Advanced Series in Mathematical Physics - Vol. 12
SOLITON EQUATIONS AND HAMILTONIAN SYSTEMS
by L A Dickey (University of Oklahoma)
The theory of soliton equations and integrable systems has developed rapidly during the last 20 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 followed one single work by Gardner, Greene, Kruskal, and Miura about the Korteweg-de Vries equation (KdV) which, had seemed to be merely an unassuming equation of mathematical physics describing waves in shallow water.
This branch of science is attractive because it is one of those which revives the interest in the basic principles of mathematics, a beautiful formula.
Contents:
- Integrable Systems Generated by Linear Differential nth Order
Operators
- Hamiltonian Structures
- Hamiltonian Structures of the KdV-Hierarchies
- The Kupershmidt-Wilson Theorem
- The KP-Hierarchy
- Hamiltonian Structure of the KP-Hierarchy
- Baker Function, t-Function
- Grassmannian, t-Function and Baker Function after Segal and Wilson. Algebraic-Geometrical Krichever's Solutions
- Matrix First-Order Operators
- KdV-Hierarchies as Reductions of Matrix Hierarchies
- Stationary Equations
- Stationary Equations of the KdV-Hierarchy in the Narrow Sense (n=2)
- Stationary Equations of the Matrix Hierarchy
- Stationary Equations of the KdV-Hierarchies
- Matrix Differential Operators Polynomially Depending on a Parameter
- Multi-time Lagrangian and Hamiltonian Formalism
- Further Examples and Applications
Readership: Applied mathematicians.
"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 the 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."
| 320pp |
Pub. date: Sept 1991 |
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