Nonlinear Equations in Physics and Mathematics

I - Dynamical Systems and Inverse Scattering Problems.- Integrable Many-Body Problems.- 1. Introduction.- 2. Many-Body Problems Solvable by the Lax Trick.- 3. Motion of Poles of Nonlinear Partial Differential Equations and Related Many-Body Problems.- 4. Motion of Zeros of Linear Evolution Equations and Related Integrable Many-Body Problems.- Inverse Scattering Problems for Nonlinear Applications.- 1. Introduction.- 2. Scattering Problems before Nonlinear Applications.- 3. Structure of the I.S.P. Solution: The Transformation Operator.- 4. Structure of the I.S.P. Solution: The Integral Equation.- 5. Construction of Nonlinear Equations.- 6. A Concluding Diagram.- Solutions of Nonlinear Equations Simulating Pair Production and Pair Annihilation.- The Two-Time Method Applied to Slowly Evolving Oscillating Systems.- 1. Introduction.- 2. The Two-Time Method.- 3. The Harmonic Approximation and Vibrational Stability Analysis.- 4. Summary.- II - Solitons.- Solitons in Physics.- 1. Introductory Mathematics.- 2. Applications of Solitons to Nonlinear Physics.- 3. Some Particular Applications of Solitons in Physics.- 4. Quantized Solitons.- Solitons and Geometry.- 1. Introduction.- 2. Geometry.- 3. Physics.- 4. Solitons.- Hirota's Method of Solving Soliton-Type Equations.- 1. The Korteweg-de Vries Equation.- 2. The Sine-Gordon Equation.- 3. The Double-Sine-Gordon Equation.- 4. A Hierarchy of KdV Equations.- 5. Polynomial Conserved Densities.- Prolongation Structure Techniques for the New Hierarchy of Korteweg-de Vries Equations.- Perturbation Theory for the Double Sine-Gordon Equation.- III - Discrete Systems and Continuum Mechanics.- Painleve Transcendents and Scaling Functions of the Two-Dimensional Ising Model.- 1. Introduction.- 2. Two-Dimensional Ising Model.- 3. Scaling Limit and Scaling Functions.- 4. Explicit Formulas for $${hat F_ pm }(x)$$.- 5. Painleve Transcendents.- Statistical Mechanics of Nonlinear Lattice Dynamic Models Exhibiting Phase Transitions.- 1. Introduction.- 2. The Models and their Continuum Limit, the Equations of Motion and Particular Solutions.- 3. Dynamic Variables, Conservation Laws and Spectral Densities.- 4. Molecular-Dynamics Technique.- 5. Molecular-Dynamics Results; Static and Dynamic Equilibrium Properties.- 6. Nonlinear Heat-Pulse Propagation.- Nonlocal Continuum Mechanics and Some Applications.- 1. Introduction.- 2. Balance Laws.- 3. Constitutive Equations.- 4. Thermodynamic Restrictions.- 5. Linear Theory.- 6. Determination of Nonlocal Elastic Moduli.- 7. Surface Waves.- 8. Screw Dislocation.- 9. Fracture Mechanics.- 10. Nonlocal Fluid Mechanics and Turbulence.- IV - Nonlinear Field Theories and Quantization.- Quantization of a Nonlinear Field Equation.- 1. The Classical Theory.- 2. Symmetry Transformations.- 3. Quantum Mechanics of the Free Field Equation.- 4. Compactification of Time.- 5. Perturbation Expansion.- 6. The Classical Scattering Theory.- 7. The Quantization.- 8. The Commutation Relations.- 9. Quantization in Analogy to the Thirring Model.- Characteristic "Quanta" of Nonlinear Field Equations.- 1. Linear Fields.- 2. Nonlinear Fields.- 3. Multicomponent Fields.- 4. Nonlinear Chiral Fields.- 5. Physical Interpretation and Applications.- 6. Choice of Nonlinear Model.- Nonlinear Schrodinger Equation with Sources: An Application of the Canonical Formalism.- 1. A General Field Theoretical Problem.- 2. An Application of the Canonical Formalism.- Nonlinear Field Equations and Collective Phenomena.- 1. Introduction.- 2. A Simple Model.- 3. Presence of Pairing Force.- 4. Conclusion.- Nonperturbative Self-Interactions, Solitary Waves and Others.- 1. Introduction.- 2. Nonperturbative, Self-Interacting Quantum Fields.- 3. Solitary Wave Propagators.- 4. Solitary Waves and Others.- 5. Concluding Remarks.- Bound States of Fermions in External and Self-Consistent Fields.- 1. Solutions of the Dirac Equation.- 2. Quantum Field Theory of Spin-1/2 Particles in Strong External Fields.- 3. Supercharged Vacuum and Klein's Paradox.- 4. Strong Fields in Quantum Field Theory.