Soliton theory is known to be an active area of research combining many different areas of mathematics and having deep connections to science and engineering. That is why most of the books on the subject are written for researchers with doctorates in mathematics or physics. In contrast with this, the book by A. Kasman assumes a minimum of mathematical prerequisites (a calculus sequence and a course in linear algebra) and aims to present the subject at a level accessible to any undergraduate math major. On the other hand, the textbook assumes that the reader has access to the computer program Mathematica. It is assumed that Mathematica will help the reader to visualize solutions to nonlinear PDEs with complicated formulae, to perform some messy computations, and even to operate with elliptic functions without defining them rigorously and proving their important properties.
The book consists of 13 chapters and 3 appendices, one of them providing a brief guide to Mathematica. Chapters 1 and 2 summarize some of the key differences between linear and nonlinear differential equations. The story of solitons is presented in Chapter 3. The connection between solitary waves and algebraic geometry is introduced in Chapter 4, multi-soliton solutions of the Korteweg-de Vries (KdV) equation are studied in Chapter 5, rules for multiplying and factoring differential operators are provided in Chapter 6. Chapters 7 and 8 deal with the isospectrality and the Lax representation giving the way to recognize other differential equations deserving to be called “soliton equations”. Chapter 9 introduces the KP equation, which is a generalization of the KdV equation involving an additional spatial dimension, and presents the Hirota bilinear version of the KP equation and techniques for solving it. The wedge product of a pair of vectors in a 4-dimensional space is introduced in Chapter 10 and used to motivate the definition of the Grassmann cone. Chapters 11 and 12 lead the reader to Sato’s theory demonstrating a complete equivalence between the soliton equations of the KP hierarchy and the infinitely many algebraic equations characterizing all possible Grassmann cones. The final chapter reviews what is covered by the book.