A Comment on the Letter by Gregory L. Eyink, Phys. Rev. Lett. 74, 3800 (1995). The authors of the Letter offer a Reply.

A Reply to the Comment by G. Falkovich and V. Lebedev.

Random banded matrices with linearly increasing diagonal elements are recently considered as an attractive model for complex nuclei and atoms. Apart from early papers by Wigner there were no analytical studies on the subject. In this Letter we present analytical and numerical results for local spectral density of states (LDOS) for a more general case of matrices with a sparsity inside the band. The crossover from the semicircle form of LDOS to that given by the Breit-Wigner formula is studied in detail.

We analyze the classical and quantum properties of the integrable dimer problem. The classical version exhibits exactly one bifurcation in phase space, which gives birth to permutational symmetry broken trajectories and a separatrix. The quantum analysis yields all tunneling rates (splittings) in leading order of perturbation. In the semiclassical regime the eigenvalue spectrum obtained by numerically exact diagonalization allows one to conclude about the presence of a separatrix and a bifurcation in the corresponding classical model.

We study quantum stochastic resonance (QSR) in the periodically driven biased spin-boson system. The amplitude and the phase of the nonlinear response function vs temperature are studied in the incoherent and coherent tunneling regimes, encompassing adiabatic and nonadiabatic driving, weak and strong Ohmic coupling $\alpha$. For $\alpha =1/2\mathrm{}$ we present exact QSR results. As a general feature, a principal maximum appears when the static asymmetry exceeds the driving frequency and strength. Moreover, the nonlinear QSR shows a noise-induced suppression of higher harmonics and a characteristic phase shift.

Diffraction corrections to the semiclassical spectral density of dispersing (Sinai) billiards, due to orbits which are almost tangent to the concave part of the boundary, are studied here for the first time. We show that most periodic orbits needed for quantization must be corrected. For orbits which just miss tangency, the corrections are of the same magnitude as the semiclassical contributions themselves. For orbits which glance at an extreme forward direction, the new theory replaces the semiclassical term that approaches $0$ at tangency with a finite one. These corrections are one of the most significant modifications of the trace formula considered so far.

Spiral waves are investigated in chemical systems whose underlying spatially homogeneous dynamics is governed by a deterministic chaotic attractor. We show how the local periodic behavior in the vicinity of a spiral defect is transformed to chaotic dynamics far from the defect. The transformation occurs by a type of period doubling as the distance from the defect increases. The change in character of the dynamics is described in terms of the phase space flow on closed curves surrounding the defect.

Stable localized states are predicted for a saturable absorber in an optical cavity. These “bullet holes” resemble 2D spatial solitons, and we demonstrate an optical memory array scheme based on them. A topological argument shows that one or more unstable localized states coexist with the stable bullet hole.

We report the first experimental observation of nonlinear propagation effects in fiber Bragg gratings, resulting in nonlinear optical pulse compression and soliton propagation. The solitons occur at frequencies near the photonic band gap of the grating; they are due to a combination of the negative dispersion of the grating, which dominates the material dispersion, and self-phase modulation. The solitons propagate at velocities well below the speed of light in the uniform medium.

Numerical study of the one-dimensional nonlinear partial differential equation, equivalent to that proposed [Recent Advances in Engineering Science (Springer-Verlag, Berlin, 1989)] to describe longitudinal seismic waves, is presented. The equation has a threshold of short-wave instability and symmetry, providing slow long-wave dynamics. It is shown that the threshold of the short-wave instability corresponds to a point of “continuous” (second order) transition from a spatially uniform state to a chaotic regime. Thus, contrary to the conventional scenarios, turbulence arises from the spatially uniform state as a result of one and the only one supercritical bifurcation.

We consider the problem of prediction and nonlinear modeling for chaotic time series and examine the effects of changing the local metric used to select nearest neighbors in the embedding space of delay register vectors. Analyzing simulated numerical data and real data, it is shown that the fit achieved for the case where the components of the metric tensor are constants over the whole attractor is improved by a proper selection of the local metric. Our results also suggest how deviations from the Euclidean case can be used as a tool to discriminate chaos from correlated noise.

A major barrier in semiclassical calculations for chaotic systems is the exponential increase in the number of terms at long times. Using an analogy with spin-chain partition functions, we overcome this “exponential wall” for the baker's map, reducing to order ${\mathrm{NT}}^{3/2}$ the number of operations needed to evolve an $N-\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}$ system for $T$ time steps. This method enables us to obtain semiclassical results up to the Heisenberg time and beyond, providing new insight as to the accuracy of the semiclassical approximation. The semiclassical result is often correct; its breakdown is nonuniform.

We show that the propagation equation for the slowly varying envelope of the electric field in a homogeneous dispersive nonisotropic medium contains terms that rotate the 3D wave packet of an optical pulse propagating as an extraordinary wave about an axis perpendicular to the propagation vector. It also possesses Fresnel diffraction coefficients that depend not only on the refractive index but also its derivatives with respect to direction. An analytic expression for the slowly varying envelope is obtained for an initial Gaussian wave packet by keeping terms up to second order in the wave equation for the slowly varying envelope.

We study the two-dimensional localization problem for (i) a classical diffusing particle advected by a quenched random mean-zero vorticity field, and (ii) a quantum particle in a quenched random mean-zero magnetic field. Through a combination of numerical and analytical techniques we argue that both systems have extended eigenstates at a special point in the spectrum invariant under particle-hole symmetry, where a sublattice decomposition obtains. In a neighborhood of this point, the Lyapunov exponents of the transfer matrices acquire ratios characteristic of conformal invariance allowing an indirect determination of $1/r$ for the typical spatial decay of eigenstates.

We studied thermal turbulence in a low Prandtl number, Pr, fluid using liquid Hg ( $\mathrm{Pr}=0.024\mathrm{}$). For Rayleigh numbers ( ${10}^6<\mathrm{Ra}<{10}^8$), the histogram of temperature fluctuations at the center of the cell (aspect ratio 1), and the scaling laws for the Nusselt number and the large scale flow, all showed He-like hard turbulence. However, the cascade range of the frequency spectrum of the temperature fluctuations was smaller than expected. Direct measurement showed that the thermal and viscous boundary layers had crossed, which may explain the peculiar scaling.

A Comment on the Letter by Narayanan Menon, Sidney R. Nagel, and David C. Venerus, Phys. Rev. Lett. 73, 963 (94). The authors of the Letter offer a Reply.

A Reply to the Comment by Claus F. Behrens, et al.

Parameters of a given model describing a (chaotic) dynamical system are estimated from scalar time series using autosynchronization where the parameter adaption process is controlled by the synchronization of the model to the given dynamics. A practical method is presented for deriving the necessary ordinary differential equations for the parameter controlling loop.

A two-photon detection scheme is used to measure three- and four-photon correlations in a light beam and study their time dependence.

We present a path-integral Monte Carlo study of dissociation in dense hydrogen ( $1.75\le r_s\le 2.2$, with $r_s$ the Wigner sphere radius). As the temperature is lowered from ${10}^5$ to 5000 K, a molecular hydrogen gas forms spontaneously from a neutral system of protons and electrons. At high density, $r_s<2.0\mathrm{}$, thermally activated dissociation is accompanied by decreasing pressure, signaling the presence of a first order transition and critical point. The decrease in electron kinetic energy during dissociation is responsible for the pressure decrease and transition. At lower density the phase transition disappears.