Preface ........................................................ xi
Part 1. History, Basic Models, and Travelling Waves ............ 1
Chapter 1. Introduction and a Brief Review of the History ...... 3
Chapter 2. Basic Models ....................................... 17
2.1. Introduction ............................................. 17
2.2. Models ................................................... 17
2.3. Comments ................................................. 22
Chapter 3. Solitary and Periodic Travelling Wave Solutions .... 25
3.1. Introduction ............................................. 25
3.2. Travelling Wave Solutions ................................ 25
3.3. Examples ................................................. 27
3.4. The Poisson Summation Theorem and Periodic Wave
Solutions ................................................ 39
3.5. Comments ................................................. 42
Part 2. Well-Posedness and Stability Definition ............... 47
Chapter 4. Initial Value Problem .............................. 49
4.1. Introduction ............................................. 49
4.2. Some Results about Well-Posedness ........................ 49
4.3. Some Results about Global Well-Posedness ................. 57
4.4. Comments ................................................. 58
Chapter 5. Definition of Stability ............................ 61
5.1. Introduction ............................................. 61
5.2. Orbital Stability ........................................ 61
5.3. Comments ................................................. 64
Part 3. Stability Theory ...................................... 67
Chapter 6. Orbital Stability-the Classical Method ............. 69
6.1. Introduction ............................................. 69
6.2. Stability of Solitary Wave Solutions for the GKdV ........ 70
6.3. "Stability of the Blow-up" for a Class of KdV
Equations ................................................ 81
6.4. Comments ................................................. 87
Chapter 7. Grillakis-Shatah-Strauss's Stability Approach ...... 91
7.1. Introduction ............................................. 91
7.2. Geometric Overview of the Theory ......................... 91
7.3. Stability of Solitary Wave Solutions ..................... 93
7.4. Stability of Solitary Waves for KdV-Type Equations ....... 98
7.5. On Albert-Bona's Spectrum Approach ....................... 99
7.6. Comments ................................................ 100
Part 4. The Concentration-Compactness Principle in Stability
Theory ........................................................ 103
Chapter 8. Existence and Stability of Solitary Waves for
the GBO ........................................... 105
8.1. Introduction ............................................ 105
8.2. Solitary Waves for the GBO .............................. 107
8.3. Stability of Solitary Waves for the GBO Equations ....... 119
8.4. Comments ................................................ 124
Chapter 9. More about the Concentration-Compactness
Principle ......................................... 127
9.1. Introduction ............................................ 127
9.2. Solitary Wave Solutions of Benjamin-Type Equations ...... 127
9.3. Stability of Solitary Wave Solutions: the GKdV
Equations ............................................... 128
9.4. Stability of Solitary Wave Solutions: the Benjamin
Equation ................................................ 129
9.5. Stability of Solitary Wave Solutions: the Fourth-Order
Equation ................................................ 133
9.6. Stability of Solitary Wave Solutions: the GKP-I
Equations ............................................... 133
9.7. Comments ................................................ 135
Chapter 10. Instability of Solitary Wave Solutions ............ 137
10.1. Introduction ............................................ 137
10.2. Instability of Solitary Wave Solutions: the GB
Equations ............................................... 139
10.3. Fifth-Order Korteweg-de Vries Equations ................. 150
10.4. A Generalized Class of Benjamin Equations ............... 152
10.5. Linear Instability and Nonlinear Instability ............ 153
10.6. Comments ................................................ 157
Part 5. Stability of Periodic Travelling Waves ............... 159
Chapter 11. Stability of Cnoidal Waves ........................ 161
11.1. Introduction ............................................ 161
11.2. Stability of Cnoidal Waves with Mean Zero for KdV
Equation ................................................ 164
11.3. Stability of Constant Solutions for the KdV Equation .... 174
11.4. Cnoidal Waves for the ID Benney-Luke Equation ........... 177
11.5. Angulo and Natali's Stability Approach .................. 183
11.6. Comments ................................................ 196
Part 6. APPENDICES ........................................... 199
Appendix A. Sobolev Spaces and Elliptic Functions ............ 201
A.l. Introduction ............................................ 201
A.2. Lebesgue Space LP(Ω) .................................... 201
A.3. The Fourier Transform in L1( n) ......................... 201
A.4. The Fourier Transform in L2(n) ......................... 202
A.5. Tempered Distributions .................................. 202
A.6. Sobolev Spaces .......................................... 204
A.7. Sobolev Spaces of Periodic Type ......................... 206
A.8. The Symmetric Decreasing Rearrangement .................. 207
A.9. The Jacobian Elliptic Functions ....................... 208
Appendix B. Operator Theory .................................. 211
B.l. Introduction ............................................ 211
B.2. Closed Linear Operators: Basic Theory ................... 211
B.3. Pseudo-Differential Operators and Their Spectrum ........ 229
B.4. Spectrum of Linear Operators Associated to Solitary
Waves ................................................... 231
B.5. Sturm-Liouville Theory .................................. 237
B.6. Floquet Theory .......................................... 240
Bibliography .................................................. 245
Index ......................................................... 255
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