Solitons

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.