Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear Pdes (Student Mathematical Library) by Alex Kasman (PDF)

Description

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.

User’s Reviews

Editorial Reviews: Review “This book challenges and intrigues from beginning to end. It would be a treat to use for a capstone course or senior seminar.” Mathematical Association of America Reviews (mathdl.maa.org/mathDL/19/?pa=reviews&sa=viewBook&bookId=71730)

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This was the book for my topics course in Undergrad mathematics. It is one of those math book that has voice. i noticed that right away, then the author came to is a lecture and it became very clear that this book was one of passion, not just to write a book. It touches on many topics, and is written in such away that it can be understood with Calc 3 and Linear algebra. Alex is still looking for suggestion on how to make it better for undergrads. From one math nerd to another, this is a good book that may introduce you to the world of Solitons, but really touches on PDEs, Algebraic Geometry, Analytic geometry, Hierarchy of Eqs, Jacobians. Did I mention it made sense to Undergrads?

⭐This is a great book on the Soliton Theory. It is perfect for after an ODE or PDE class. The Chapter problems are insightful and the use of Mathematica to illustrate the behavior of the phenomena helps build an intuition for Solitons. Highly recommended as a first book to approach Solitons.

⭐Excellent book. It confirms my view that many complex topics in mathematics originate from a set of neat ideas that when exposed properly, are quite comprehensible. (Another good example of such a presentation is Levy-Civita’s book on absolute differential calculus). Fun to read and a wonderful starting point for further research. One of the few things I missed was a separate chapter on inverse scattering problems.

⭐Excellent book. I continue to learn from it. The only disadvantage (or maybe it is an advantage) is that you need access to Mathematica in order to properly go through the book.

⭐This book is really marvelous. A true introduction in one of the most fascinating ad interdisciplinary topic of Mathematics. Everything is in there: history, phylosophy, and mathematics, of course: from Geometry (algebraic/differential) to algebra and analysis. But what it really made me happy is to understand the author’s democratic conception of knowledges. He does not try to hid his knowledge or the subtleties of the subject behind technics, but he really want to share with the reader what he knows about the subject (although he knows much more than he wrote on it). Thanks Prof. Alex for this amazing piece of mathematical exposition

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