Nonlinear dispersive equations

272 Pages·2009·1.634 MB·English

by  Jaime Angulo Pava

Mathematical
Surveys
and
Monographs
Volume 156
Nonlinear Dispersive
Equations
Existence and Stability
of Solitary and Periodic
Travelling Wave Solutions
Jaime Angulo Pava
American Mathematical Society
Nonlinear Dispersive
Equations
Existence and Stability
of Solitary and Periodic
Travelling Wave Solutions

Mathematical
Surveys
and
Monographs
Volume 156
Nonlinear Dispersive
Equations
Existence and Stability
of Solitary and Periodic
Travelling Wave Solutions
Jaime Angulo Pava
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
Jerry L. Bona Michael G. Eastwood
Ralph L. Cohen, Chair J. T. Stafford
Benjamin Sudakov
2000 Mathematics Subject Classification. Primary 76B25, 35Q53, 35Q55, 37K45, 76B15,
45M15;Secondary 76B55, 35B10,34D20, 35A15, 47A10, 47A75.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-156
Library of Congress Cataloging-in-Publication Data
Pava,JaimeAngulo,1962–
Nonlineardispersiveequations: existenceandstabilityofsolitaryandperiodictravellingwave
solutions/JaimeAnguloPava.
p.cm. —(Mathematicalsurveysandmonographs;v.156)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-4897-5(alk.paper)
1.Nonlinearwaves. 2.Waveequation—Numericalsolution. 3.Stability. I.Title.
QA927.A54 2009
531(cid:1).1133—dc22
2009022821
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A mi mujer Martha y mi hija Victoria Mel, por supuesto.

Contents
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
vii
viii CONTENTS
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 1D 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.1. Introduction 201
A.2. Lebesgue Space Lp(Ω) 201
A.3. The Fourier Transform in L1(Rn) 201
A.4. The Fourier Transform in L2(Rn) 202
A.5. Tempered Distributions 202
CONTENTS ix
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.1. 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