| Preface |
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xi | |
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Part
1. History, Basic Models, and Travelling Waves |
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1 | (46) |
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Introduction and a Brief Review of the History |
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3 | (14) |
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17 | (8) |
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17 | (1) |
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17 | (5) |
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22 | (3) |
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Solitary and Periodic Travelling Wave Solutions |
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25 | (22) |
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25 | (1) |
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Travelling Wave Solutions |
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25 | (2) |
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27 | (12) |
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The Poisson Summation Theorem and Periodic Wave Solutions |
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39 | (3) |
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42 | (5) |
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Part
2. Well-Posedness and Stability Definition |
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47 | (20) |
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49 | (12) |
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49 | (1) |
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Some Results about Well-Posedness |
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49 | (8) |
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Some Results about Global Well-Posedness |
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57 | (1) |
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58 | (3) |
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61 | (6) |
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61 | (1) |
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61 | (3) |
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64 | (3) |
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67 | (36) |
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Orbital Stability---the Classical Method |
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69 | (22) |
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69 | (1) |
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Stability of Solitary Wave Solutions for the GKdV |
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70 | (11) |
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``Stability of the Blow-up'' for a Class of KdV Equations |
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81 | (6) |
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87 | (4) |
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Grillakis-Shatah-Strauss's Stability Approach |
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91 | (12) |
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91 | (1) |
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Geometric Overview of the Theory |
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91 | (2) |
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Stability of Solitary Wave Solutions |
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93 | (5) |
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Stability of Solitary Waves for KdV-Type Equations |
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98 | (1) |
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On Albert-Bona's Spectrum Approach |
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99 | (1) |
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100 | (3) |
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Part
4. The Concentration-Compactness Principle in Stability Theory |
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103 | (56) |
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Existence and Stability of Solitary Waves for the GBO |
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105 | (22) |
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105 | (2) |
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Solitary Waves for the GBO |
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107 | (12) |
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Stability of Solitary Waves for the GBO Equations |
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119 | (5) |
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124 | (3) |
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More about the Concentration-Compactness Principle |
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127 | (10) |
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127 | (1) |
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Solitary Wave Solutions of Benjamin-Type Equations |
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127 | (1) |
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Stability of Solitary Wave Solutions: the GKdV Equations |
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128 | (1) |
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Stability of Solitary Wave Solutions: the Benjamin Equation |
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129 | (4) |
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Stability of Solitary Wave Solutions: the Fourth-Order Equation |
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133 | (1) |
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Stability of Solitary Wave Solutions: the GKP-I Equations |
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133 | (2) |
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135 | (2) |
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Instability of Solitary Wave Solutions |
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137 | (22) |
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137 | (2) |
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Instability of Solitary Wave Solutions: the GB Equations |
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139 | (11) |
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Fifth-Order Korteweg-de Vries Equations |
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150 | (2) |
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A Generalized Class of Benjamin Equations |
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152 | (1) |
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Linear Instability and Nonlinear Instability |
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153 | (4) |
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157 | (2) |
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Part
5. Stability of Periodic Travelling Waves |
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159 | (40) |
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Stability of Cnoidal Waves |
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161 | (38) |
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161 | (3) |
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Stability of Cnoidal Waves with Mean Zero for KdV Equation |
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164 | (10) |
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Stability of Constant Solutions for the KdV Equation |
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174 | (3) |
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Cnoidal Waves for the 1D Benney-Luke Equation |
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177 | (6) |
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Angulo and Natali's Stability Approach |
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183 | (13) |
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196 | (3) |
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199 | (46) |
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Appendix A. Sobolev Spaces and Elliptic Functions |
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201 | (10) |
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201 | (1) |
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201 | (1) |
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The Fourier Transform in L1 (Rn) |
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201 | (1) |
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The Fourier Transform in L2 (Rn) |
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202 | (1) |
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202 | (2) |
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204 | (2) |
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Sobolev Spaces of Periodic Type |
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206 | (1) |
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The Symmetric Decreassing Rearrangement |
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207 | (1) |
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The Jacobian Elliptic Functions |
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208 | (3) |
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Appendix B. Operator Theory |
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211 | (34) |
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211 | (1) |
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Closed Linear Operators: Basic Theory |
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211 | (18) |
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Pseudo-Differential Operators and Their Spectrum |
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229 | (2) |
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Spectrum of Linear Operators Associated to Solitary Waves |
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231 | (6) |
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237 | (3) |
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240 | (5) |
| Bibliography |
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245 | (10) |
| Index |
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255 | |
Lisainfo e-raamatute kohta