Streamers often constitute the first stage of dielectric breakdown in strong electric fields: a nonlinear ionization wave transforms a nonionized medium into a weakly ionized nonequilibrium plasma. New understanding of this old phenomenon can be gained through modern concepts of (interfacial) pattern formation. As a first step towards an effective interface description, we determine the front width, solve the selection problem for planar fronts, and calculate their properties. Our results are in good agreement with many features of recent three-dimensional numerical simulations.

We numerically investigate the critical properties of nonequilibrium continuous phase transitions in two-dimensional, synchronously updated lattices of coupled chaotic maps. A finite-size scaling analysis provides evidence for the existence of a new universality class, characterized by a correlation-length exponent $\nu =0.89\mathrm{}\pm 0.03<\mathrm{}{\nu }_{\mathrm{Ising}}=1.0\mathrm{}$, while the exponent ratios $\beta /\nu$, $\gamma /\nu$, and the amplitude ratio $U^{*}$ are consistent with the 2D Ising universality class. The standard value of $\nu$ is recovered for asynchronous updating rules.

A new method for the stabilization and manipulation of unstable states of a pattern forming system is presented. The technique is applied to an optical system where unstable homogeneous solutions, rolls, squares, hexagons, and honeycombs are all stabilized and tracked. The control consists of a small spatial modulation to the input pump field, which is derived from the Fourier transform of the output electric field. Once stabilization is achieved, the control vanishes. The method can be used as a numerical tool for pattern forming systems to determine the existence and stability of solutions.

We demonstrate the first experimental investigation of the interaction between black and gray solitons in an optical fiber. The fiber length equals 8 soliton distances. The repulsive character of the interaction is observed during the walk-off of the gray soliton through the black one inside the finite-width background carrier pulse. Opposite cases of the gray soliton phase profile are studied, corresponding to gray pulses with a group velocity higher and smaller than that of the carrier pulse. Numerical simulations presented show good agreement with experimental data.

Existence of a saturated steady solution of a nonlinear evolution equation subject to a boundary condition at $x=0\mathrm{}$, called a nonlinear global mode, is illustrated on the real subcritical Ginzburg-Landau model. Such a nonlinear global mode is shown to exist whereas the flow is linearly stable, convectively unstable, or absolutely unstable. If the linearized evolution operator is absolutely unstable, then a global mode exists but the converse is false. This result relies only on the existence of a structurally unstable heteroclinic orbit in the phase space and is likely to be generic as demonstrated by the supercritical Ginzburg-Landau and the van der Pol-Duffing equations.

We demonstrate the stabilization of unstable periodic orbits whose trajectories in phase space are distant from the unperturbed dynamics in a convective flow experiment. A model independent, nonlinear control algorithm uses temperature measurements near the free surface of a convecting liquid bridge to compute control perturbations which are applied by a thermoelectric element. The algorithm employs a time series reconstruction of a nonlinear control surface to alter the system dynamics.

We investigate experimentally the nonlinear dynamics of a femtosecond pulse undergoing self-focusing in a normally dispersive medium. Above a certain threshold power, we observe that the pulse splits into two pulses, and measurements of the pulse spectra provide evidence that supercontinuum generation is closely connected with the temporal splitting. We also observe that the split pulses can undergo additional splittings. Our observations are in good agreement with the predictions of the theoretical model based on the nonlinear Schrödinger equation describing self-focusing in dispersive media.

We demonstrate highly efficient transmission of light around sharp corners in photonic band-gap waveguides. Numerical simulations reveal complete transmission at certain frequencies, and very high transmission $(>95\mathrm{}\%)$ over wide frequency ranges. High transmission is observed even for 90° bends with zero radius of curvature, with a maximum transmission of 98% as opposed to 30% for analogous conventional dielectric waveguides. We propose a simple one-dimensional scattering theory model with a dynamic frequency-dependent well depth to describe the transmission properties.

The region of parameter space (acoustic pressure $P_a$, bubble radius $R_0$) in which stable single bubble sonoluminescence (SBSL) occurs in an air-water system is a small fraction of that which is accessible. This is due to the existence of an island of dissolution at high $P_a$ and small $R_0$. For dissolved gas concentrations above 50% of saturation, the region lies above the threshold for shape oscillations and is unobservable. Below 50%, an oscillating bubble is stabilized on the boundary of the island which lies below the shape threshold. SBSL is shown to exist exclusively along this boundary.

A new geometric framework connecting scale distributions to coverage statistics is employed to analyze level sets arising in turbulence as well as in other phenomena. A 1D formalism is described and applied to Poisson, lognormal, and power-law statistics. A $d$-dimensional generalization is also presented. Level sets of 2D spatial measurements of jet-fluid concentration in turbulent jets are analyzed to compute scale distributions and fractal dimensions. Lognormal statistics are used to model the level sets at inner scales. The results are in accord with data from other turbulent flows.

High-resolution direct numerical simulation data for three-dimensional Navier-Stokes turbulence in a periodic box are used to study the scaling behavior of low-order velocity structure functions with positive and negative powers. Similar to high-order statistics, the low-order relative scaling exponents exhibit unambiguous departures from the Kolmogorov 1941 theory and agree well with existing multiscaling models. No transition from normal scaling to anomalous scaling is observed.

A solid-state laser resonator is demonstrated which incorporates a real-time holographic element that corrects dynamically for both intracavity phase and polarization distortions. The results are compared with those from a holographic laser which corrects for phase distortions but does not include polarization correction. In comparison to the latter system the polarization correcting laser displays superior performance and notably, an output beam quality and energy that is insensitive to changes in intracavity polarization distortions.

The stability of twisted straight rods is described within the framework of the time dependent Kirchhoff equations for thin elastic filaments. A perturbation method is developed to study the linear stability of this problem and find the dispersion relations. A nonlinear analysis results in a new amplitude equation, describing the deformation of the rod beyond the instability, which takes the form of a pair of nonlinear, second-order evolution equations coupling the local deformation amplitude to the twist density. Various solutions, such as solitary waves, are presented.

It is shown that the idea that scaling behavior in turbulence is limited by one outer length $L$ and one inner length η is untenable. Every $n\mathrm{th}$ order correlation function of velocity differences ${\mathcal{F}}_n({\mathit{R}}_1,{\mathit{R}}_2,\dots )$ exhibits its own crossover length ${\eta }_n$ to dissipative behavior as a function of $R_1$. This depends on $n$ and on the remaining separations $R_2{,R}_3,\dots$. One result is that when separations are of the same order $R$, this scales as ${\eta }_n\left(R\right)\sim \eta (R/L{)}^{x_n}$ with $x_n=({\zeta }_n-{\zeta }_{n+1\mathrm{}}+{\zeta }_3-{\zeta }_2)/(2-{\zeta }_2)$, ${\zeta }_n$ the scaling exponent of the $n\mathrm{th}$ order structure function. We derive an infinite set of scaling relations that bridge the exponents of correlations of gradient fields to the exponents ${\zeta }_n$, including the “bridge relation” for the scaling exponent of dissipation fluctuations $\mu =2-{\zeta }_6$.

Radiation transport through high-opacity materials can be described using the Rosseland mean opacity of the medium, which is dominated by low-opacity regions in the frequency-dependent opacity. By mixing gold and gadolinium, we can fill in low-opacity regions of one material with high-opacity regions of another material, resulting in a material with a Rosseland mean opacity $1.5\times$ higher than either of the constituents. For a given laser energy, this can raise the temperature of the laser heated hohlraums, or for a given desired temperature, require less laser energy.

Near-field optical holography, i.e., a technique that allows one to record and reconstruct not only far- but also near-field components of optical fields, is conceptually formulated. A near-field hologram can be recorded with a reference wave, which is totally internally reflected by a surface with quasi-two-dimensional objects on it, and reconstructed with methods of near-field optical microscopy. Subwavelength resolution of the proposed technique is demonstrated by using both numerical simulations and experimental observations.

Interaction of a light beam with matter reveals extreme sensitivity to transverse motions. Spatio-temporal delocalization of interaction and diffusive transport of light-induced perturbations is revealed through a new family of diffraction patterns. The response of the medium to the transverse motion of the light enhances near the second order phase transition point. The phenomenon has large application potential for the study of motions that are beyond the power of Doppler-effect-based methods, and suggests fundamental problems about critical processes with nonstationary influences.

A crisis in chaotic dynamical systems is characterized by the conversion of a nonattracting, Cantor-set-like chaotic saddle into a chaotic attractor. The gaps in between various pieces of the chaotic saddle are densely filled after the crisis. We give a quantitative scaling theory for the growth of the topological entropy for a major class of crises, the interior crisis. The theory is confirmed by numerical experiments.

We present a theoretical model of photorefractive parametric oscillation that covers, for the first time, to our knowledge, the occurrence of the whole spectrum of parametric processes from transversal over degenerate to longitudinal parametric oscillation. It is shown that inclusion of so-called noneigenwaves is essential for completing the model. We report on the first experiment that shows the transition from transversal over degenerate to longitudinal parametric oscillation. The experimental observations agree well with the theoretical predictions.

Experiments on continuously sheared granular materials (glass spheres) with diameters $1.0\le d\le 5\mathrm{mm}$ show large fluctuations in the normal stress, $\sigma \left(t\right)$. Experiments are carried out in an annular Couette geometry for rotation rates $0.003<\mathrm{}\dot{\theta }<1\mathrm{}\mathrm{rad}/\mathrm{s}$. Power spectra from $\sigma \left(t\right)$, $P\left(\omega \right)$, show rate invariance in the fluctuations: $\dot{\theta }P\left(\omega \right)$ is a function only of $\omega /\dot{\theta }$, independent of $\dot{\theta }$. $\dot{\theta }P\left(\omega \right)$ depends relatively weakly on $d$. The distributions of stresses ${\rho }_{\sigma }$ are similar to recent predictions for static arrays, but the width of ${\rho }_{\sigma }$ varies only weakly with $d$, suggesting stronger spatial correlation effects than expected from theory.