Nonlinear Equations in Physics and Mathematics

This is the third Volume in a series of books devoted to the interdisciplinary area between mathematics and physics, all ema­ nating from the Advanced Study Institutes held in Istanbul in 1970, 1972 and 1977. We believe that physics and mathematics can develop best in harmony and in close communication and cooper­ ation with each other and are sometimes inseparable. With this goal in mind we tried to bring mathematicians and physicists together to talk and lecture to each other-this time in the area of nonlinear equations. The recent progress and surge of interest in nonlinear ordi­ nary and partial differential equations has been impressive. At the same time, novel and interesting physical applications mul­ tiply. There is a unifying element brought about by the same characteristic nonlinear behavior occurring in very widely differ­ ent physical situations, as in the case of "solitons," for exam­ ple. This Volume gives, we believe, a very good indication over all of this recent progress both in theory and applications, and over current research activity and problems. The 1977 Advanced Study Institute was sponsored by the NATO Scientific Affairs Division, The University of the Bosphorus and the Turkish Scientific and Technical Research Council. We are deeply grateful to these Institutions for their support, and to lecturers and participants for their hard work and enthusiasm which created an atmosphere of lively scientific discussions.

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.- Painlevé 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. Painlevé 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 Schrödinger 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.