**Student Mathematical Library**

Volume: 54;
2010;
304 pp;
Softcover

MSC: Primary 35; 37; 14; 15;

Print ISBN: 978-0-8218-5245-3

Product Code: STML/54

List Price: $49.00

Individual Price: $39.20

**Electronic ISBN: 978-1-4704-1637-9
Product Code: STML/54.E**

List Price: $49.00

Individual Price: $39.20

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#### Supplemental Materials

# Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear PDEs

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*Alex Kasman*

Solitons are explicit solutions to nonlinear partial differential equations exhibiting particle-like 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 fiber-optic communication networks.

*Glimpses of Soliton Theory* addresses some of the hidden
mathematical connections in soliton theory which have been revealed over
the last half-century. It aims to convince the reader that, like the
mirrors and hidden pockets used by magicians, the underlying
algebro-geometric 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 thought-provoking
but not overly-difficult 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 self-study, one-semester special topics,
or “capstone” course.

#### Table of Contents

# Table of Contents

## Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear PDEs

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface ix10 free
- Differential equations 118 free
- Developing PDE intuition 2340
- The story of solitons 4562
- Elliptic curves and KdV traveling waves 6784
- KdV 𝑛-solitons 95112
- Multiplying and factoring differential operators 113130
- Eigenfunctions and isospectrality 133150
- Lax form for KdV and other soliton equations 149166
- The KP equation and bilinear KP equation 173190
- The Grassmann cone Γ_{2,4} and the bilinear KP equation 197214
- Pseudo-differential operators and the KP hierarchy 219236
- The Grassman cone Γ_{𝑘,𝑛} and the bilinear KP hierarchy 235252
- Concluding remarks 251268
- Mathematica guide 257274
- Complex numbers 269286
- Ideas for independent projects 275292
- References 289306
- Glossary of symbols 297314
- Index 301318 free
- Back Cover Back Cover1322

#### Readership

Undergraduate and graduate students interested in nonlinear PDEs; applications of algebraic geometry to differential equations.

#### 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 worked-out 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