A Reply to the Comment by Helmut R. Brand and Harald Pleiner.

We present an $ħ\to 0$ approximation for the quasienergy spectrum of a periodically kicked top, valid under conditions of both regular and chaotic classical motion. In contrast to conventional periodic-orbit theory we implement the semiclassical limit for each matrix element of the Floquet operator rather than for the trace of the propagator. Even though a classical looking action is involved, the approximate matrix elements are specified in terms of complex ghost trajectories instead of real classical orbits. Our mean error for the quasienergies is a surprisingly small 3% of the mean spacing.

For the first time the statistics of the spatial correlations of eigenfunctions in a random system is investigated. It is rigorously shown that the wave functions corresponding to different energy levels are uncorrelated in space. Within a given eigenstate the joint distribution of the electron density at two different space points is described by a one-parameter universal law, where the parameter is the Friedel function. It is found that this distribution is decoupled in the region of small fluctuations of the electron density and reveals an increasing spatial correlation for the large fluctuations.

We derive return maps and phase diagrams to identify mechanisms of synchronization of pulse-coupled oscillators and emphasize the importance of temporal delays and inhibitory coupling. Optimum synchronization between two oscillators is obtained for inhibitory coupling, where delays give rise to stable in-phase synchronization; those with excitatory coupling only synchronize with a phase lag. In large ensembles of globally coupled oscillators the delayed interaction leads to new collective phenomena like synchronization in multistable clusters of common phases for inhibitory coupling; for excitatory coupling a mechanism of emerging and decaying synchronized clusters prevails.

We report the observation of a gain of approximately $\exp \left(11\right)$ at 41.8 nm in 8-times-ionized xenon. This extreme ultraviolet (XUV) laser is driven by a 10-Hz, 70-mJ circularly polarized femtosecond laser pulse. The laser is focused into Xe at pressures ranging from 5 to 12 torr. The laser is collisionally excited, with both the ions and electrons produced by field induced tunneling.

In a binary mixture of water and ethanol, convection sets in in the form of traveling waves. Fluctuating patterns are observed below the convection threshold and interpreted as the result of thermal fluctuations. The study presented here is the first to identify these fluctuations below the convective instability, and, in particular, a recently predicted crossover in the divergence of the intensity of fluctuating traveling waves is observed for the first time.

Beyond the primary bifurcation, hydrodynamic pattern-forming systems often show a transition from a structureless base state to a pattern of parallel lines. At the next bifurcation it is commonly observed that these lines become unstable by the Eckhaus or the zigzag mechanism. In open flow systems, one usually has to distinguish between the absolute and the convective thresholds of instability. This paper investigates how an external flow affects these secondary instabilities. It determines when a spatially periodic line pattern becomes absolutely or convectively unstable with respect to Eckhaus or zigzag perturbations. Experiments to test our theoretical predictions are suggested.

For helium, the accuracy of calculated transport properties and virial coefficients based on an accurate ab initio potential now exceeds that of the best measurements. The ab initio results should be used to calibrate measuring apparatus.

A new class of chaotic systems is generated in a practical, nonlinear, mutually coupled phase-locked loop (PLL) circuit. Theoretical results make it possible to understand experimental results of such PLL's on the onset of chaos using the geometry of the invariant manifolds, while the resultant simple geometry and complex dynamics is expected to have applications in other areas, e.g., power systems or interacting bar magnets. The 1D map projected from a numerically obtained attractor indicates the existence of a piece-wise linear structure having a sensitive dependence on the bifurcation parameter $f_{02}$.

In the study of wave transport, a randomly layered medium is generally considered one dimensional due to its localizing character in the layering direction. However, we show that when the system is perturbed by the addition of a small amount of isotropic randomness, wave propagation in the direction parallel to the layers can exhibit significant leakage out of the localizing layers. Numerical study of this phenomenon in 2D anisotropic models reveals the existence of wave-energy tunneling via evanescent waves in the layering direction, which gives rise to finite-size scaling behavior intermediate between 1D and 2D.

Following Einsteins's argument for the diffusion coefficient of colloidal dispersions, we postulate that the steady state concentration profile in a suspension of noncolloidal monodisperse particles reflects the dynamic equilibrium resulting from a balance between gravity-driven convection and hydrodynamic dispersion. Using an acoustic technique, the steady state concentration profile of a counterflow-stabilized suspension (a fluidized bed) as well as stationary propagating sedimentation fronts inside the bed were determined. From these profiles we can derive the concentration dependence of the hydrodynamic dispersion coefficient.

A Comment on the Letter by R. Blümel and B. Esser Phys. Rev. Lett. 72, 3658 (1994).

A Reply to the Comment by Paolo Allegrini et al.

The spectrum of hydrogen in a magnetic field is a paradigm of quantum chaos and may be analyzed accurately by periodic-orbit-type theories. In nonhydrogenic atoms, the core induces pure quantum effects, especially additional spectral modulations, which cannot be analyzed reliably in terms of classical orbits and their stability parameters. Provided core-scattered waves are included consistently, core-scattered modulations as well as corrected amplitudes for primitive orbits are in excellent agreement with quantum results. We consider whether these systems correspond to quantum chaos.

Absolute instabilities in the gyrotron traveling wave amplifier are investigated with a simulation approach which models electron and wave dynamics in the sever and distributed-loss section. Distributed wall losses are shown to be far more effective than the sever in stabilizing these instabilities. Physical interpretations are given and theoretical predictions are verified by a $K_a$-band experiment which achieved 62 kW peak power with 12% bandwidth, 21% efficiency, and 33 dB saturated gain through the use of a mechanically adjustable magnetron injection gun.

Numerical simulations have been used to determine the influence of stochastic noise on the lifetimes of chaotic transients. The general literature on the subject of noise and chaos appears to suggest that there should be a significant noise dependency, but remarkably none was found for the model systems reported here. These null results are confirmed by direct measurements from an electronic analog of one of the systems.

Through large-scale simulations of a three-dimensional, deterministic cellular automaton, we demonstrate the existence of algebraic spatial and temporal correlations in nonequilibrium systems that break a continuous time-translation symmetry to produce coherent periodic oscillations. Our results provide the first numerical support for a recent hypothesis that phase fluctuations in such systems are described by the Kardar-Parisi-Zhang equation. The coefficient of the nonlinear term in the equation is determined numerically.

Using 1D short-wavelength (8 μm) Compton free-electron-laser (FEL) equations with slippage, we explore propagation of a high-power pulse down a tapered undulator FEL traveling-wave amplifier. For an appropriate taper, a short pulse (∼300 fsec FWHM) with regular features will propagate self-similarly as it grows in power, slipping through a much longer electron pulse. The power spectrum of the pulse is nearly Gaussian with no sidebands. The electron energy is depressed by ∼33%, but slippage causes the peak pulse intensity to be about the same as the electron-beam power density, ∼10 TW/${\mathrm{cm}}^2$.

We show that the critical behavior of a driven interface, depinned from quenched random impurities, depends on the isotropy of the medium. In anisotropic media the interface is pinned by a bounding (conducting) surface characteristic of a model of mixed diodes and resistors. Different universality classes describe depinning along a hard and a generic direction. The exponents in the latter (tilted) case are highly anisotropic, and obtained exactly by a mapping to growing surfaces. Various scaling relations are proposed in the former case which explain a number of recent numerical observations.