This new edition has been thoroughly revised, expanded and contain some updates function of the novel results and shift of scientific interest in the topics. The book has a Foreword by Jerry L. Bona and Hongqiu Chen. The book is an introduction to nonlinear waves and soliton theory in the special environment of compact spaces such a closed curves and surfaces and other domain contours. It assumes familiarity with basic soliton theory and nonlinear dynamical systems.
The first part of the book introduces the mathematical concept required for treating the manifolds considered, providing relevant notions from topology and differential geometry. An introduction to the theory of motion of curves and surfaces - as part of the emerging field of contour dynamics - is given.
The second and third parts discuss the modeling of various physical solitons on compact systems, such as filaments, loops and drops made of almost incompressible materials thereby intersecting with a large number of physical disciplines from hydrodynamics to compact object astrophysics.
This book is intended for graduate students and researchers in mathematics, physics and engineering.
Author(s): Andrei Ludu
Series: Springer Series in Synergetics
Edition: 3
Publisher: Springer
Year: 2022
Language: English
Pages: 582
City: Cham
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
