You are here

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

Alex Kasman
Publication Date: 
Number of Pages: 
Student Mathematical Library
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Bill Satzer
, on
This is the second edition of a book first reviewed here. It continues to be a very attractive text. Its subject is at the crossroads of an amazing collection of mathematical areas including partial differential equations, elliptic curves, and the algebra of differential operators, as well as striking applications in science and engineering. All of this is made accessible to undergraduates with multivariable calculus and linear algebra. The author sets out the history of the area clearly and motivates his approach very well.
This second edition does many of the things a second edition often does: refines some of the exposition and adds new exercises and references. It also provides new projects that augment the many that were present in the first edition. The main addition in this new edition is a considerably expanded treatment of KdV multisolitons. This is a more advanced topic that adds some additional depth to the book and suggests opportunities for more research. It begins to explore the mystery of how a nonlinear equation with several solitons can behave asymptotically as if its solution were a linear combination of several one-soliton solutions.
The word “glimpses” in the title is well-chosen. It’s the author’s idea to introduce the subject with selected topics, explanations and examples with a lot of details, and technology (Mathematica) used when it has something special to offer. This would be an ideal book for a guided undergraduate seminar, for a capstone course, and perhaps also for self-study. Some level of guidance from an instructor would be valuable.
Bill Satzer (, now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.