Introduction 1
References 7
Part One．The Nonlinear Schr？dinger Equation(NS Model) 9
Chapter Ⅰ．Zero Curvature Representation 11
1．Formulation of the NS Model 11
2．Zero Curvature Condition 20
3．Properties of the Monodromy Matrix in the Quasi-Periodic Case 26
4．Local Integrals of the Motion 33
5．The Monodromy Matrix in the Rapidly Decreasing Case 39
6．Analytic Properties of Transition Coefficients 46
7．The Dynamics of Transition Coefficients 51
8．The Case of Finite Density.Jost Solutions 55
9．The Case of Finite Density.Transition Coefficients 62
10．The Case of Finite Density.Time Dynamics and Integrals of the Motion 72
11．Notes and References 78
References 80
Chapter Ⅱ．The Riemann Problem 81
1．The Rapidly Decreasing Case.Formulation of the Riemann Problem 81
2．The Rapidly Decreasing Case.Analysis of the Riemann Prob-lem 89
3．Application of the Inverse Scattering Problem to the NS Model 108
4．Relationship Between the Riemann Problem Method and the Gelfand-Levitan-Marchenko Integral Equations Formulation 114
5．The Rapidly Decreasing Case.Soliton Solutions 126
6．Solution of the Inverse Problem in the Case of Finite Density.The Riemann Problem Method 137
7．Solution of the Inverse Problem in the Case of Finite Density.The Gelfand-Levitan-Marchenko Formulation 146
8．Soliton Solutions in the Case of Finite Density 165
9．Notes and References 177
References 182
Chapter Ⅲ．The Hamiltonian Formulation 186
1．Fundamental Poisson Brackets and the r-Matrix 186
2．Poisson Commutativity of the Motion Integrals in the Quasi-Periodic Case 194
3．Derivation of the Zero Curvature Representation from the Fun-damental Poisson Brackets 199
4．Integrals of the Motion in the Rapidly Decreasing Case and in the Case of Finite Density 205
5．The ？-Operator and a Hierarchy of Poisson Structures 210
6．Poisson Brackets of Transition Coefficients in the Rapidly Decreasing Case 222
7．Action-Angle Variables in the Rapidly Decreasing Case 229
8．Soliton Dynamics from the Hamiltonian Point of View 241
9．Complete Integrability in the Case of Finite Density 249
10．Notes and References 267
References 274
Part Two．General Theory of Integrable Evolution Equations 279
Chapter Ⅰ．Basic Examples and Their General Properties 281
1．Formulation of the Basic Continuous Models 281
2．Examples of Lattice Models 292
3．Zero Curvature Representation as a Method for Constructing Integrable Equations 305
4．Gauge Equivalence of the NS Model(x=-1)and the HM Model 315
5．Hamiltonian Formulation of the Chiral Field Equations and Related Models 321
6．The Riemann Problem as a Method for Constructing Solutions of Integrable Equations 333
7．A Scheme for Constructing the General Solution of the Zero Curvature Equation.Concluding Remarks on Integrable Equa-tions 339
8．Notes and References 345
References 350
Chapter Ⅱ．Fundamental Continuous Models 356
1．The Auxiliary Linear Problem for the HM Model 356
2．The Inverse Problem for the HM Model 370
3．Hamiltonian Formulation of the HM Model 384
4．The Auxiliary Linear Problem for the SG Model 393
5．The Inverse Problem for the SG Model 407
6．Hamiltonian Formulation of the SG Model 431
7．The SG Model in Light-Cone Coordinates 446
8．The Landau-Lifshitz Equation as a Universal Integrable Model with Two-Dimensional Auxiliary Space 457
9．Notes and References 463
References 467
Chapter Ⅲ．Fundamental Models on the Lattice 471
1．Complete Integrability of the Toda Model in the Quasi-Peri-odic Case 471
2．The Auxiliary Linear Problem for the Toda Model in the Rap-idly Decreasing Case 475
3．The Inverse Problem and Soliton Dynamics for the Toda Model in the Rapidly Decreasing Case 489
4．Complete Integrability of the Toda Model in the Rapidly Decreasing Case 499
5．The Lattice LL Model as a Universal Integrable System with Two-Dimensional Auxiliary Space 508
6．Notes and References 519
References 521
Chapter Ⅳ．Lie-Algebraic Approach to the Classification and Analysis of Integrable Models 523
1．Fundamental Poisson Brackets Generated by the Current Alge-bra 523
2．Trigonometric and Elliptic r-Matrices and the Related Funda-mental Poisson Brackets 533
3．Fundamental Poisson Brackets on the Lattice 540
4．Geometric Interpretation of the Zero Curvature Representation and the Riemann Problem Method 543
5．The General Scheme as Illustrated with the NS Model 558
6．Notes and References 566
References 573
Conclusion 577
List of Symbols 579
Index 585