Nonlinear Phenomena in Physics

It was almost four hundred years ago that Galileo wrote in Il Saggiatore that the "Book of Nature is written in mathema ti ca 1 characters". Thi s sentence, i nspi red at the dawn of physics has proved with the passage of time to contain a deep truth and also a warning: in order to understand Nature, first we must learn to read mathema­ tical characters. Indeed, writing physical law in such characters has proved not as hard as unraveling the content of the resulting equations. In particular, the lack of knowledge in the field of nonlinear mathematics has been a severe limita­ tion in the past. Thus the solution to equations such as the Navier-Stokes equation in fluid dynamics has remained elusive. The recent advent of fast computers and some important analytical and numerical results in the study of bifurcations and nonlinear waves have encouraged work both in theory and experiment involving non­ linear phenomena. An explosive growth in the specialized literature penetrating most research areas in physics in the last few years has ensued. This book contains the most recent advances in nonlinear physics in various fields including astrophysics, gravitation, particle physics, quantum optics, fluid dynamics and the mathematics underlying the phenomena of chaos and nonlinear waves. It presents a selection from the lectures delivered at the XXI '_atin American School of Physics held in Santiago, Chile in July-August 1984 (EtAF'84).

Inhalt

I Mathematical Methods and General.- Dynamics for Golden Mean Rotation.- 1. Introduction.- 2. Golden Mean Circle Maps.- 3. Trajectory Scaling.- 4. The Golden Mean Power Spectrum.- 5. Conclusion.- Nonlinear Phenomena in Dissipative Systems.- 1. Stability Analysis of Equilibria.- 2. Steady Non Equilibrium Behavior and Pattern Formation.- 3. Time Dependent Periodic Phenomena.- 4. Quasi periodic Phenomena.- 5. Chaotic Phenomena.- References.- Statistical Mechanics of the sine-Gordon Field: Part I.- 1. Introduction.- 2. Elementary Aspects of Action-Angle Variables: Action-Angle Variables for the Klein-Gordon Equation.- 3. The Ferromagnetic CsNiF3; A Physical Example of the sine-Gordon Equation.- 4. Complete Integrability of the Classical sine-Gordon Equation.- 5. Functional Integrals for the Partition Function Z.- 6. The Partition Function for the linear Klein-Gordon Equation.- 7. Transfer Integral Method for the Classical Partition Function of the sine-Gordon Equation.- Statistical Mechanics of the sine-Gorden Field: Part II.- 8. Evaluation of the Partition Function Z for the sine-Gordon Field in Terms of Action-Angle Variables: Construction of the Measure.- 9. Floquet Theory for Periodic Problems.- 10. Partition Function for the sinh-Gordon Equation.- 11. Sine-Gordon Statistical Mechanics Without Phonon-Phonon Phase-Shifts.- 12. Sine-Gordon Statistical Mechanics with Phonon-Phonon Phase-Shifts Included.- 13. Classical Fermions.- 14. Unsolved Problems.- 15. Conclusion.- Appendices.- References.- Probabilistic Cellular Automata.- 1. Introduction.- 2. Automata Dynamics.- 3. The Stick-Slip Behavior of Faults.- References.- II Quantum Optics.- Chaos, Generalized Multistability and Low Frequency Spectra in Quantum Optics.- 1. Introduction: Order and Chaos in Quantum Optics.- 2. Physics of Stimulated Emission Processes.- 3. The Maxwell-Bloch Equations.- 4. Onset of Chaos, Generalized Multistability and low Frequency Spectra in Quantum Optic s.- References.- From Optical Bistability to Chaos.- 1. Introduction.- 2. The Ring Cavity.- 3. The Maxwell-Bloch Equations.- 4. The Linear Approximation.- 5. Optical Bistability.- 6. Semiclassical Theory: Steady State.- 7. Discussion of the State Equation.- 8. Linear Stability Analysis.- 9. Dispersive Optical Bistability.- 10. Quantum Theory of Optical Bistability.- 11. Spectrum of the Transmitted Light.- 12. Photon Statistics.- 13. Side Mode Instabilities.- 14. Mullistability.- 15. Ikeda Instability. Appraoch to Chaos.- References.- III Fluids.- Recent Developments in Rayleigh-Benard Convection and Interface Dynamics.- 1. The Rayieigh-Benard Instability.- 2. Fluid-Fluid Interface, the Saffman-Taylor Problem.- 3. Solid-Liquid Interface, from Cells to Dendrites.- References.- IV Astrophysics and General Relativity.- Nonlinear Problems in Stellar Dynamics.- 1. Introduction.- 2. Stellar Motions on the Plane of Symmetry of a Spiral Galaxy.- 3. Stellar Motions on the Meridian Plane of an Axisymmetric Galaxy.- 4. Transition from Ordered to Stochastic Motion.- 5. Bifurcations and Gaps.- 6. Three-Dimensional Systems.- References.- Solitons in General Relativity and Supergravity.- Lecture 1: Black Hole Solitons in Ungauged Extended Supergravity.- 1. The Soliton Concept for Gravity.- 2. Absence of Solitons Without Horizons.- 3. Black Holes and the Breakdown of Supersymmetry at Finite Temperature.- 4. Extended Supersymmetry: Central Charges and Multiplet Shortening.- 5. Black Holes in N=2 Supergravity: Charge Without Charge.- 6. The Bogomolny Inequality.- 7. The Zero Modes.- 8. Conclusion.- References.- Lecture 2: Supersymmetric Monopoles in Kaluza-Klein Theory.- 1. Kaluza-Klein Theory.- 2. Gross-Perry-Sorkin Monopoles.- 3. N = 8 Supergravity in 5-Dimensions.- 4. Supersymmetries and Zero Models.- 5. Electric Magnetic Duality.- 6. Monopole-Pyrgon Duality.- References.- The Dynamics of Spacetime Curvature: Nonlinear Aspects.- 1. Introduction.- 2. Gravitational Waves.- 3. Geons.- 4. Black Holes.- 5. White Holes.- 6. Wormholes.- 7. Singularities.- 8. Techniques of Future Research: Soliton Theory and Numerical Relativity.- References.- V High Energy Physics.- Unification, Supersymmetry and Cosmology.- 1. The Standard Model.- 2. From the Standard Model to Grand Unification.- 3. Simple GUT Model s.- 4. New Interactions in GUTs.- 5. The Hierarchy Problem.- 6. Supersymmetric GUTs.- 7. Supersymmetry at Low Energies.- 8. Supersymmetry and CERN Collider Data.- 9. GUTs and Cosmology.- 10. Inflation and Supersymmetric Cosmology.- References.- Lectures on Quark Flavor Mixing in the Standard Model.- 1. Overview of the Standard Model.- 2. Phenomenological Determination of the Cabibbo-Kobayashi-Maskawa Matrix Elements.- 3. The ?S = 1 Effective Hamiltonian for Non-Leptonic Decays in the Standard Model.- 4. Violation of CP-Invariance in the Standard Model.- References.- Index of Contributors.