Preface ........................................................ ix
Chapter 1. Introduction ....................................... xi
Chapter 2. Preliminaries ....................................... 1
1. Laplace transform ................................... 1
2. Sobolev spaces Hk(R+) ............................... 6
Chapter 3. General Theory ...................................... 9
1. Pseudodifferential Operator on a Half-Line .......... 9
2. Boundary Value Problem on a Half-Line .............. 12
Chapter 4. Nonlinear Schrodinger Type Equations ............... 29
1. Setting of the problem ............................. 29
2. Linear problem ..................................... 34
3. Local existence for nonlinear problem .............. 45
4. Asymptotics determined by the boundary data ........ 47
5. Asymptotics determined by nonlinearity ............. 57
Chapter 5. Whitham Equation ................................... 71
1. Introduction ....................................... 71
2. Linear problem ..................................... 73
3. Preliminaries ...................................... 75
4. Local existence .................................... 81
5. Large time asymptotics ............................. 85
Chapter 6. Korteweg-de Vries-Burgers Equation ................. 93
1. Introduction ....................................... 93
2. Linear problem ..................................... 95
3. Preliminaries ...................................... 99
4. Local existence for linear case ................... 111
5. Local existence for nonlinear problem ............. 114
6. Large time asymptotics ............................ 116
Chapter 7. Large Initial Data ................................ 121
1 Introduction ...................................... 121
2. Linear problem .................................... 122
3. Local existence ................................... 126
4. Preliminary estimates ............................. 128
5. Global existence .................................. 132
Chapter 8. KdV-B Type Equation ............................... 139
1. Setting of the problem ............................ 139
2. Linear problem .................................... 140
3. Energy estimate ................................... 143
4. Local existence ................................... 144
5. Large time asymptotics ............................ 145
Chapter 9. Dirichlet Problem for KdV Equation ................ 149
1. Introduction ...................................... 149
2. Linear problem .................................... 153
3. Preliminaries ..................................... 153
4. Global existence .................................. 161
Chapter 10. Neumann Problem for KdV Equation .................. 171
1. Introduction ...................................... 171
2. Linear problem .................................... 174
3. Preliminaries ..................................... 179
4. Local existence ................................... 194
5. Proof of Theorem 31 ............................... 197
Chapter 11. Landau-Ginzburg Equations ......................... 205
1. Introduction ...................................... 205
2. Preliminaries ..................................... 208
3. Proof of Theorem 34 ............................... 214
Chapter 12. Burgers Equation with Pumping ..................... 221
1. Introduction ...................................... 221
2. Rarefaction wave .................................. 224
3. Shock wave ........................................ 230
4. Zero boundary conditions .......................... 233
Chapter 13. KdVB Equation on a Segment ........................ 245
1. Introduction ...................................... 245
2. Linear problem .................................... 246
3. Local existence for the nonlinear problem ......... 255
4. Large time asymptotics ............................ 267
5. Large initial data ................................ 259
Chapter 14. NLS Equation on Segment ........................... 261
1. Introduction ...................................... 261
2. Linear problem .................................... 265
3 Global existence ................................... 271
Chapter 15. Periodic Problem .................................. 275
1. Introduction ...................................... 275
2. Preliminary estimates ............................. 285
3. Proof of theorems ................................. 292
Bibliography .................................................. 309
Index ......................................................... 317
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