《孤立子理论中的哈密顿方法英文》讲述了：The book is based on the Hamiltonian
interpretation of the method, hence the title. Methods of
differential geometry and Hamiitonian formalism in particular are
very popular in modern mathematical physics. It is precisely the
general Hamiltonian formalism that presents the inverse scattering
method in its most elegant form. Moreover, the Hamiltonian
formalism provides a link between classical and quantum mechanics.
So the book is not only an introduction to the classical soliton
theory but also the groundwork for the quantum theory of solitons,
to be discussed in another volume.
The book is addressed to specialists in mathematical physics.
This has determined the choice of material and the level of
mathematical rigour. We hope that it will also be of interest to
mathematicians of other specialities and to theoretical physicists
as well. Still, being a mathematical treatise it does not contain
applications of soliton theory to specific physical
phenomena.
目錄：
Introduction
References
Part One The Nonlinear Schrodinger Equation NS Model
Chapter Ⅰ Zero Curvature Representation
1.Formulation of the NS Model
2.Zero Curvature Condition
3.Properties of the Monodromy Matrix in the Quasi-Periodic
Case
4.Local Integrals of the Motion
5.The Monodromy Matrix in the Rapidly Decreasing Case
6.Analytic Properties of Transition Coefficients
7.The Dynamics of Transition Coefficients
8.The Case of Finite Density.Jost Solutions
9.The Case of Finite Density.Transition Coefficients
10.The Case of Finite Density.Time Dynamics and Integrals of the
Motion
1.Notes and References
References
Chapter Ⅱ The Riemann Problem
1.The Rapidly Decreasing Case.Formulation of the Riemann
Problem
2.The Rapidly Decreasing Case.Analysis of the Riemann Problem
3.Application of the Inverse Scattering Problem to the NS
Model
4.Relationship Between the Riemann Problem Method and the
Gelfand-Levitan-Marchenko Integral Equations Formulation
5.The Rapidly Decreasing Case.Soliton Solutions
6.Solution of the Inverse Problem in the Case of Finite Density.The
Riemann Problem Method
7.Solution of the Inverse Problem in the Case of Finite Density.The
Gelfand-Levitan-Marchenko Formulation
8.Soliton Solutions in the Case of Finite Density
9.Notes and References References
Chapter Ⅲ The Hamiltonian Formulation
1.Fundamental Poisson Brackets and the Matrix
2.Poisson Commutativity of the Motion Integrals in the
Quasi-Periodic Case
3.Derivation of the Zero Curvature Representation from the
Fundamental Poisson Brackets
4.Integrals of the Motion in the Rapidly Decreasing Case and in the
Case of Finite Density
5.The A-Operator and a Hierarchy of Poisson Structures
6.Poisson Brackets of Transition Coefficients in the Rapidly
Decreasing Case
7.Action-Angle Variables in the Rapidly Decreasing Case
8.Soliton Dynamics from the Hamiltonian Point of View
9.Complete Integrability in the Case of Finite Density
10.Notes and References
References
Part Two General Theory of Integrable Evolution Equations
Chapter Ⅰ Basic Examples and Their General Properties
1.Formulation of the Basic Continuous Models
2.Examples of Lattice Models
3.Zero Curvature Representation''s a Method for Constructing
Integrable Equations
4.Gauge Equivalence of the NS Model #=-1 and the HM Model
5.Hamiltonian Formulation of the Chiral Field Equations and Related
Models
6.The Riemann Problem as a Method for Constructing Solutions of
Integrable Equations
7.A Scheme for Constructing the General Solution of the Zero
Curvature Equation. Concluding Remarks on Integrable
Equations
8.Notes and References
References
Chapter Ⅱ Fundamental Continuous Models
1.The Auxiliary Linear Problem for the HM Model
2.The Inverse Problem for the HM Model
3.Hamiltonian Formulation of the HM Model
4.The Auxiliary Linear Problem for the SG Model
5.The Inverse Problem for the SG Model
6.Hamiltonian Formulation of the SG Model
7. The SG Model in Light-Cone Coordinates
8. The Landau-Lifshitz Equation as a Universal Integrable Model
with Two-Dimensional Auxiliary Space
9. Notes and References
References
Chapter Ⅲ Fundamental Models on the Lattice
1. Complete Integrability of the Toda Model in the Quasi-Peri-odic
Case
2. The Auxiliary Linear Problem for the Toda Model in the Rap-idly
Decreasing Case
3. The Inverse Problem and Soliton Dynamics for the Toda Model in
the Rapidly Decreasing Case
4. Complete Integrability of the Toda Model in the Rapidly
Decreasing Case
5. The Lattice LL Model as a Universal Integrable System with
Two-Dimensional Auxiliary Space
6. Notes and References
References
Chapter Ⅳ Lie-Algebraic Approach to the Classification and
Analysisof lntegrable Models
1. Fundamental Poisson Brackets Generated by the Current
Alge-bra
2. Trigonometric and Elliptic r-Matrices and the Related
Funda-mental Poisson Brackets
3. Fundamental Poisson Brackets on the Lattice
4. Geometric Interpretation of the Zero Curvature Representation
and the Riemann Problem Method
5. The General Scheme as Illustrated with the NS Model
6. Notes and References
References
Conclusion
List of Symbols
Index
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