Nonlinear Waves and Solitons on Contours and Closed Surfaces

Foreword
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Contents
Symbols
1 Introduction
	1.1 Intuitive Introduction to Nonlinear Waves and Solitons
	1.2 Integrability
	1.3 Algebraic and Geometric Approaches
	1.4 A List of Useful Derivatives in Finite Dimensional Spaces
	References
Part I Mathematical Prerequisites
2 Topology and Algebra
	2.1 What Is Topology
		2.1.1 Topological Spaces and Separation
		2.1.2 Compactness and Weierstrass-Stone Theorem
		2.1.3 Connectedness and Homotopy
		2.1.4 Separability and Metric Spaces
	2.2 Elements of Homology
	2.3 Group Action
	References
3 Vector Fields, Differential Forms, and Derivatives
	3.1 Manifolds and Maps
	3.2 Differential and Vector Fields
	3.3 Existence and Uniqueness Theorems: Differential Equation Approach
	3.4 Existence and Uniqueness Theorems: Flow Box Approach
	3.5 Compact Supported Vector Fields
	3.6 Differential Forms and the Lie Derivative
	3.7 Differential Systems, Integrability and Invariants
	3.8 Poincaré Lemma
	3.9 Fiber Bundles and Covariant Derivative
		3.9.1 Principal Bundle and Frames
		3.9.2 Connection Form and Covariant Derivative
	3.10 Tensor Analysis
	3.11 The Mixed Covariant Derivative
	3.12 Curvilinear Orthogonal Coordinates
	3.13 Special Two-Dimensional Nonlinear Orthogonal Coordinates
	3.14 Problems
	References
4 The Importance of the Boundary
	4.1 The Power of Compact Boundaries: Representation Formulas
		4.1.1 Representation Formula for n=1: Taylor Series
		4.1.2 Representation Formula for n=2: Cauchy Formula
		4.1.3 Representation Formula for n=3: Green Formula
		4.1.4 Representation Formula in General: Stokes Theorem
	4.2 Comments and Examples
	References
Part II Curves and Surfaces
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
	References
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 mathbbR3
		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
	6.6 Problems
	References
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
	References
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
	References
Part III Solitons and Nonlinear Waves on Closed Curves and Surfaces
9 Kinematics of Fluids
	9.1 Lagrangian Verses Eulerian Frames
		9.1.1 Introduction
		9.1.2 Geometrical Picture for Lagrangian Verses 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
	References
10 Hydrodynamics
	10.1 Momentum Conservation: Euler and Navier–Stokes Equations
	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 mathbbR3
		10.6.2 Decomposition Formula for Transversal Isotropic Vector Fields
		10.6.3 Solenoidal–Toroidal Decomposition Formulas
	10.7 Problems
	References
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
	References
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 Moving Boundary
	12.4 Oscillations in Two-Dimensional Liquid Drops
	12.5 Contours Described by Quartic Closed Curves
	12.6 Nonlinear Waves in Rotating Leidenfrost Drops
	References
13 Dynamics of Two-Dimensional Fluid in Bounded Domain via Conformal Variables (A. Chernyavsky  and S. Dyachenko)
	13.1 Introduction
	13.2 Mechanics of Droplet and the Conformal Map
		13.2.1 The Hamiltonian, Momentum and Angular Momentum
		13.2.2 The Center of Mass
	13.3 The Complex Equations of Motion
		13.3.1 Kinematic Equation
		13.3.2 Dynamic Condition
	13.4 Traveling Waves Around a Disk
	13.5 Linear Waves
	13.6 Numerical Simulation
	13.7 Series Solution
	13.8 Nonlinear Waves
	13.9 Conclusion
	References
14 Nonlinear Surface Waves in Three Dimensions
	14.1 Oscillations of Inviscid Drops: The Linear Model
		14.1.1 Drop Immersed in Another Fluid
		14.1.2 Drop with Rigid Core
		14.1.3 Moving Core
		14.1.4 Drop Volume
	14.2 Oscillations of Viscous Drops: The Linear Model
		14.2.1 Model 1
	14.3 Nonlinear Three-Dimensional Oscillations of Axisymmetric Drops
		14.3.1 Nonlinear Resonances in Drop Oscillation
	14.4 Other Nonlinear Effects in Drop Oscillations
	14.5 Solitons on the Surface of Liquid Drops
	14.6 Problems
	References
15 Other Special Nonlinear Compact Systems
	15.1 Solitons on Interfaces of Layered Fluid Droplet (Written by A. S. Carstea)
	15.2 Nonlinear Compact Shapes and Collective Motion
	15.3 The Hamiltonian Structure for Free Boundary Problems on Compact Surfaces
	References
Part IV Physical Nonlinear Systems at Different Scales
16 Filaments, Chains, and Solitons
	16.1 Vortex Filaments
		16.1.1 Gas Dynamics Filament Model and Solitons
		16.1.2 Special Solutions
		16.1.3 Integration of Serret–Frenet Equations for Filaments
		16.1.4 The Riccati Form of the Serret–Frenet Equations
	16.2 Soliton Solutions on the Vortex Filament
		16.2.1 Constant Torsion Vortex Filaments
		16.2.2 Vortex Filaments and the Nonlinear Schrödinger Equation
	16.3 Closed Curves Solitons
	16.4 Nonlinear Dynamics of Stiff Chains
	16.5 Problems
	References
17 Solitons on the Boundaries of Microscopic Systems
	17.1 Solitons as Elementary Particles
	17.2 Quantization of Solitons on a Closed Contour and Instantons
	17.3 Clusters as Solitary Waves on the Nuclear Surface
	17.4 Nonlinear Schrödinger Equation Solitons on Quantum …
	17.5 Solitons and Quasimolecular Structure
	17.6 Soliton Model for Heavy Emitted Nuclear Clusters
	17.7 Quintic Nonlinear Schrödinger Equation for Nuclear Cluster Decay
	17.8 Contour Solitons in the Quantum Hall Liquid
	References
18 Nonlinear Contour Dynamics in Macroscopic Systems
	18.1 Plasma Vortex
		18.1.1 Effective Surface Tension in Magnetohydrodynamics and Plasma Systems
		18.1.2 Trajectories in Magnetic Field Configurations
		18.1.3 Magnetic Surfaces in Static Equilibrium
	18.2 Elastic Spheres
	18.3 Curvature Dependent Nonlinear Diffusion on Closed Surfaces
	18.4 Nonlinear Evolution of Oscillation Modes in Neutron Stars
	References
19 Mathematical Appendix
	19.1 Differentiable Manifolds
	19.2 Riccati Equation
	19.3 Special Functions
	19.4 One-Soliton Solutions for the KdV, MKdV, and Their Combination
	19.5 Scaling and Nonlinear Dispersion Relations1
	References
Index