14 Problems 007Download PDF (411.8 KB)fulltext Chapter 5: Geometry of Curves 5.1 Elements of Differential Geometry of Curves 5.2 Closed Curves 5.3 Curves Lying on a Surface 5.4 Problems 008Download PDF (1.8 MB)fulltext Chapter 6: Geometry of Surfaces 6.1 Elements of Differential Geometry of Surfaces 6.2 Covariant Derivative and Connections 6.3 Geometry of Parameterized Surfaces Embedded in R3 6.3.1 Christoffel Symbols and Covariant Differentiation for Hybrid Tensors 6.4 Compact Surfaces 6.5 Surface Differential Operators 6.5.1 Surface Gradient 6.5.2 Surface Divergence 6.5.3 Surface Laplacian 6.5.4 Surface Curl 6.5.5 Integral Relations for Surface Differential Operators 6.5.6 Applications 6.5.6.1 Cylindrical Surfaces 6.5.6.2 Spherical Surfaces 6.5.6.3 Toroidal Surfaces 6.5.6.4 Closed Surfaces 6.6 Problems 009Download PDF (739.0 KB)fulltext Chapter 7: Motion of Curves and Solitons 7.1 Kinematics of Two-Dimensional Curves 7.2 Mapping Two-Dimensional Curve Motion into Nonlinear Integrable Systems 7.3 The Time Evolution of Length and Area 7.4 Cartan Theory of Three-Dimensional Curve Motion 7.5 Kinematics of Three-Dimensional Curves 7.6 Mapping Three-Dimensional Curve Motion into Nonlinear Integrable Systems 7.7 Problems 010Download PDF (308.5 KB)fulltext Chapter 8: Theory of Motion of Surfaces 8.1 Differential Geometry of Surface Motion 8.2 Coordinates and Velocities on a Fluid Surface 8.3 Kinematics of Moving Surfaces 8.4 Dynamics of Moving Surfaces 8.5 Boundary Conditions for Moving Fluid Interfaces 8.6 Dynamics of the Fluid Interfaces 8.7 Problems 011Download PDF (50.1 KB)front-matter Part II Solitons and Nonlinear Waves on Closed Curves and Surfaces 012Download PDF (1.1 MB)fulltext Chapter 9: Kinematics of Hydrodynamics 9.1 Lagrangian vs. Eulerian Frames 9.1.1 Introduction 9.1.2 Geometrical Picture for Lagrangian vs. Eulerian 9.2 Fluid Fiber Bundle 9.2.1 Introduction 9.2.2 Motivation for a Geometrical Approach 9.2.3 The Fiber Bundle 9.2.4 Fixed Fluid Container 9.2.5 Free Surface Fiber Bundle 9.2.6 How Does the Time Derivative of Tensors Transform from Euler to Lagrange Frame? 9.3 Path Lines, Stream Lines, and Particle Contours 9.4 Eulerian–Lagrangian Description for Moving Curves 9.5 The Free Surface 9.6 Equation of Continuity 9.6.1 Introduction 9.6.2 Solutions of the Continuity Equation on Compact Intervals 9.7 Problems 013Download PDF (1.2 MB)fulltext Chapter 10: Dynamics of Hydrodynamics 10.1 Momentum Conservation: Euler and Navier–StokesEquations 10.2 Boundary Conditions 10.3 Circulation Theorem 10.4 Surface Tension 10.4.1 Physical Problem 10.4.2 Minimal Surfaces 10.4.3 Application 10.4.4 Isothermal Parametrization 10.4.5 Topological Properties of Minimal Surfaces 10.4.6 General Condition for Minimal Surfaces 10.4.7 Surface Tension for Almost Isothermal Parametrization 10.5 Special Fluids 10.6 Representation Theorems in Fluid Dynamics 10.6.1 Helmholtz Decomposition Theorem in R3 10.6.2 Decomposition Formula for Transversal Isotropic Vector Fields 10.6.3 Solenoidal–Toroidal Decomposition Formulas 10.7 Problems 014Download PDF (703.7 KB)fulltext Chapter 11: Nonlinear Surface Waves in One Dimension 11.1 KdV Equation Deduction for Shallow Waters 11.2 Smooth Transitions Between Periodic and Aperiodic Solutions 11.3 Modified KdV Equation and Generalizations 11.4 Hydrodynamic Equations Involving Higher-Order Nonlinearities 11.4.1 A Compact Version for KdV 11.4.2 Small Amplitude Approximation 11.4.3 Dispersion Relations 11.4.4 The Full Equation 11.4.5 Reduction of GKdV to Other Equations and Solutions 11.4.6 The Finite Difference Form 11.5 Boussinesq Equations on a Circle 015Download PDF (1.1 MB)fulltext Chapter 12: Nonlinear Surface Waves in Two Dimensions 12.1 Geometry of Two-Dimensional Flow 12.2 Two-Dimensional Nonlinear Equations 12.3 Two-Dimensional Fluid Systems with Boundary 12.4 Oscillations in Two-Dimensional Liquid Drops 12.5 Contours Described by Quartic Closed Curves 12.6 Surface Nonlinear Waves in Two-Dimensional Liquid Nitrogen Drops 016Download PDF (4.7 MB)fulltext Chapter 13: Nonlinear Surface Waves in Three Dimensions 13.1 Oscillations of Inviscid Drops: The Linear Model 13.1.1 Drop Immersed in Another Fluid 13.1.2 Drop with Rigid Core 13.1.3 Moving Core 13.1.4 Drop Volume 13.2 Oscillations of Viscous Drops: The Linear Model 13.2.1 Model 1 13.3 Nonlinear Three-Dimensional Oscillations of Axisymmetric Drops 13.3.1 Nonlinear Resonances in Drop Oscillation 13.4 Other Nonlinear Effects in Drop Oscillations 13.5 Solitons on the Surface of Liquid Drops 13.6 Problems 017Download PDF (216.1 KB)fulltext Chapter 14: Other Special Nonlinear Compact Systems 14.1 Nonlinear Compact Shapes and Collective Motion 14.2 The Hamiltonian Structure for Free Boundary Problems on Compact Surfaces 018Download PDF (42.0 KB)front-matter Part III Physical Nonlinear Systems at Different Scales 019Download PDF (676.3 KB)fulltext Chapter 15: Filaments, Chains, and Solitons 15.1 Vortex Filaments 15.1.1 Gas Dynamics Filament Model and Solitons 15.1.2 Special Solutions 15.1.3 Integration of Serret–Frenet Equations for Filaments 15.1.4 The Riccati Form of the Serret–Frenet Equations 15.2 Soliton Solutions on the Vortex Filament 15.2.1 Constant Torsion Vortex Filaments 15.2.2 Vortex Filaments and the Nonlinear Schrödinger Equation 15.3 Closed Curves Solitons 15.4 Nonlinear Dynamics of Stiff Chains 15.5 Problems 020Download PDF (992.6 KB)fulltext Chapter 16: Solitons on the Boundaries of Microscopic Systems 16.1 Solitons as Elementary Particles 16.2 Quantization of Solitons on a Closed Contour and Instantons 16.3 Clusters as Solitary Waves on the Nuclear Surface 16.4 Solitons and Quasimolecular Structure 16.5 Soliton Model for Heavy Emitted Nuclear Clusters 16.5.1 Quintic Nonlinear Schrödinger Equation for Nuclear Cluster Decay 16.6 Contour Solitons in the Quantum Hall Liquid 16.6.1 Perturbative Approach 16.6.2 Geometric Approach 021Download PDF (1.1 MB)fulltext Chapter 17: Nonlinear Contour Dynamics in Macroscopic Systems 17.1 Plasma Vortex 17.1.1 Effective Surface Tension in Magnetohydrodynamics and Plasma Systems 17.1.2 Trajectories in Magnetic Field Configurations 17.1.3 Magnetic Surfaces in Static Equilibrium 17.2 Elastic Spheres 17.3 Curvature Dependent Nonlinear Diffusion on Closed Surfaces 17.4 Nonlinear Evolution of Oscillation Modesin Neutron Stars 022Download PDF (127.1 KB)fulltext Chapter 18: Mathematical Annex 18.1 Differentiable Manifolds 18.2 Riccati Equation 18.3 Special Functions 18.4 One-Soliton Solutions for the KdV, MKdV, and Their Combination 18.5 Scaling and Nonlinear Dispersion Relations 023Download PDF (165.5 KB)back-matter References Index