Hardcover ISBN:  9780821848975 
Product Code:  SURV/156 
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AMS Member Price:  $70.40 
Electronic ISBN:  9781470413835 
Product Code:  SURV/156.E 
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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 selfcontained 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 Kortewegde Vries, BenjaminOno, 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.

Table of Contents

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

Wellposedness and stability definition

4. Introduction to part 2: wellposedness 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. GrillakisShatahStrauss’s stability approach

The ConcentrationCompactness Principle in stability theory

10. Introduction to part 4: the concentrationcompactness principle in the stability theory

11. Existence and stability of solitary waves for the GBO equations

12. More about the ConcentrationCompactness 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


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This book provides a selfcontained 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 Kortewegde Vries, BenjaminOno, 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

Wellposedness and stability definition

4. Introduction to part 2: wellposedness 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. GrillakisShatahStrauss’s stability approach

The ConcentrationCompactness Principle in stability theory

10. Introduction to part 4: the concentrationcompactness principle in the stability theory

11. Existence and stability of solitary waves for the GBO equations

12. More about the ConcentrationCompactness 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