In the 25 years of its existence Soliton Theory has drastically expanded our understanding of “integrability” and contributed a lot to the reunification of Mathematics and Physics in the range from deep algebraic geometry and modern representation theory to quantum field theory and optical transmission lines.
The book is a systematic introduction to the Soliton Theory with an emphasis on its background and algebraic aspects. It is the first one devoted to the general matrix soliton equations, which are of great importance for the foundations and the applications.
Differential algebra (local conservation laws, Bäcklund–Darboux transforms), algebraic geometry (theta and Baker functions), and the inverse scattering method (Riemann–Hilbert problem) with well-grounded preliminaries are applied to various equations including principal chiral fields, Heisenberg magnets, Sin–Gordon, and Nonlinear Schrödinger equation.
Sample Chapter(s)
Chapter 1: Introduction (701 KB)
Contents:
- Introduction:
- Chiral Fields and Sin-Gordon Equation
- Generalized Heisenberg Magnet and VNS Equation
- Conservation Laws and Algebraic-Geometric Solutions:
- Local Conservation Laws
- Generalized Lax Equations
- Algebraic-Geometric Solutions of Basic Equations
- Algebraic-Geometric Solutions of Sin-Gordon, NS, etc
- Bäcklund Tranforms and Inverse Problem:
- Bäcklund Transformations
- Introduction to the Scattering Theory
- Applications of the Inverse Problem Method
Readership: Mathematicians, mathematical physicists and graduate students familiar with basic notions from analysis and algebraic geometry.
