Glimpses of soliton theory pdf

Skip to content. Glimpses of Soliton Theory. 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. KP Solitons and the Grassmannians. Katayama, Kronecker's limit formulas and Monthly , 41— Glimpses of soliton theory.

The algebra and geometry of nonlinear PDEs. Student Mathematical Library , Klain, G. 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 -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.

As a capstone course, or for independent study, soliton theory ties together several important applications to science and engineering with an extraordinary range of mathematical topics from PDEs to elliptic curves, differential algebras and Grassmanians.

Partial differential equations and algebraic geometry meet in a most remarkable and unexpected way. After an introductory review of differential equations that emphasizes the differences between linear and nonlinear equations, the author tells the story of solitons. He begins with James Scott Russell and continues to twentieth century developments and applications.

These include examples in telecommunications where solitons travel down optical fibers and biology where solitons play a role in DNA transcription and energy transfer.

The connection to elliptic curves and algebraic geometry begins here. It is also used throughout for a variety of straightforward, if messy, calculations and for animations of wave dynamics.

This innovative use of Mathematica works well here where the object is to offer glimpses of a broad and subtle theory. It does, however, tie the book to the software — it would be unsatisfying to read the book without having access to Mathematica , and difficult to derive full benefit from it. After dipping into algebraic geometry, the author goes on to discuss the n-soliton version of the KdV equation and its solutions that look asymptotically like linear combinations of solutions to one-soliton equations.

The next few chapters try to explain the special nature of the KdV equations, and along the way discuss the algebra of differential operators, isospectral matrices, and the Lax form for KdV and other soliton equations. We then finally get a glimmer of the geometry of the solution space, and a way to describe it using the Grassman cone. This book challenges and intrigues from beginning to end. It would be a treat to use for a capstone course or senior seminar.