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| Hardcover ISBN: | 978-0-8218-4897-5 |
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| Hardcover ISBN: | 978-0-8218-4897-5 |
| Product Code: | SURV/156 |
| List Price: | $129.00 |
| MAA Member Price: | $116.10 |
| AMS Member Price: | $103.20 |
| eBook ISBN: | 978-1-4704-1383-5 |
| Product Code: | SURV/156.E |
| List Price: | $125.00 |
| MAA Member Price: | $112.50 |
| AMS Member Price: | $100.00 |
| Hardcover ISBN: | 978-0-8218-4897-5 |
| eBook ISBN: | 978-1-4704-1383-5 |
| Product Code: | SURV/156.B |
| List Price: | $254.00 $191.50 |
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Book DetailsMathematical Surveys and MonographsVolume: 156; 2009; 256 ppMSC: Primary 76; 35; 37; 45; Secondary 34; 47
This book provides a self-contained presentation of classical and new methods for studying wave phenomena that are related to the existence and stability of solitary and periodic travelling wave solutions for nonlinear dispersive evolution equations. Simplicity, concrete examples, and applications are emphasized throughout in order to make the material easily accessible. The list of classical nonlinear dispersive equations studied includes Korteweg-de Vries, Benjamin-Ono, and Schrödinger equations. Many special Jacobian elliptic functions play a role in these examples.
The author brings the reader to the forefront of knowledge about some aspects of the theory and motivates future developments in this fascinating and rapidly growing field. The book can be used as an instructive study guide as well as a reference by students and mature scientists interested in nonlinear wave phenomena.
ReadershipGraduate students and research mathematicians interested in nonlinear wave phenomena.
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Table of Contents
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History, basic models, and travelling waves
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1. Introduction and a brief review of the history
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2. Basic models
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3. Solitary and periodic travelling wave solutions
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Well-posedness and stability definition
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4. Introduction to part 2: well-posedness and stability definition
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5. Initial value problem
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6. Definition of stability
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Stability theory
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7. Introduction to part 3: stability theory
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8. Orbital stability—the classical method
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9. Grillakis-Shatah-Strauss’s stability approach
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The Concentration-Compactness Principle in stability theory
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10. Introduction to part 4: the concentration-compactness principle in the stability theory
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11. Existence and stability of solitary waves for the GBO equations
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12. More about the Concentration-Compactness Principle
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13. Instability of solitary wave solutions
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Stability of periodic travelling waves
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14. Introduction to part 5: stability of periodic travelling waves
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15. Stability of cnoidal waves
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Appendices
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16. Introduction to part 6: appendix
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17. Sobolev spaces and elliptic functions
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18. Operator theory
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Additional Material
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RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Additional Material
- Requests
This book provides a self-contained presentation of classical and new methods for studying wave phenomena that are related to the existence and stability of solitary and periodic travelling wave solutions for nonlinear dispersive evolution equations. Simplicity, concrete examples, and applications are emphasized throughout in order to make the material easily accessible. The list of classical nonlinear dispersive equations studied includes Korteweg-de Vries, Benjamin-Ono, and Schrödinger equations. Many special Jacobian elliptic functions play a role in these examples.
The author brings the reader to the forefront of knowledge about some aspects of the theory and motivates future developments in this fascinating and rapidly growing field. The book can be used as an instructive study guide as well as a reference by students and mature scientists interested in nonlinear wave phenomena.
Graduate students and research mathematicians interested in nonlinear wave phenomena.
-
History, basic models, and travelling waves
-
1. Introduction and a brief review of the history
-
2. Basic models
-
3. Solitary and periodic travelling wave solutions
-
Well-posedness and stability definition
-
4. Introduction to part 2: well-posedness and stability definition
-
5. Initial value problem
-
6. Definition of stability
-
Stability theory
-
7. Introduction to part 3: stability theory
-
8. Orbital stability—the classical method
-
9. Grillakis-Shatah-Strauss’s stability approach
-
The Concentration-Compactness Principle in stability theory
-
10. Introduction to part 4: the concentration-compactness principle in the stability theory
-
11. Existence and stability of solitary waves for the GBO equations
-
12. More about the Concentration-Compactness Principle
-
13. Instability of solitary wave solutions
-
Stability of periodic travelling waves
-
14. Introduction to part 5: stability of periodic travelling waves
-
15. Stability of cnoidal waves
-
Appendices
-
16. Introduction to part 6: appendix
-
17. Sobolev spaces and elliptic functions
-
18. Operator theory
