孤立子理论中的哈密顿方法

IntroductionReferencesPart One The Nonlinear Schrodinger Equation (NS Model)Chapter Ⅰ Zero Curvature Representation1.Formulation of the NS Model2.Zero Curvature Condition3.Properties of the Monodromy Matrix in the Quasi-PeriodicCase4.Local Integrals of the Motion5.The Monodromy Matrix in the Rapidly Decreasing Case6.Analytic Properties of Transition Coefficients7.The Dynamics of Transition Coefficients8.The Case of Finite Density.Jost Solutions9.The Case of Finite Density.Transition Coefficients10.The Case of Finite Density.Time Dynamics and Integrals of theMotion1.Notes and ReferencesReferencesChapter Ⅱ The Riemann Problem1.The Rapidly Decreasing Case.Formulation of the RiemannProblem2.The Rapidly Decreasing Case.Analysis of the Riemann Problem3.Application of the Inverse Scattering Problem to the NSModel4.Relationship Between the Riemann Problem Method and theGelfand-Levitan-Marchenko Integral Equations Formulation5.The Rapidly Decreasing Case.Soliton Solutions6.Solution of the Inverse Problem in the Case of Finite Density.TheRiemann Problem Method7.Solution of the Inverse Problem in the Case of Finite Density.TheGelfand-Levitan-Marchenko Formulation8.Soliton Solutions in the Case of Finite Density9.Notes and References ReferencesChapter Ⅲ The Hamiltonian Formulation1.Fundamental Poisson Brackets and the Matrix2.Poisson Commutativity of the Motion Integrals in theQuasi-Periodic Case3.Derivation of the Zero Curvature Representation from theFundamental Poisson Brackets4.Integrals of the Motion in the Rapidly Decreasing Case and in theCase of Finite Density5.The A-Operator and a Hierarchy of Poisson Structures6.Poisson Brackets of Transition Coefficients in the RapidlyDecreasing Case7.Action-Angle Variables in the Rapidly Decreasing Case8.Soliton Dynamics from the Hamiltonian Point of View9.Complete Integrability in the Case of Finite Density10.Notes and ReferencesReferencesPart Two General Theory of Integrable Evolution EquationsChapter Ⅰ Basic Examples and Their General Properties1.Formulation of the Basic Continuous Models2.Examples of Lattice Models3.Zero Curvature Representation's a Method for ConstructingIntegrable Equations4.Gauge Equivalence of the NS Model (#=-1) and the HM Model5.Hamiltonian Formulation of the Chiral Field Equations and RelatedModels6.The Riemann Problem as a Method for Constructing Solutions ofIntegrable Equations7.A Scheme for Constructing the General Solution of the ZeroCurvature Equation. Concluding Remarks on IntegrableEquations8.Notes and ReferencesReferencesChapter Ⅱ Fundamental Continuous Models1.The Auxiliary Linear Problem for the HM Model2.The Inverse Problem for the HM Model3.Hamiltonian Formulation of the HM Model4.The Auxiliary Linear Problem for the SG Model5.The Inverse Problem for the SG Model6.Hamiltonian Formulation of the SG Model7. The SG Model in Light-Cone Coordinates8. The Landau-Lifshitz Equation as a Universal Integrable Modelwith Two-Dimensional Auxiliary Space9. Notes and ReferencesReferencesChapter Ⅲ Fundamental Models on the Lattice1. Complete Integrability of the Toda Model in the Quasi-Peri-odicCase2. The Auxiliary Linear Problem for the Toda Model in the Rap-idlyDecreasing Case3. The Inverse Problem and Soliton Dynamics for the Toda Model inthe Rapidly Decreasing Case4. Complete Integrability of the Toda Model in the RapidlyDecreasing Case5. The Lattice LL Model as a Universal Integrable System withTwo-Dimensional Auxiliary Space6. Notes and ReferencesReferencesChapter Ⅳ Lie-Algebraic Approach to the Classification andAnalysisof lntegrable Models1. Fundamental Poisson Brackets Generated by the CurrentAlge-bra2. Trigonometric and Elliptic r-Matrices and the RelatedFunda-mental Poisson Brackets3. Fundamental Poisson Brackets on the Lattice4. Geometric Interpretation of the Zero Curvature Representationand the Riemann Problem Method5. The General Scheme as Illustrated with the NS Model6. Notes and ReferencesReferencesConclusionList of SymbolsIndex

The book is addressed to specialists in mathematical physics.This has determined the choice of material and the level ofmathematical rigour. We hope that it will also be of interest tomathematicians of other specialities and to theoretical physicistsas well. Still, being a mathematical treatise it does not containapplications of soliton theory to specific physicalphenomena.