Part I Mathematical Prerequisites
Introduction .................................................... 5
1.1 Introduction to Soliton Theory .......................... 5
1.2 Algebraic and Geometric Approaches ...................... 6
1.3 A List of Useful Derivatives ............................ 8
2 Mathematical Prerequisites .................................. 11
2.1 Elements of Topology ................................... 11
2.1.1 Separation Axioms ............................... 12
2.1.2 Compactness ..................................... 15
2.1.3 Weierstrass Stone Theorem ....................... 17
2.1.4 Connectedness, Connectivity, and Homotopy ....... 18
2.1.5 Separability and Basis .......................... 20
2.1.6 Metric and Normed Spaces ........................ 20
2.2 Elements of Homology ................................... 21
3 The Importance of the Boundary .............................. 23
3.1 The Power of Compact Boundaries:
Representation Formulas ................................ 23
3.1.1 Representation Formula for n = 1:
Taylor Series ................................... 24
3.1.2 Representation Formula for n = 2:
Cauchy Formula .................................. 24
3.1.3 Representation Formula for n = 3:
Green Formula ................................... 25
3.1.4 Representation Formula in General:
Stokes Theorem .................................. 26
3.2 Comments and Examples .................................. 28
4 Vector Fields, Differential Forms,
and Derivatives ............................................. 31
4.1 Manifolds and Maps ..................................... 32
4.2 Differential and Vector Fields ......................... 35
4.3 Existence and Uniqueness Theorems:
Differential Equation Approach ......................... 39
4.4 Existence and Uniqueness Theorems: Flow
Box Approach ........................................... 45
4.5 Compact Supported Vector Fields ........................ 47
4.6 Lie Derivative and Differential Forms .................. 47
4.7 Invariants ............................................. 52
1.8 Fiber Bundles ......................................... 54
4.9 Poincare Lemma ......................................... 57
4.10 Tensor Analysis. Covariant Derivative,
and Connections ........................................ 58
4.11 The Mixed Covariant Derivative ......................... 60
4.12 Curvilinear Orthogonal Coordinates ..................... 62
4.12.1 Gradient ........................................ 64
4.12.2 Divergence ...................................... 64
4.12.3 Curl ............................................ 65
4.12.4 Laplacian ....................................... 65
4.12.5 Special Two-Dimensional Nonlinear
Orthogonal Coordinates .......................... 66
4.13 Problems ............................................... 67
5 Geometry of Curves .......................................... 69
5.1 Elements of Differential Geometry of
Curves ................................................. 69
5.2 Closed Curves .......................................... 76
5.3 Curves Lying on a Surface .............................. 78
5.4 Problems ............................................... 79
6 Motion of Curves and Solitons ............................... 81
6.1 Nonlinear Kinematics of Two-Dimensional
Curves and Solitons .................................... 82
6.1.1 The Time Evolution of Length and
Area in General .................................. 94
6.2 Kinematics of Curve Motion: Three Dimension ........... 101
6.3 Problems .............................................. 102
7 Geometry of Surfaces ....................................... 103
7.1 Elements of Differential Geometry of Surfaces ......... 105
7.2 Covariant Derivative and Connections .................. 112
7.3 Geometry of Parametrized Surfaces Embedded
in R3 ................................................. 116
7.3.1 Christ off el Symbols and Covariant
Differentiation for Hybrid Tensors ............. 118
7.4 Compact Surfaces ...................................... 120
7.5 Surface Differential Operators ........................ 122
7.5.1 Surface Gradient ............................... 123
7.5.2 Surface Divergence ............................. 125
7.5.3 Surface Laplacian .............................. 126
7.5.4 Surface Curl ................................... 127
7.5.5 Integral Relations for Surface
Differential Operators ......................... 129
7.5.6 Applications ................................... 131
7.6 Problems ............................................... 134
8 Theory of Motion of Surfaces ............................... 137
8.1 Coordinates and Velocities on a
Fluid Surface ......................................... 137
8.2 Geometry of Moving Surfaces ........................... 143
8.3 Dynamics of Moving Surfaces ........................... 145
8.4 Boundary Conditions for Moving Fluid
Interfaces ............................................ 148
8.5 Dynamics of the Fluid Interfaces ...................... 149
8.6 Problems .............................................. 151
Part II Solitons and Nonlinear Waves on Closed
Curves and Surfaces
9 Kinematics of Hydrodynamics ................................ 157
9.1 Lagrangian vs. Eulerian Frames ........................ 157
9.1.1 Introduction ................................... 158
9.1.2 Geometrical Picture for Lagrangian
vs. Eulerian ................................... 159
9.2 Fluid Fiber Bundle .................................... 161
9.2.1 Introduction ................................... 161
9.2.2 Motivation for a Geometrical Approach .......... 164
9.2.3 The Fiber Bundle ............................... 167
9.2.4 Fixed Fluid Container .......................... 168
9.2.5 Free Surface Fiber Bundle ...................... 172
9.2.6 How Does the Time Derivative of Tensors
Transform from Euler to Lagrange Frame? ........ 174
9.3 Path Lines. Stream Lines, and Particle
Contours .............................................. 178
9.4 Eulerian Lagrangian Description for
Moving Curves ......................................... 184
9.5 The Free Surface ...................................... 184
9.6 Equation of Continuity ................................ 186
9.6.1 Introduction ................................... 186
9.6.2 Solutions of the Continuity Equation
on Compact Intervals ........................... 192
9.7 Problems .............................................. 198
10 Dynamics of Hydrodynamics ................................. 201
10.1 Momentum Conservation: Euler and Navier Stokes
Equations ............................................. 201
10.2 Boundary Conditions ................................... 204
10.3 Circulation Theorem ................................... 206
10.4 Surface Tension ....................................... 212
10.4.1 Physical Problem ............................... 212
10.4.2 Minimal Surfaces ............................... 214
10.4.3 Application .................................... 216
10.4.4 Isothermal Parametrization ..................... 219
10.4.5 Topological Properties of
Minimal Surfaces ............................... 222
10.4.6 General Condition for Minimal
Surfaces ....................................... 224
10.4.7 Surface Tension for Almost
Isothermal Parametrization ..................... 225
10.5 Special Fluids ........................................ 228
10.6 Representation Theorems in Fluid Dynamics ............. 228
10.6.1 Helmholtz Decomposition Theorem
in R3 .......................................... 228
10.6.2 Decomposition Formula for Transversal
Isotropic Vector Fields ........................ 231
10.6.3 Solenoidal-Toroidal Decomposition
Formulas ....................................... 234
10.7 Problems .............................................. 234
11 Nonlinear Surface Waves in One Dimension .................. 237
11.1 KdV Equation Deduction for Shallow Waters ............. 237
11.2 Smooth Transitions Between Periodic and
Aperiodic Solutions ................................... 242
11.3 Modified KdV Equation and Generalizations ............. 246
11.4 Hydrodynamic Equations Involving Higher-Order
Nonlinearities ........................................ 249
11.4.1 A Compact Version for KdV ..................... 249
11.4.2 Small Amplitude Approximation .................. 252
11.4.3 Dispersion Relations ........................... 254
11.4.4 The Full Equation .............................. 255
11.4.5 Reduction of GKdV to Other Equations
and Solutions .................................. 257
11.4.6 The Finite Difference Form ..................... 261
11.5 Boussinesq Equations on a Circle ...................... 264
12 Nonlinear Surface Waves in Two Dimensions .................. 267
12.1 Geometry of l"wo-Dimensional Flow ..................... 267
12.2 Two-Dimensional Nonlinear Equations ................... 275
12.3 Two-Dimensional Fluid Systems with Boundary ........... 278
12.4 Oscillations in Two-Dimensional Liquid Drops .......... 281
12.5 Contours Described by Quartic Closed Curves ........... 283
12.6 Surface Nonlinear Waves in Two-Dimensional
Liquid Nitrogen Drops ................................. 284
13 Nonlinear Surface Waves in Three Dimensions ................ 289
13.1 Oscillations of Inviscid Drops: The Linear
Model ................................................. 291
13.1.1 Drop Immersed in Another Fluid ................. 293
13.1.2 Drop with Rigid Core ........................... 295
13.1.3 Moving Core .................................... 301
13.1.4 Drop Volume .................................... 305
13.2 Oscillations of Viscous Drops: The Linear
Model ................................................. 307
13.2.1 Model 1 ........................................ 308
13.3 Nonlinear Three-Dimensional Oscillations
of Axisymmetric Drops ................................. 322
13.3.1 Nonlinear Resonances in Drop Oscillation ....... 330
13.4 Other Nonlinear Effects in Drop Oscillations .......... 340
13.5 Solitons on the Surface of Liquid Drops ............... 344
13.6 Problems .............................................. 353
14 Other Special Nonlinear Compact Systems .................... 355
14.1 Nonlinear Compact Shapes and Collective Motion ........ 355
14.2 The Hamiltonian Structure for Free Boundary
Problems on Compact Surfaces .......................... 359
Part III Physical Nonlinear Systems at Different Scales
15 Filaments, Chains, and Solitons ............................ 367
15.1 Vortex Filaments ...................................... 367
15.1.1 Gas Dynamics Filament Model and Solitons ....... 372
15.1.2 Special Solutions .............................. 375
15.1.3 Integration of Serret-Frenet Equations for
Filaments ...................................... 377
15.1.4 The Riccati Form of the
Serret-Frenet Equations ........................ 380
15.1.5 Soliton Solutions on the
Vortex Filament ................................ 381
15.1.6 Vortex Filaments and the Nonlinear
Schrodinger Equation ........................... 384
15.2 Nonlinear Dynamics of Stiff Chains .................... 387
15.3 Problems .............................................. 390
16 Solitons on the Boundaries of Microscopic Systems .......... 391
16.1 Field Theory Model on a Closed Contour
and Instantons ........................................ 392
16.1.1 Quantization: Excited States ................... 394
16.1.2 Quantization: Instantons and
Tunneling ...................................... 394
16.2 Clusters as Solitary Waves on the Nuclear
Surface ............................................... 396
16.3 Solitons and Quasimolecular Structure ................. 404
16.4 Soliton Model for Heavy Emitted Nuclear
Clusters .............................................. 406
16.4.1 Quintic Nonlinear Schrodinger Equation
for Nuclear Cluster Decay ...................... 408
16.5 Contour Solitons in the Quantum Hall Liquid ........... 411
16.5.1 Perturbative Approach .......................... 414
16.5.2 Geometric Approach ............................. 417
17 Nonlinear Contour Dynamics in Macroscopic
Systems .................................................... 423
17.1 Plasma Vortex ......................................... 423
17.1.1 Effective Surface Tension in
Magnetohydrodynamics and Plasma
Systems ........................................ 423
17.1.2 Trajectories in Magnetic Field
Configurations ................................. 424
17.1.3 Magnetic Surfaces in Static
Equilibrium .................................... 433
17.2 Elastic Spheres ....................................... 440
17.2.1 Nonlinear Evolution of Oscillation
Modes in Neutron Stars ......................... 441
18 Mathematical Annex ......................................... 445
18.1 Differentiable Manifolds .............................. 445
18.2 Riccati Equation ...................................... 446
18.3 Special Functions ..................................... 446
18.4 One-Soliton Solutions for the KdV, MKdV,
and Their Combination ................................. 448
18.5 Scaling and Nonlinear Dispersion
Relations ............................................. 450
References .................................................... 453
Index ......................................................... 461 |
