CONTENTS -- PREFACE -- INTRODUCTORY CHAPTERS -- 1. INTRODUCTION -- References for Section 1 -- 2. MATHEMATICAL THEORY OF SOLITONS: AN OUTLINE -- 2.1 Introduction -- 2-2 Introductory Bibliography Notes on the 1ST Method -- 2-1 Introduction -- 2-3 The Estabrook-Wahlquist Prolongation Method -- 2-4 The Symmetry Approach -- References for Section 2 -- References for Section 2 - Books -- 3. SOLITONS IN PARTICLE PHYSICS:A GUIDE THROUGH THE LITERATURE -- 3-1 One-dimensional Systems -- 3-2 Vortices and Two-dimensional Systems -- 3-3 Monopoles and Three-dimensional Solitons -- 3-4 Four-dimensional Solitons: Instantons -- References for Section 3 -- References for Section 3 - Books and Reviews -- REPRINTED PAPERS -- INTERACTION OF "SOLITONS" IN A COLLISIONLESS PLASMA AND THE RECURRENCE OF INITIAL STATES -- METHOD FOR SOLVING THE KORTEWEG-deVRIES EQUATION* -- Integrals of Nonlinear Equations of Evolution and Solitary Waves* -- Bibliography -- Method for Solving the Sine-Gordon Equation* -- General Derivation of Backlund Transformations from Inverse Scattering Problems* -- ON RELATIVISTIC INVARIANT FORMULATION OF THE INVERSE SCATTERING TRANSFORM METHOD -- 1. INTRODUCTION -- 2. THE SINE-GORDON MODEL -- 3. THE MASSIVE THIRRING MODEL -- 4. CONCLUSION -- REFERENCES -- Canonical Structure of Soliton Equations via Isospectral Eigenvalue Problems (*). -- 1. - Introduction. -- 2. - A canonical Hamiltonian structure. -- 3. - Connection of the canonical structure of the soliton equations with the associated spectral problem. -- 4. - Proof of the completeness conjecture in the 2 x2 Zakharov-Shabat spectral problem, -- 5. - Applications -- Coupled Nonlinear Evolution Equations Solvable Via the Inverse Spectral Transform, and Solitons that Come Back: the Boomeron.

Solution by the Spectral-Transform Method of a Nonlinear Evolution Equation Including as a Special Case the Cylindrical KdV Equation. -- EXACT THEORY- OF TWO-DIMENSIONAL SELF-FOCUSING AND ONE-DIMENSIONALSELF-MODULATION OF WAVES IN NONLINEAR MEDIA -- 1. THE DIRECT SCATTERING PROBLEM -- 2. THE INVERSE SCATTERING PROBLEM -- 3. N-SOLITON SOLUTIONS (EXPLICIT FORMULA) -- 4. N-SOLITON SOLUTIONS (ASYMPTOTIC FORM AS t ± -- 5. BOUND STATES AND MULTIPLE EIGENVALUES -- 6. STABILITY OF SOLITONS -- 7. QUASICLASSICAL APPROXIMATI -- 8. CONSERVATION LAWS -- Reiativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method -- INTRODUCTION -- 1. RELATIVISTICALLY INVARIANT INTEGRABLESY STEMS IN TWO-DIMENSIONAL SPACE-TIME -- 2. PRINCIPAL CHiRAL FIELDS -- 3. CHIRAL FIELDS AND THE REDUCTIONS PROBLEM -- 4. THE METHOD OF THE INVERSE SCATTERING PROBLEM -- 5. SOLITON SOLUTIONS -- 6. INTEGRATION OF CHIRAL FIELDS ON GRASSMANN MANIFOLDS -- CONCLUSION -- Backlund Transformation for Solutions of the Korteweg-de Vries Equation* -- Prolongation structures of nonlinear evolution equations* -- I. INTRODUCTION -- II. THE KORTEWEG-DE VRIES EQUATION -- III. CONSERVATION LAWS AND POTENTIALS -- V. PROLONGATION STRUCTURE OF THE KdV EQUATION -- IV. MULTIPLE PROLONGATION AND PSEUDOPOTENTIALS -- VI. SOLUTION TECHNIQUES -- VII. CONCLUSION -- Prolongation structures of nonlinear evolution equations. II -- I. INTRODUCTION -- II. THE PS OF THE NONLINEAR SCHRODINGER EQUATION -- III. THE PSEUDOPOTENTIAL -- IV. TRANSFORMATIONS OF THE PROLONGED IDEAL -- V. BACKLUND TRANSFORMATIONS -- VI. THE THEOREM OF PERMUTABILITY -- APPENDIX -- Prolongation analysis of the cylindrical Korteweg-de Vries equation -- I. INTRODUCTION -- II. THE PROLONGATION CALCULATION -- IV. THE THEOREM OF PERMUTABILITY.

V. EXAMPLES OF EXPLICIT SOLUTIONS -- III. BACKLUND TRANSFORMATIONS -- VI. CONCLUDING REMARKS -- APPENDIX A -- APPENDIX B -- APPENDIX C -- Nonlinear evolution equations and nonabelian prolongat -- I. INTRODUCTION -- II. BASIC EQUATIONS -- III. CASE WHERE Fu AND Fuu ARE LINEARLYINDEPENDENT -- IV. CASE WHERE Fu AND Fuu ARE LINEARLY DEPENDENT -- V. CONCLUDING REMARKS -- ACKNOWLEDGME -- APPENDIX A -- APPENDIX B -- APPENDIX C -- BScklund transformations and the symmetries of the Yang equations -- 1. INTRODUCTION -- 2: THE "BACKLUND TRANSFORMATION" AS A GENERALIZED SYMMETRY -- 3. SYMMETRIES OF THE YANG POTENTIALS -- 4. COORDINATE SYMMETRIES -- ACKNOWLEDGMENTS -- Inverse scattering problems in higher dimensions: Yang-Mills fields and the supersymmetric sine-Gordon equation -- 1. INTRODUCTION -- 2. LINEAR PROLONGATION FORMS IN THREE DIMENSIONS -- 3. LINEAR PROLONGATION FORMS IN FOUR DIMENSIONS -- 4. THE SELF-DUAL YANG-MILLS EQUATIONS -- 5. THE SUPERSYMMETRIC SINE-GORDON EQUATION -- ACKNOWLEDGMENTS -- "iST-solvable" nonlinear evolution equations and existence-An extension of Lax's methoda) -- INTRODUCTION -- 1. RESUME OF THE 1ST, LAX'S METHOD, AND ITS EXTENSI -- A. The inverse problem -- B. The inverse scattering transform (IST) -- C. Lax's method and the present generalization -- D. The reduced £-dependent forms and anaiyticity in E -- E. Obtaining the evolution equation -- 2. THE NEED TO PROVE EXISTENCE -- 3. THE ONE-DIMENSIONAL SCHRODINGER CASE -- A. OutlineHere -- B. Lax's method -- C. The reduced form of BClearly, when applied -- D. The reduced commutator equation and the auxiliary equation -- E. Evolution of the data and the dispersion relation -- F. The evolution equation tor the potential -- G. Dispersion relations which are ratios of entire functions -- 4. PROOF OF EXISTENCE, UNIQUENESS, AND ANALYTICITY OF THE SCHRODINGER AUXILIARY FUNCTION.

5. NONLINEAR EVOLUTION EQUATIONS FORLINEAR FIRST ORDER n X n MATRIX EQUATIONSOF THE ZAKHAROV-SHABAT TYPE -- A. The linear matrix equation -- B. Hie 2 X 2 cases and self-adjointness -- C. The evolution equation and off-diagonalness of O -- D. Derivation of the reduced commutator equation -- E. The auxiliary equation -- F. Vector form of the auxiliary equation -- G. The evolution equation obtained from the auxiliary equation -- H. Relation to the scattering data -- I. The AKNS data as the 2 X 2 case -- D. Derivation of the reduced commutator equation -- J. The AKNS evolution equation as the 2 X 2 case -- 6. CONCLUSION -- ACKNOWLEDGMENTS -- APPENDIX: CERTAIN 2 X 2 CASES AND THESCHRODINGER EQUATION -- Evolution equations possessing infinitely many symmetries -- SYMPLECTIC STRUCTURES, THEIR BACKLUND TRANSFORMATIONS AND HEREDITARY SYMMETRIES -- 1. Introduction and basic notions -- 2. Symplectic-implectic operators, Hamiltoniansy stems and Magri's results -- 3. Hereditary symmetries admitting a symplectic-implectic factorization. -- 4. Motivation and illustrations of the basicnotions -- 5. Backlund transformations for implecti coperators -- 6. Examples and applications -- 6.1. Some implectic operators -- 6.2. Construction of hereditary symmetries and more implectic operators -- 6.3. The corresponding Hamiltonian systems,their symmetries, conservation laws and soliton solutions -- 6.3.1. The symmetries -- 6.3.2 The Noether operators and the fact thatthe conserved covariants are gradients -- 6.3.3. An orthogonality property -- 6.3.4. The conservation laws -- 6.4. The complex case -- 6.5. A new class of exactly solvable third order equations -- 6.6 Some fifth order equations -- Acknowledgement -- References -- A simple model of the integrable Hamiltonian equation** -- INTRODUCTION -- 1. SYMMETRIES AND CONSERVATION LAWS OF HAMILTONIAN EVOLUTION EQUATIONS.

A. Symmetries -- B. Conservation laws -- C. Connecting the conservation laws with the symmetries -- 2. AN EXAMPLE: THE KORTEWEG-de VRIES EQUATION -- 3. INTEGRABLE HAMILTONIAN EQUATIONS -- 4. SUMMARY -- 5. APPLICATIONS -- APPENDIX A -- APPENDIX B -- Some New Conservation Laws -- I. INTRODUCTION -- II. ILLUSTRATIVE EXAMPLE: THE "STRIP" -- III. ILLUSTRATIVE EXAMPLE: GENERAL RELATIVITY -- IV. THE GENERAL CASE -- V. ON THE SPACE THEORY OF MATTER -- REFERENCES -- Kinks* -- I. CLASSICAL FIELDS -- II. PARTICLE NUMBERS -- III. QUANTUM FIELDS -- IV. EXAMPLES OF HOMOTOPY CONSERVATION -- V. SPIN AND DOUBLE VALUEDNESS -- VI. EXAMPLES OF SPIN CALCULATIONS -- 1. Vector fi -- 2. Unit 3-vector field -- 3. Unit 4-vector field -- 4. Rotator field -- 5. General Relativity -- 6. Displacement field -- Sine-Gordon Equation -- 1. INTRODUCTION -- 2. THE ONE-DIMENSIONAL MODEL -- 3. BOUND STATES -- 4. INTERACTION BETWEEN KINKS -- 5. SMALL OSCILLATIONS -- 6. QUANTIZATION -- 7. TRANSPARENCY OF KINKS -- 8. RELATION TO THE KORTEWEG-DE VRIES EQUATION -- ACKNOWLEDGMENT -- APPENDIX -- Nonperturbative methods and extended-hadron models in field theory.II. Two-dimensional models and extended hadrons* -- I. INTRODUCTION -- II. REVIEW OF THE SEMICLASSICAL METHOD -- III. BOUND STATES IN A TWO-DIMENSIONAL FIELD-THEORY MODEL -- IV. ADDING FERMIONS -- ACKNOWLEDGMENTS -- APPENDIX -- Quantization of nonlinear waves* -- I. INTRODUCTION -- II. CLASSICAL SOLUTIONS AND THE TRANSLATION MODE -- III. QUANTUM THEORY USING THE EFFECTIVE ACTION -- IV. THE KERMAN-KLEIN METHOD -- V. FURTHER ASPECTS OF THE THEORY -- VI. CONCLUSION -- Soliton quantization -- Quantum sine-Gordon equation as the massive Thirring model* -- I. INTRODUCTION -- II. THE HAMILTONIAN -- III. A VARIATIONAL COMPUTATION -- IV. PERTURBATION THEORY -- V. PUZZLING QUESTIONS, BUT NOT BEYOND ALL CONJECTURE -- A. Bounds on coupling constants.

B. Metamorphosis of fermions into bosons.