Populations of phase oscillators interacting globally through a general coupling function $f\left(x\right)$ have been considered. We analyze the conditions required to ensure the existence of a Lyapunov functional giving close expressions for it in terms of a generating function. We have also proposed a family of exactly solvable models with singular couplings showing that it is possible to map the synchronization phenomenon into other physical problems. In particular, the stationary solutions of the least singular coupling considered, $f\left(x\right)=\mathrm{sgn}\left(x\right)$, have been found analytically in terms of elliptic functions. This last case is one of the few nontrivial models for synchronization dynamics which can be analytically solved.

We present hitherto unknown forms of soliton dynamics in the forbidden frequency gap of a Bragg reflector, modified by periodic layers of near-resonant two-level systems (TLS). Remarkably, even extremely low TLS densities create an allowed band within the forbidden gap. This spectrum gives rise, for any Bragg reflectivity, to a vast family of stable gap solitons, both standing and moving, having a unique analytic form, an arbitrary pulse area, and inelastic collision properties. These findings suggest new possibilities of transmission control, noise filtering, or “dynamical cavities” (self-traps) for both weak and strong signal pulses.

Convection structures in binary fluid mixtures are investigated for positive Soret coupling in the driving regime where solutal and thermal contributions to the buoyancy forces compete. Bifurcation properties of stable and unstable stationary square, roll, and crossroll structures and the oscillatory competition between rolls and squares are determined numerically as a function of fluid parameters. A novel type of subharmonic bifurcation cascade where the oscillation period grows in integer steps as $n\frac{2\pi }{\omega }$ is found and elucidated to be an entrainment process.

We study statistical properties of the eigenvectors of non-Hermitian random matrices, concentrating on Ginibre's complex Gaussian ensemble, in which the real and imaginary parts of each element of an $N\times N$ matrix, $J$, are independent random variables. Calculating ensemble averages based on the quantity $〈L_{\alpha }|L_{\beta }〉〈R_{\beta }|R_{\alpha }〉$, where $〈L_{\alpha }|$ and $|R_{\beta }〉$ are left and right eigenvectors of $J$, we show for large $N$ that eigenvectors associated with a pair of eigenvalues are highly correlated if the two eigenvalues lie close in the complex plane. We examine consequences of these correlations that are likely to be important in physical applications.

We describe an iterative approach to computing long-time semiclassical dynamics in the presence of chaos, which eliminates the need for summing over an exponentially large number of classical paths, and has good convergence properties even beyond the Heisenberg time. Long-time semiclassical properties can be compared with those of the full quantum system. The method is used to demonstrate semiclassical dynamical localization in one-dimensional classically diffusive systems, showing that interference between classical paths is a sufficient mechanism for limiting long-time phase space exploration.

Using the Lorenz equations, we have investigated whether unstable periodic orbits (UPOs) associated with a strange attractor may predict the occurrence of the robust sharp peaks in histograms of some experimental chaotic time series. Histograms with sharp peaks occur for the Lorenz parameter value $r=60.0\mathrm{}$ but not for $r=28.0\mathrm{}$, and the sharp peaks for $r=60.0\mathrm{}$ do not correspond to a histogram derived from any single UPO. However, we show that histograms derived from the time series of a non-Axiom- $A$ chaotic system can be accurately predicted by an escape-time weighting of UPO histograms.

We show how the analytical approach of Zakharov and Rubenchik [Sov. Phys. JETP 38, 494 (1974)] to modulational instability (MI) of solitary waves in the nonlinear Schrödinger equation can be generalized for models with two phase symmetries. MI of three-wave parametric spatial solitons due to group velocity dispersion (GVD) is investigated as a typical example of such models. We reveal a new branch of neck instability, which dominates the usual snake type MI found for normal GVD. The resultant nonlinear evolution is thereby qualitatively different from cases with only a single phase symmetry.

We report the observation of discrete spatial optical solitons in an array of 41 waveguides. Light was coupled to the central waveguide. At low power, the propagating field spreads as it couples to more waveguides. When sufficient power was injected, the field was localized close to the input waveguides and its distribution was successfully described by the discrete nonlinear Schrödinger equation.

We study the dynamics of the classical Euler buckling instability of compressed objects such as flexible molecular chains and thin rods. We reveal that this dynamics is a coarsening process self-similar in time. We relate this process to phase ordering phenomena such as spinodal decomposition.

We extend the Layzer-type approach to unstable interfacial fluid mixing, applied up to now only to vacuum bubbles, to spikes and derive the analytical solutions of the model for the positions, velocities, accelerations, and curvatures at the tips of the bubble and spike over all times. The analytical predictions are in good agreement with the results from numerical simulations for both spikes and bubbles. We give the first analytical prediction for the asymptotic growth rate of a spike at the Richtmyer-Meshkov unstable interface. We predict that, in contrast to the asymptotic bubble growth rate, the asymptotic growth rate of a spike at the Richtmyer-Meshkov unstable interface is a constant and depends on the initial condition.

The evolution of macroscopic material closed filaments in a time-periodic chaotic 2D flow is simulated for cases with large, small, and very small islands of regular motion using an algorithm that preserves spatial continuity. The length of the stretched filament increases much faster than predicted by the Liapunov exponent. In chaotic regions, the filament asymptotically evolves into a self-similar structure with permanent spatial nonuniformities in density. Filament densities and local length scales corresponding to different times are described by families of frequency distributions with invariant shape that can be collapsed onto a single curve by means of a simple scaling.

Band distributions (BDs) are introduced describing quantization in a toral phase space. A BD is the uniform average of an eigenstate phase-space probability distribution over a band of toral boundary conditions. A general explicit expression for the Wigner BD is obtained. It is shown that the Wigner functions for all of the band eigenstates can be reproduced from the Wigner BD. Also, BDs are shown to be closer to classical distributions than eigenstate distributions. Generalized BDs, associated with sets of adjacent bands, are used to extend in a natural way the Chern-index characterization of the classical-quantum correspondence on the torus to arbitrary rational values of the scaled Planck constant.

We have built a thermoacoustic engine consisting of a differentially heated stack of plates in a looped tube and observed spontaneous gas oscillations of the traveling wave mode running around the loop. Stability boundary and thermally produced acoustic power are compared with those for the engine tested in a resonator. The engine in a looped tube acts as a traveling wave power amplifier, whose onset temperature ratios are significantly smaller than those for the engine in a resonator.

Motivated by recent experimental work, I numerically investigated the possibility of obtaining Taylor vortices in a spherical shell of aspect ratio 0.336. It is found that Taylor vortices can exist for Reynolds numbers in the range $415\le \mathrm{Re}\le 2040$. With increasing Reynolds number, the initially equatorially symmetric vortices become asymmetric at $\mathrm{Re}=1390\mathrm{}$. Increasing Re still further, these asymmetric vortices become time dependent at $\mathrm{Re}=1940\mathrm{}$, followed by a period-doubling cascade to chaos at $\mathrm{Re}\approx 2035$. For $\mathrm{Re}>2040\mathrm{}$ the chaotic solution collapses back to the basic state having no Taylor vortices.

Classical Stokes drift is the small time-averaged drift velocity of suspended nondiffusing particles in a fluid due to the presence of a wave. We consider the effect of adding diffusion to the motion of the particles and show in particular that a nonzero time-averaged drift velocity exists in general even when the classical Stokes drift is zero. Our results are obtained from a general procedure for calculating ensemble-averaged Lagrangian mean velocities for motion that is close to Brownian and are verified by numerical simulations in the case of sinusoidal forcing.

We develop an iterative approach, from which an exact systematic calculation of the moments of the probability density of test particles, under a generalized Fickian evolution and in the presence of a stratified random velocity field, can be calculated. The type of fractional diffusion-advection equation studied is useful for transport in anisotropic and heterogeneous media. Our method stresses the relevance of the coupling between anomalous diffusion and convection by stratified random velocity fields.

We have measured the probability density $|\psi \left(r\right){|}^2$ in the semiclassical limit of a classically chaotic square well potential with and without time reversal symmetry, and compared our findings with theoretical predictions. We find that wave functions with time-reversal symmetry have larger fluctuations than those without time-reversal symmetry. To quantify the degree of these fluctuations, eigenmodes both with and without time-reversal symmetry are statistically analyzed and the two-point spatial correlation function and the probability density distribution function of the eigenmodes are found to agree with theoretical predictions.

We suggest a technique for producing subfemtosecond pulses of radiation. The technique is based on using electromagnetically induced transparency to produce a strongly driven molecular coherence. This coherence results in a Raman spectrum with Bessel function amplitudes and phases corresponding to a frequency modulated signal, thereby allowing compression by the group velocity dispersion of the same medium.

Dynamics of particles advected by fast rotating incompressible turbulent fluid flow is studied. Fast rotation and particle inertia imply the divergent particle velocity field and result in both intermittency in spatial distribution of particles and formation of the large-scale inhomogeneous structures. A nonzero mean helicity of fluid flow causes an additional mean nondiffusive turbulent flux of inertial particles. Intermittency in the systems with and without external pumping is studied. Fast rotation causes anomalous scaling already in the second moment of inertial particle number density and may result in excitation of a small-scale instability of inertial particle distribution, which leads to the formation of small-scale particle clusters. We discuss the relevance of our results for atmospheric, astrophysical, and industrial turbulent rotating flows.

The following wavelength doubling cascade from a uniform state to a turbulent pattern is observed in a 3D anisotropic liquid: First oscillating convective rolls followed by a stationary stripe pattern, then Möbius defect creation stabilizing an unusual curved roll pattern that eventually becomes turbulent. Once created, Möbius defects are topologically trapped and the initial uniform state is not recovered. The system experimentally investigated is electroconvection above a highly nonlinear base state.