We study solitary wave solutions of the higher order nonlinear Schrödinger equation for the propagation of short light pulses in an optical fiber. Using a scaling transformation we reduce the equation to a two-parameter canonical form. Solitary wave (1-soliton) solutions always exist provided easily met inequality constraints on the parameters in the equation are satisfied. Conditions for the existence of $N$-soliton solutions ( $N\ge 2$) are determined; when these conditions are met the equation becomes the modified Korteweg–de Vries equation. A proper subset of these conditions meet the Painlevé plausibility conditions for integrability.

The evolution of density profiles within a shock accelerated gaseous interface was investigated via a three-directional laser absorption diagnostic technique for negative, close to zero, and positive initial density jump of the interface. Results show that the direction of the shock wave acceleration could have a non-negligible effect on the mixing process development. Furthermore, the turbulent diffusion within the induced mixing zone is clearly identified from the thickening of the density profiles with time.

The Pyragas method for controlling chaos is investigated in detail from the experimental as well as theoretical point of view. We show by an analytical stability analysis that the revolution around an unstable periodic orbit governs the success of the control scheme. Our predictions concerning the transient behavior of the control signal are confirmed by numerical simulations and an electronic circuit experiment.

We calculate all four nonzero Lyapunov exponents for a three-dimensional, dilute, random Lorentz gas by combining dynamical systems and Boltzmann equation methods. In the presence of an external field and a Gaussian thermostat the Lyapunov exponents, calculated up to second order in the applied field, satisfy a conjugate pairing rule. Agreement of the results obtained here with those of computer simulations of Dellago and Posch [following Letter, Phys. Rev. Lett. 78, 211 (1997)].

We report on the numerical computation of the Lyapunov exponents and the Kolmogorov-Sinai entropy for a three-dimensional, dilute, random Lorentz gas both in equilibrium and in nonequilibrium steady states. In our method the phase- and tangent-space dynamics are treated exactly. Furthermore, we propose a highly efficient stochastic technique for the computation of Lyapunov spectra of dilute systems. Our results are in excellent agreement with analytical results of Latz, van Beijeren, and Dorfman [preceding Letter, Phys. Rev. Lett. 78, 207 (1997)].

We consider the statistics of wave function intensities in disordered and classically chaotic systems. We present numerical results for the two-dimensional tight-binding Anderson model of localization and for the hydrogen atom in a strong magnetic field. At low disorder or strong classical chaos, respectively, we find extended states which show a Porter-Thomas distribution of intensities according to the universal predictions of random matrix theory. Localization due to disorder or weakly unstable regions in classical phase space causes characteristic deviations from universality.

We report on the discovery of a transition in rings of coupled electronic circuits in the chaotic regime to a collective periodic state characterized by a time scale that is between two and three orders faster than that corresponding to an isolated circuit. This transition arises from a linear instability in the uniform synchronized state of the ring through a symmetric Hopf bifurcation. The same type of transition is also observed for other coupled chaotic systems, e.g., a ring of Lorenz attractors.

We demonstrate that the $(1+1)$ dimensional, normal-dispersion, nonlinear Schrödinger equation with an “internal viscosity” has a stable “dark” shock wave (SW) solution, which is the invasion of the empty (dark) domain into the energy-carrying one. It may be interpreted as an optical SW in a loss-compensated nonlinear optical fiber. We predict that it can be created experimentally with a temporal width of a few picoseconds at a carrier-wave background power about 10 W. We develop a theoretical analysis that captures the physics of the SW propagation. The prediction that the SW velocity has a constant value in the limit of small viscosity, and scales as the square root of the viscosity in the large viscosity limit, are confirmed by the full dynamics simulations.

A theory for stable bubble oscillations in high pressure sound fields is presented. It is based on the strong influence of the surface tension on the dynamics of small bubbles and takes into account rectified diffusion and the resonancelike response of small bubbles to very strong acoustic pressure amplitudes. This theory provides an explanation for the existence of small, stably oscillating bubbles that have been observed in experiments on sonoluminescence.

We propose a simple model for arch formation in silos. We show that small pertubations (such as the thermal expansion of the beads) may lead to giant stress fluctuations on the bottom plate of the silo. The relative amplitude $\Delta$ of these fluctuations is found to be power-law distributed, as ${\Delta }^{-\tau }$, $\tau \simeq 1$. These fluctuations are related to large-scale “static avalanches,” which correspond to long-range redistributions of stress paths within the silo.

The appearance of a new type of fast nonlinear traveling wave states in binary fluid convection with increasing Soret effect is elucidated and the parameter range of their bistability with the common slower ones is evaluated numerically. The bifurcation behavior and the significantly different spatiotemporal properties of the different wave states—e.g., frequency, flow structure, and concentration distribution—are determined and related to each other and to a convenient measure of their nonlinearity. This allows one to derive a limit for the applicability of small amplitude expansions. Additionally, a universal scaling behavior of frequencies and mixing properties is found.

An experimental investigation was conducted to determine the scaling of the mean velocity profile for a fully developed, smooth pipe flow. Measurements of the mean velocity profiles and static pressure gradients were performed at 26 different Reynolds numbers between $31\times {10}^3$ and $35\times {10}^6$. The profiles indicate two overlap regions: one which scales as a power law and one which scales as a log law, where the log law is only evident when the Reynolds number exceeds approximately $300\times {10}^3$. It is proposed that the velocity scales for the inner and outer regions are different, which is contrary to commonly accepted beliefs.

A moment method is used to solve the Enskog equation for the steady-state distribution under conditions of uniform shear flow. Comparison to nonequilibrium molecular dynamics demonstrates that the lowest order solution gives a good quantitative description of nonlinear effects such as shear thinning and normal stresses in a moderately dense fluid. The results are used as a basis in the formulation of a simple and quantitatively accurate kinetic model of the Enskog equation.

A Comes on the Letter by James E. Martin and Judy Odinek, Phys. Rev. Lett. 75, 2827 (1995). The authors of the Letter offer a Reply.

A Reply to the Comment by Liang Fu and Lorenzo Resca.

A large enhancement of the optical Kerr nonlinearity associated with light-induced molecular orientation is observed in a liquid of anisotropic organic molecules when it is doped with a small amount of dye. The effect is ascribed to a photoinduced intermolecular force field, whose orienting action adds to the direct optical field action. The experimental results are in agreement with a model based on such interpretation. The proposed mechanism is in principle very general, and should work in all absorbing liquids made of anisotropic molecules.

Optical nonlinearity of liquid crystals (LC) in the isotropic phase can be enhanced by 1 order of magnitude by dissolving 0.1% of anthraquinone dye in the LC. The enhancement decreases by $\sim 30\%$ when the LC transforms into the nematic phase. The same guest-host effect also exists in non-LC liquids. It can be explained by a model based on the change of guest-host interaction induced by optical excitations of the dye.

Optical second harmonic generation from Si(111)-Si${\mathrm{O}}_2$ interface shows a strong nonmonotonic dependence on the oxide thickness between 2 and 300 nm. The Brewster angle of incidence, $p$-in, $p$-out combination of polarizations and strong, uniform suppression of this effect by near-index matching fluid exclude trivial multiple reflections and microscopic interface effects. The observation can be interpreted to originate from the optical (Casimir) nonlocality stemming from the thickness-dependent electron-electron interaction via virtual photons of the quantized electromagnetic field.

Many mixtures of granular materials tend to segregate when tumbled in a rotating horizontal cylinder, with the different components separating into bands of relatively pure single concentration along the rotational axis. The use of magnetic resonance imaging (MRI) to study concentration variations within a mixture of different sized spheres segregated in this manner indicates a more complicated structure within the bulk than can be observed or even inferred from surface measurements. Moreover, when the rotation speed is reduced and the homogeneous mixed state appears to be restored on the surface, the MRI data reveals that the radially segregated state persists.

The symbolic analysis introduced in this paper allows quantitative description of dynamical coupling between different time signals. In order to demonstrate how this method works we applied it to the explicit examples of chaotic signals. Our results appear to be quite robust when external noise is added.