
Book DetailsStudent Mathematical LibraryVolume: 54; 2010; 304 ppMSC: Primary 35; 37; 14; 15
Now available in Second Edition: STML/100
Solitons are explicit solutions to nonlinear partial differential equations exhibiting particlelike behavior. This is quite surprising, both mathematically and physically. Waves with these properties were once believed to be impossible by leading mathematical physicists, yet they are now not only accepted as a theoretical possibility but are regularly observed in nature and form the basis of modern fiberoptic communication networks.
Glimpses of Soliton Theory addresses some of the hidden mathematical connections in soliton theory which have been revealed over the last halfcentury. It aims to convince the reader that, like the mirrors and hidden pockets used by magicians, the underlying algebrogeometric structure of soliton equations provides an elegant and surprisingly simple explanation of something seemingly miraculous.
Assuming only multivariable calculus and linear algebra as prerequisites, this book introduces the reader to the KdV Equation and its multisoliton solutions, elliptic curves and Weierstrass \(\wp\)functions, the algebra of differential operators, Lax Pairs and their use in discovering other soliton equations, wedge products and decomposability, the KP Equation and Sato's theory relating the Bilinear KP Equation to the geometry of Grassmannians.
Notable features of the book include: careful selection of topics and detailed explanations to make this advanced subject accessible to any undergraduate math major, numerous worked examples and thoughtprovoking but not overlydifficult exercises, footnotes and lists of suggested readings to guide the interested reader to more information, and use of the software package Mathematica® to facilitate computation and to animate the solutions under study. This book provides the reader with a unique glimpse of the unity of mathematics and could form the basis for a selfstudy, onesemester special topics, or “capstone” course.
ReadershipUndergraduate and graduate students interested in nonlinear PDEs; applications of algebraic geometry to differential equations.

Table of Contents

Chapters

Chapter 1. Differential equations

Chapter 2. Developing PDE intuition

Chapter 3. The story of solitons

Chapter 4. Elliptic curves and KdV traveling waves

Chapter 5. KdV $n$solitons

Chapter 6. Multiplying and factoring differential operators

Chapter 7. Eigenfunctions and isospectrality

Chapter 8. Lax form for KdV and other soliton equations

Chapter 9. The KP equation and bilinear KP equation

Chapter 10. The Grassmann cone $\Gamma _{2,4}$ and the bilinear KP equation

Chapter 11. Pseudodifferential operators and the KP hierarchy

Chapter 12. The Grassman cone $\Gamma _{k,n}$ and the bilinear KP hierarchy

Chapter 13. Concluding remarks

Appendix A. Mathematica guide

Appendix B. Complex numbers

Appendix C. Ideas for independent projects


Additional Material

Reviews

This book challenges and intrigues from beginning to end. It would be a treat to use for a capstone course or senior seminar.
William J. Satzer, MAA Reviews 
[T]his introduction to soliton theory is ideal for precisely the type of course for which it is intended  a .single semester special topics class' or a 'capstone experience . . . course.' . . . One of the delightful bonuses found in the text is the list of sources for additional reading found at the end of each chapter. In addition, the appendix, Ideas for Independent Projects,' provides both the student and the teacher many options for even more connections and/or more depth in numerous areas of study. Recommended.
J. T. Zerger, CHOICE 
The book is well written and contains numerous workedout examples as well as many exercises and a guide to the literature for further reading. In particular, I feel that it serves its intended purpose quite well.
Gerald Teschl, Mathematical Reviews


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Now available in Second Edition: STML/100
Solitons are explicit solutions to nonlinear partial differential equations exhibiting particlelike behavior. This is quite surprising, both mathematically and physically. Waves with these properties were once believed to be impossible by leading mathematical physicists, yet they are now not only accepted as a theoretical possibility but are regularly observed in nature and form the basis of modern fiberoptic communication networks.
Glimpses of Soliton Theory addresses some of the hidden mathematical connections in soliton theory which have been revealed over the last halfcentury. It aims to convince the reader that, like the mirrors and hidden pockets used by magicians, the underlying algebrogeometric structure of soliton equations provides an elegant and surprisingly simple explanation of something seemingly miraculous.
Assuming only multivariable calculus and linear algebra as prerequisites, this book introduces the reader to the KdV Equation and its multisoliton solutions, elliptic curves and Weierstrass \(\wp\)functions, the algebra of differential operators, Lax Pairs and their use in discovering other soliton equations, wedge products and decomposability, the KP Equation and Sato's theory relating the Bilinear KP Equation to the geometry of Grassmannians.
Notable features of the book include: careful selection of topics and detailed explanations to make this advanced subject accessible to any undergraduate math major, numerous worked examples and thoughtprovoking but not overlydifficult exercises, footnotes and lists of suggested readings to guide the interested reader to more information, and use of the software package Mathematica® to facilitate computation and to animate the solutions under study. This book provides the reader with a unique glimpse of the unity of mathematics and could form the basis for a selfstudy, onesemester special topics, or “capstone” course.
Undergraduate and graduate students interested in nonlinear PDEs; applications of algebraic geometry to differential equations.

Chapters

Chapter 1. Differential equations

Chapter 2. Developing PDE intuition

Chapter 3. The story of solitons

Chapter 4. Elliptic curves and KdV traveling waves

Chapter 5. KdV $n$solitons

Chapter 6. Multiplying and factoring differential operators

Chapter 7. Eigenfunctions and isospectrality

Chapter 8. Lax form for KdV and other soliton equations

Chapter 9. The KP equation and bilinear KP equation

Chapter 10. The Grassmann cone $\Gamma _{2,4}$ and the bilinear KP equation

Chapter 11. Pseudodifferential operators and the KP hierarchy

Chapter 12. The Grassman cone $\Gamma _{k,n}$ and the bilinear KP hierarchy

Chapter 13. Concluding remarks

Appendix A. Mathematica guide

Appendix B. Complex numbers

Appendix C. Ideas for independent projects

This book challenges and intrigues from beginning to end. It would be a treat to use for a capstone course or senior seminar.
William J. Satzer, MAA Reviews 
[T]his introduction to soliton theory is ideal for precisely the type of course for which it is intended  a .single semester special topics class' or a 'capstone experience . . . course.' . . . One of the delightful bonuses found in the text is the list of sources for additional reading found at the end of each chapter. In addition, the appendix, Ideas for Independent Projects,' provides both the student and the teacher many options for even more connections and/or more depth in numerous areas of study. Recommended.
J. T. Zerger, CHOICE 
The book is well written and contains numerous workedout examples as well as many exercises and a guide to the literature for further reading. In particular, I feel that it serves its intended purpose quite well.
Gerald Teschl, Mathematical Reviews