Advanced Series in Mathematical Physics - Vol. 26
SOLITON EQUATIONS AND HAMILTONIAN SYSTEMS
Second Edition
by L A Dickey (University of Oklahoma, USA)
The theory of soliton equations and integrable systems has developed rapidly during the last 30 years with numerous applications in mechanics and physics. For a long time, books in this field have not been written but the flood of papers was overwhelming: many hundreds, maybe thousands of them. All this output followed one single work by Gardner, Green, Kruskal, and Mizura on the Korteweg-de Vries equation (KdV), which had seemed to be merely an unassuming equation of mathematical physics describing waves in shallow water.
Besides its obvious practical use, this theory is attractive also because it satisfies the aesthetic need in a beautiful formula which is so inherent to mathematics.
The second edition is up-to-date and differs from the first one considerably. One third of the book (five chapters) is completely new and the rest is refreshed and edited.
Contents:
- Integrable Systems Generated by Linear Differential nth Order
Operators
- Hamiltonian Structures
- Hamiltonian Structure of the GD Hierarchies
- Modified KdV and GD. The Kupershmidt–Wilson Theorem
- The KP Hierarchy
- Baker Function, t-Function
- Additional Symmetries, String Equation
- Grassmannian. Algebraic-Geometrical Krichever Solutions
- Matrix First-Order Operator, AKNS-D Hierarchy
- Generalization of the AKNS-D Hierarchy: Single-Pole and Multi-Pole Matrix Hierarchies
- Isomonodromic Deformations and the Most General Matrix Hierarchy
- Tau Functions of Matrix Hierarchies
- KP, Modified KP, Constrained KP, Discrete KP, and q-KP
- Another Chain of KP Hierarchies and Integrals Over Matrix Varieties
- Transformational Properties of a Differential Operator under Diffeomorphisms and Classical W-Algebras
- Further Restrictions of the KP, Stationary Equations
- Stationary Equations of the Matrix Hierarchy
- Field Lagrangian and Hamiltonian Formalism
- Further Examples and Applications
Readership: Applied mathematicians and mathematical physicists.
"The author wrote in the introduction to the first edition that he 'tried to make this book available to beginners in this area having only basic training in algebra and analysis'. In the reviewer's opinion, he definitely achieved this goal. Moreover, this book can also be very useful for the researchers already active in the field of soliton equations and integrable systems."
Reviews of the First Edition: "There is a bibliography of 112 items. This book is pedagogically written and is highly recommended for its detailed description of the resolvent method for soliton equations."
"The book of L A Dickey presents one more point of view on the mathematical theory of solitons or, in other words, on the theory of nonlinear partial differential equations ... The series of joint papers of I M Gelfand and L A Dickey in the middle of seventies was an important step in the development of the mathematical theory of nonlinear integrable equations ... As a whole the book presents a very good exposition of the important part of the soliton theory."
| 420pp |
Pub. date: Jan 2003 |
|
