Abstract: A summary of the spectral theory for quasiperiodic sine‐ and sinh‐Gordon equations is given. Analogies with whole‐line solitons and scattering theory motivates the discussion. The relation between the ingredients in the inverse spectral solution of the periodic sine‐Gordon equation and physical characteristics of sine‐Gordon waves is emphasized. The explicit topics covered are summarized in the table of contents in the Introduction.

Abstract: A quantum mechanical mnemonic which identifies the energy levels corresponding to those obtained by quantization of the classical quasiperiodic motion in coupled anaharmonic oscillator systems is presented. The mnemonic reproduces the known classical regular to irregular behavior as a function of the energy. The mnemonic is based on the definition of a suitable effective Hamiltonian for each level. Quasiperiodic levels are those in which the wave function has a weight greater than 50% on a degenerate subspace of any separable part of the Hamiltonian. This condition ensures that there exists an effective Hamiltonian defined on this degenerate subspace which commutes with the separable Hamiltonian; the latter is therefore an effective constant of motion for the total Hamiltonian. In some cases, depending on the degenerate subspace, other effective constants of motion exist. There is a correlation between the semiclassical quantization methods of quasiperiodic trajectories and the present effective Hamiltonian approach.

Abstract: In this paper, we investigate the quantum dynamics of wave packets of bound states in the nonlinear Henon–Heiles (HH) system. The time evolution of a wave packet was separated into two processes involving (a) the motion of its first moments and (b) its spreading. On the time scale when wave packet spreading can be neglected, we were able to establish relations between wave packet quantum dynamics expressed in terms of the initial‐state population probability P(t) and classical trajectory dynamics. We have demonstrated that the initial states, which are characterized by a periodic time evolution of P(t) over many periods, are related to the five elliptic periodic orbits in the HH system. The decay of P(t) over a few vibrational periods is related to the classical features of quasiperiodic trajectories and can be accounted for in terms of overlap reduction effects. When the center of the initial wave packet is moved further away from the classical periodic orbits, P(t) is characterized by a single initial p...

Abstract: Quasiperiodic solutions of perturbed integrable Hamiltonian equations such as weakly coupled harmonic oscillators can be found by constructing an appropriate coordinate transformation which leads to a "small divisor problem". However the numerical difficulties are not merely caused by the "small divisors" but rather by the appearence of ghost solutions, which appear in any reasonable discretization of the problem. Our numerical treatment, based on a Newton-type iteration, guarantees an approximation of the relevant solution of the nonlinear problem. Numerical solutions are found up to a critical value of the coupling constant, which is much larger than the coupling constants allowed by the existence theory available so far.

Abstract: The stability of the null state for a nonlinear Burgers system is examined. The results include (i) an energy estimate for global stability for states involving arbitrary modulation in time, and (ii) an analysis of the bifurcation from the null state for slow modulations. For the slow modulations it is determined that the amplitude A(tau) of the bifurcated disturbance velocity satisfies a Landau-type equation with time-dependent growth rate theta(tau). Particular attention is given to periodic and quasiperiodic modulations of the system, which lead to analogous behavior in theta(tau). For each of these oscillatory-type modulations, it is found that A/sup 2/(tau) has the same long-time mean value as the unmodulated case, implying no alteration of the final mean kinetic energy. Applications to various fluid-dynamical phenomena are discussed.

Abstract: A line shape formalism is derived classically which considers molecules undergoing both collisional and intramolecular relaxation. Results for a model system demonstrate a significant effect on the line shapes when chaos sets in and the intramolecular relaxation becomes important. An experiment on polyatomic molecules is proposed and should isolate the two effects.

Abstract: Zahlreiche technische Regelungsprobleme ζ. Β. die Blindstromkompensation und die aktive Filterung in der Starkstromtechnik bedingen durch die technische Ausführung des Stellgliedes einen speziellen Verlauf der Stellgrößen, welcher sich stückweise durch endliche Fourierreihen darstellen läßt. Im folgenden wird gezeigt, wie sich dieses Problem auf ein lineares und zeitinvariantes, zeitdiskretes Mehrgrößenregelungsproblem zurückfuhren läßt, wenn man die Fourierkoeffizienten selbst als Stellgrößen interpretiert und die entsprechenden Meßgrößen durch gleitende Mittelung erzeugt.

Abstract: A numerical study on the spectra of the vibrational motion of the isotopic species of water molecule is performed. At low energy the classical motion is quasiperiodic and the spectra show well resolved lines whose frequencies can be identified with the transition frequencies between vibrational states. As energy is increased, the motion becomes chaotic and the spectra are broadened. The effects of isotopic substitution is investigated. Freezing the bending mode affects the motion considerably. A cross-correlation function is introduced and its spectral features studied.

Abstract: Anharmonic Properties near Structural Phase Transitions.- Intention and Summary.- I. From early Experiments to the New View.- a) Classical and Nonclassical Behaviour.- b) Central-Peak Research.- c) Computer Simulation, and Anharmonic Theory.- d) The New View.- II. Recent Consolidating Results.- a) Anharmonicity at TC in SrTiO3.- b) Displacive-Order-Disorder Crossover in Ferroelectric Oxides.- c) Evidence for Precursor Order in RbCaF3 and KMnF3.- d) Dynamic Components in SrTiO3.- e) Critical Dyamics in KMnF3.- III. Incommensurate Systems.- The Theory of Structural Phase Transitions: Universality and Quasi-Elastic Scattering Phenomena.- Preamble.- 1. Background.- 1.1 A Simple Model and the Basic Observables.- 1.2 General Theory of Scaling Susceptibilities.- 2. Novel Universal Quantities.- 2.1 Adiabatic and Isothermal Susceptibilities.- 2.2 The Onset of Superlattice Scattering.- 3. Universal Patterns of Short Range Order.- 3.1 Introduction.- 3.2 General Theory of the Block Coordinate p.d.f.- 3.3 One Dimension: Exact Results.- 3.4 Two and Three Dimensions: Approximate Results.- 3.5 Discussion.- 3.6 Summary and Prospects.- Dynamic Correlations in the Ordered Phase of Perovskites.- I. Introduction.- II. Model.- III. Dynamic Correlations below TC.- IV. Results and Conclusions.- Phason Light Scattering in BaMnF4.- Nonlinear Excitations in Some Anharmonic Lattice Models.- 1) Introduction.- 2) Nonlinear Excitations in a model Ferroelectric gl Static Solutions 33 Stationary Solutions.- Stability of Static Periodons.- The Periodon-Phonon Coupling.- 3) Some Remarks on Toda-Lattice.- Solitons in The One-Dimensional Planar Ferromagnet CsNiF3.- 1. Abstract.- 2. Introduction.- 3. Experiment.- 4. Data evalutation.- On The Possibility to Create Nonthermal Solitons in a One-Dimensional Magnetic Sine Gordon System.- 1. Introduction.- 2. The Model.- 3. Results.- 4. Discussion.- Transport and Fluctuations in Linear Arrays of Multistable Systems.- I. Classification of Multistable Systems.- II. Open Systems.- Thermal Instabilities in Electrical Conductors.- Chains of Phase Couples and Externally Synchronized Oscillators.- III. Transport in The Driven Sine-Gordon Chain.- Propagation Velocity of Driven Kinks.- The Steady State Density of Kinks.- IV. Fluctuations in The Sine-Gordon Chain.- The Kink Counting Approach.- The Hydrodynamic Approach.- Universality.- Appendix: The Simplicity of Transport in the Smoluchowski Equation.- Non-Linear Thermal Convection.- 1. Introduction.- 2. The Basic Equations and Boundary Conditions.- 3. The Linear Theory.- a. The Horizontally Unbounded Layer.- b. The Effect of Lateral Walls.- 4. Non-Linear Convection.- 4.1 Perturbation Approach.- 4.2 Numerical Approaches.- 4.3 Comparison with Experiments.- 4.4 The mean field theory.- 5. Convection in a Small Prandtl Number Fluid.- Nematic Instability Induced by An Elliptical Shear.- I. Experimental Apparatus.- II. Instability Mechanism.- III. Modification of The Convective Structure on Increasing the Control Parameter N.- IV. The Time Evolution of The Convective Structure at Fixed N.- Instabilities and Fluctuations.- 1. Role of Fluctuations.- 2. Description of Random Processes.- 3. Phase Transitions.- 4. The Laser Instability.- 5. The Transition to Chaos in Optics.- 6. Instability Transients.- 7. Fluctuations in Hydrodynamic Instabilities.- Steady States, Limit Cycles and The Onset of Turbulence. A Few Model Calculations and Exercises.- 1. Introduction: From Steady States to Chaos in The Time Evolution of Non-Linear Systems.- 1.1 Background.- 1.2 Landau Picture of The Onset of (Quasiperiodic Gaussian) Turbulence.- 1.3 Lorenz Model, Discrete Maps and Related Matters.- 2. Steady States, Limit Cycles and Phase Transition picture.- 2.1 Steady States and Their Stability.- 2.2 Phase Transition Picture: An Illustration of Phase Coexistence. Tricritical points, Triple Points, Etc. in Convective Instability.- 2.3 Limit Cycles and Their Stability.- 3. Strange Attractors.- 3.1 Fractal Dimension: An Illustration.- 3.2 Lyapunov Characteristic Exponents.- 3.3 Exercise: Lorenz Model.- 3.4 Exercise: Two-Component Lorenz Model.- 4. Discrete Maps: Feigenbaum's Cascade and Pomeau's Intermittencies.- 4.1 Period-Doubling Cascades.- 4.2 Pomeau's Intermittencies.- The Physical Mechanism of Oscillatory and Finite Amplitude Instabilities in Systems with Competing Effects.- I. Introduction.- II. Oscillatory Instability.- III. Finite Amplitude Instability.- IV. Concluding Remarks.- Investigation of Fluctuations and Oscillatory States in Rayleigh-Benard Systems by Neutron Scattering.- 1. Introduction.- 2. Observations of Oscillatory States in a Homeotropic Nematic Sample.- A Rayleigh-Benard Experiment: Helium in a Small Box.- I. The Experiment.- II. Basics of Convection: Busse Theory.- III. The Wavenumber Selection: Effect of Side Walls and The Dynamics of a 3 Rolls to 2 rolls Transition.- IV. The Oscillatory Instability.- V. Two Oscillators: Entrainment and Locking.- VI. Routes to Turbulent Convection.- Period Doubling Bifurcation Route to Chaos.- Space-Time Symmetry in Doubly Periodic Circular Couette Flow.- 1. Introduction.- 2. Review of Experimental Results.- 3. Theory.- The Structure and Dynamics of Non-Stationary Taylor-Vortex Flow.- Pattern Formation During Crystal Growth: Theory.- 1. Introduction.- 2. Dendrites.- 3. Directional Solidification.- 4. Eutectics.- Electrothermal Instabilities at Magnetic Critical Points.- Abstract.- 1. Electronic Transport Coefficients at TC.- 2. Differential Negative Resistivity Conditions near TC.- 3. Continuously Forced Ballast Resistor.- Instabilities During Crystal Growth: Experiments.- 1. Introduction.- 2. Normal Growth and Lateral Growth.- The Nonfaceted - Faceted Transition.- The Solid-Vapor Interface.- The Solid-Solution Interface.- The Solid-Melt Interface.- 3. Sharp Interface-Diffuse Interface.- Light Scattering at the Solid Liquid Interface.- Quasi Eleastic Light Scattering Technique.- Light Scattering at the Ice-Water Interface.- Intensity Hysteresis and Dynamics.- Transient Phenomena.- Light Scattering at the Solid-Liquid Interfaces of D2O and Salol.- 4. Growth Into Supercooled Melt.- Growth of Dendrites.- 5. The Dynamics of Freezing and Melting.- Periodic Fluctuations at The Solid Liquid Interface of Salol.- 1. Introduction.- 2. Experimental.- 3. Observations.- 4. Oscillatory Scattering.- 5. Diffusive Scattering.- On The Dynamics of Epitaxial Phase Transformations.- 1. Surface Molecular Dynamics (SMD).- 2. Surface-Liquid Systems.- The Pinning of a Domain Wall by Weakened Bonds in Two Dimensions.- 1. Introduction.- 2. Solution of The Model.- 3. Discussion.- Melting in Two Dimensions.- I. Introduction.- Theoretical Background.- II. Dislocation Mediated Melting.- Dislocation Mediated Melting at a Smooth Substrate.- Periodic Substrate Potentials.- III. Fluid Phases.- Experiments and Computer Simulations.- IV. Doslocation Theory of Liquid Crystal Films.- Molecular Dynamics Study of 2-D Melting: Long Range Potentials.- 1. Introduction.- 2. Molecular Dynamics (MD).- 3. Results.- a) Coulomb System.- b) Dipolar System.- Experimental Studies of Two Dimensional Melting.- 1. Introduction.- 2. Crytalline Order and Melting in 2D.- 3. Exoerimental 2D Phase Diagrams.- 4. Diffraction Lineshapes in the 2D Solid Phase.- 5. Incommensurate Melting.- 6. Commensurate Melting.- 7. Conclusion.- An Unusual Polymorphism in the 2D Melting: The Smectic F and I Phases.- 1. Structure of the SG Phase.- 2. Structure of the 2d SF and SI Phase.- 3. Analysis of the 2d Order.- 4. The SI and SF Phases and the 2d Melting Theory.- Participants.

Abstract: A pair of uncoupled linear oscillators have natural frequencies $\delta $ and $\gamma $ and damping coefficient proportional to $\lambda $. When they are coupled through a class of cubic nonlinearities, the null solution bifurcates at $\lambda = \lambda _c $. The bifurcating solutions are analyzed near the strong resonance $\delta = \gamma $ by relating these, $\delta - \gamma = \frac{1}{2}k\varepsilon ^2 $, to the amplitude $\varepsilon $ and using multiple scale theory. The resulting approximation is uniformly valid even for the far-from-resonance case $k = O( \varepsilon ^{ - 2} )$. An entrainment parameter $\rho $ is defined, $\rho = 2( \delta - 2 )/( \lambda - \lambda _c )$, for which a critical value $\rho _c $ exists. For $| \rho |\leqq \rho _c $, there are mixed-mode frequency-locked periodic oscillations. $| \rho | > \rho _c $, the oscillations depend on two frequencies and so are periodic or quasiperiodic depending on the ratio $\delta /\gamma $.

Abstract: Spectral properties of Schrodinger operators of the typeHɛ=−Δ+ɛV, where Δ is the Laplacian,V a quasiperiodic potential and ɛ a coupling constant, are developed.V is taken to be finite sum of exponentials with generic frequencies. For small ɛ a strong stability is shown. On the other hand, examples (in the finite diffeence case) are given, for which a transition in the type of spectrum occurs, as ɛ is increased.

Abstract: We discuss the possibility that, besides periodic and quasiperiodic crystals, there exist turbulent crystals as thermodynamic equilibrium states at non-zero temperature. Turbulent crystals would not be invariant under translation, but would differ from other crystals by the fuzziness of some diffraction peaks. Turbulent crystals could appear by breakdown of long range order in quasiperiodic crystals with two independent modulations.