Solitons

1 The Soliton and Its History.- 1.1 Russell's Discovery of the 'Great Solitary Wave'.- 1.2 Definition of a Soliton: N-Soliton Solutions of Nonlinear Evolution Equations.- 1.3 Bäcklund Transformations and Conserved Densities.- 1.4 Other Physical Problems and the Discovery of the Inverse Method.- 1.5 Operator Pair Formulation of Nonlinear Evolution Equations.- 1.6 Discovery of Some Other N-Soliton Solutions: The AKNS-Zakharov-Shabat 2 × 2 Scattering Scheme and Its Geometry.- 1.7 Further Progress on Inverse Scattering Methods.- References.- 2. Aspects of Soliton Physics.- 2.1 Historical Remarks and Summary.- 2.2 Model Interacting Systems.- 2.3 Inverse Spectral Transformation and Motion Invariants.- Appendix A: Formal Derivation of the Marchenko Equations.- References.- 3. The Double Sine--Gordon Equations: A Physically Applicable System of Equations.- 3.1 Physical Background.- 3.2 Theory of Degenerate SIT.- 3.3 Spin Waves in Liquid 3He.- 3.4 Perturbation Theory for the Double Sine-Gordon Equation.- References.- 4. On a Nonlinear Lattice (The Toda Lattice).- 4.1 Nonlinear Lattices.- 4.2 The Exponential Interaction.- 4.3 Matrix Formalism.- 4.4 The Continuum Limit.- 4.5 Bäcklund Transformations.- 4.6 Concluding Remarks.- References.- 5. Direct Methods in Soliton Theory.- 5.1 Preliminaries.- 5.2 Properties of the D Operator.- 5.3 Solutions of the Bilinear Differential Equations.- 5.4 N-Soliton Solution of KdV-Type Equations.- 5.5 Bäcklund Transformations in Bilinear Form.- References.- 6. The Inverse Scattering Transform.- 6.1 General Discussion.- 6.2 The Generalized Zakharov-Shabat Eigenvalue Problem.- 6.3 Evolution of the Scattering Data.- 6.4 The Squared Eigenfunctions and Fourier Expansions.- 6.5 Evolution Equations of Class I.- 6.6 Hamiltonian Structure of theEquations of Class I.- 6.7 Systems with Two Dispersion Relations.- 6.8 Coherent Pulse Propagation.- 6.9 Moving Eigenvalues.- 6.10 The Sine-Gordon Equation.- 6.11 Schrödinger Equation.- 6.12 A Singular Perturbation Theory.- 6.13 Conclusion.- References.- 7. The Inverse Scattering Method.- 7.1 Preliminary Remarks.- 7.2 The Method of Finding "L-A" Pairs.- 7.3 Elementary Multidimensional Generalisation.- 7.4 Dressing " $$ \hat L - \hat A $$" Pairs.- 7.5 The Problem of Reduction and the Physical Interpretation of Examples.- 7.6 Two Dimensional Instability of Solitons.- 7.7 Exact Solutions of Equations in Nonlinear Optics.- 7.8 " $$ \hat L,\hat A,\hat B $$" Triad.- 7.9 The Conservation of the Spectrum of Operator Families.- 7.10 The "Dressing" of Operator Families.- References.- 8. Generalized Matrix Form of the Inverse Scattering Method.- 8.1 Historical Remarks.- 8.2 The Inverse Scattering Problem.- 8.3 The Inverse Scattering Method and Solvable Equations.- 8.4 Extension to Lattice Problems.- 8.5 Concluding Remarks.- References.- 9. Nonlinear Evolution Equations Solvable by the Inverse Spectral Transform Associated with the Matrix Schrödinger Equation.- 9.1 Direct and Inverse Matrix Schrödinger Problem; Notation.- 9.2 Generalized Wronskian Relations; Basic Formulae.- 9.3 Nonlinear Evolution Equations Solvable by the Inverse Spectral Transform; Solitons.- 9.4 The Boomeron Equation and Other Solvable Nonlinear Evolution Equations Related to it; Boomerons.- 9.5 Bäcklund Transformations.- 9.6 Nonlinear Superposition Principle.- 9.7 Conserved Quantities.- 9.8 Generalized Resolvent Formula.- 9.9 Nonlinear Operator Identities.- References.- 10. A Method of Solving the Periodic Problem for the KdV Equation and Its Generalizations.- 10.1 One-Dimensional SystemsAdmitting a Lax Representation; Their Stationary Solutions.- 10.2 Finite-Zoned Linear Operators.- 10.3 Hamiltonian Formalism for KdV in Stationary and Nonstationary Problems.- 10.4 The Akhiezer Function and Its Applications.- References.- 11. A Hamiltonian Interpretation of the Inverse Scattering Method.- 11.1 The Hamiltonian Formulation.- 11.2 Complete Integrability of the Nonlinear Schrödinger Equation.- 11.3 Applications to the Quantization Problem.- References.- 12. Quantum Solitons in Statistical Physics.- 12.1 Preliminary Remarks.- 12.2 Quantization and Quantum Solitons.- 12.3 Continuum Field Equations.- 12.4 Eigenvalue Spectrum.- References.- Further Remarks on John Scott Russel and on the Early History of His Solitary Wave.- Note Added in Proof (Chapter 1).- Additional References with Titles.