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.

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-PeriodicCase
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 theMotion
1.Notes and References
References
Chapter Ⅱ The Riemann Problem
1.The Rapidly Decreasing Case.Formulation of the RiemannProblem
2.The Rapidly Decreasing Case.Analysis of the Riemann Problem
3.Application of the Inverse Scattering Problem to the NSModel
4.Relationship Between the Riemann Problem Method and theGelfand-Levitan-Marchenko Integral Equations Formulation
5.The Rapidly Decreasing Case.Soliton Solutions
6.Solution of the Inverse Problem in the Case of Finite Density.TheRiemann Problem Method
7.Solution of the Inverse Problem in the Case of Finite Density.TheGelfand-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 theQuasi-Periodic Case
3.Derivation of the Zero Curvature Representation from theFundamental Poisson Brackets
4.Integrals of the Motion in the Rapidly Decreasing Case and in theCase of Finite Density
5.The A-Operator and a Hierarchy of Poisson Structures
6.Poisson Brackets of Transition Coefficients in the RapidlyDecreasing 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 ConstructingIntegrable Equations
4.Gauge Equivalence of the NS Model (#=-1) and the HM Model
5.Hamiltonian Formulation of the Chiral Field Equations and RelatedModels
6.The Riemann Problem as a Method for Constructing Solutions ofIntegrable Equations
7.A Scheme for Constructing the General Solution of the ZeroCurvature Equation. Concluding Remarks on IntegrableEquations
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 Modelwith 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-odicCase
2. The Auxiliary Linear Problem for the Toda Model in the Rap-idlyDecreasing Case
3. The Inverse Problem and Soliton Dynamics for the Toda Model inthe Rapidly Decreasing Case
4. Complete Integrability of the Toda Model in the RapidlyDecreasing Case
5. The Lattice LL Model as a Universal Integrable System withTwo-Dimensional Auxiliary Space
6. Notes and References
References
Chapter Ⅳ Lie-Algebraic Approach to the Classification andAnalysisof lntegrable Models
1. Fundamental Poisson Brackets Generated by the CurrentAlge-bra
2. Trigonometric and Elliptic r-Matrices and the RelatedFunda-mental Poisson Brackets
3. Fundamental Poisson Brackets on the Lattice
4. Geometric Interpretation of the Zero Curvature Representationand the Riemann Problem Method
5. The General Scheme as Illustrated with the NS Model
6. Notes and References
References
Conclusion
List of Symbols
Index