Classical and quantum properties of a discontinuous perturbed twist map are investigated. Different classical diffusive regimes, quasilinear and slow, respectively, are observed. The regime of slow classical diffusion gives rise to two distinct quantal regimes, one marked by dynamical localization, the other by quasi-integrable localization due to classical cantori. In both cases the resulting quantum stationary distributions are algebraically localized.

We present the first experimental observation of multihump multimode solitons in nature. In a broader perspective, this is the first observation of a multihumped multimode self-trapped wave packet propagating in a dispersive nonlinear medium. Double- and triple-humped spatial profiles are found by incoherent population of two modes of the beams' self-induced waveguide (self-induced potential well).

We study the nonlinear dynamics of hydrothermal waves produced by a surface-tension-driven convective flow in a long and thin annular channel heated from the side. Above onset, the supercritical traveling wave pattern undergoes a secondary instability: a supercritical Eckhaus instability. This leads to a small-wave-number phase-modulated nonlinear mode, and shows the first experimental evidence of a nonlinearly saturated phase instability mode for traveling wave patterns. At higher forcing level, this secondary pattern is subject to a tertiary instability. This mode is an amplitude mode characterized by traveling hole patterns, i.e., space-time defects that change the wave number.

The short wave instability on corotating vortex filaments is investigated. It is shown that the short wave instability always occurs on corotating vortex filaments of fixed core structure. Moreover, when the interfilament distance is smaller than or comparable to the core size, vortex filaments produce short wave unstable modes which lead to wild stretching and folding; when vortex filaments are far apart from each other, unstable modes are bounded by a fraction of the core size and the vortex stretching remains bounded. These findings may be used to explain the smooth behavior of superfluid vortices. They also explain the stretching seen in numerical calculations.

A spatial perturbation method for the stabilization of an unstable state of a pattern forming system is presented. The perturbation, as a weak driving signal, mimics the target pattern. Stabilization, selection, and tracking of unstable rolls, squares, hexagons, and honeycombs are demonstrated by numerical simulations through the example of an optical system. The application of an external spatial perturbation also results in critical slowing down and resonant phenomenon.

We report an investigation of white-light continuum generation during self-focusing in extended transparent media using 140-fs Ti:sapphire laser pulses. A band-gap threshold is found above which the width of the continuum tends to increase with increasing band gap and below which there is no continuum generation. This is, to our knowledge, the first report of a parameter predicting the width of the continuum in condensed media. Multiphoton excitation of electrons into the conduction band is proposed as the primary mechanism reponsible for the observations.

We investigate the steady state properties of a randomly driven dissipative gas. As the number of particles is increased power-law spatial correlations in the density and velocity are seen to develop, and the exponents are found to be independent of the model details. We show how the critical properties of this system can be understood in terms of a one particle model which is amenable to exact analysis.

Morphological instabilities such as encountered in solidification or in cellular flame propagation are frequently subject to spatial forcing due to external flow or imperfections of the system. A seemingly generic feature of spatial forcing is the localization of patterns in a neighborhood of inward surface stagnation points on the front.

Statistical properties of energy levels and eigenfunctions in a ballistic system with diffusive surface scattering are investigated. The two-level correlation function, the level number variance, the correlation function of wave function intensities, and the inverse participation ratio are calculated.

Exact calculations of transmission and reflection coefficients in surface randomly corrugated optical waveguides are presented. As the length of the corrugated part of the waveguide increases, there is a strong preference to forward coupling through the lowest mode. An oscillating behavior of the enhanced backscattering as a function of the wavelength is predicted. Although the transport is strongly nonisotropic, the analysis of the probability distributions of the transmitted waves confirms in this configuration distributions predicted by random matrix theory for volume disorder.

The effect of dispersion or diffraction on zero-velocity solitons in studied for the generalized massive Thirring model describing a nonlinear optical fiber with grating or parallel-coupled planar waveguides with misaligned axes. The Thirring solitons existing at zero dispersion/diffraction are shown numerically to be separated by a finite gap from three isolated soliton branches. Inside the gap, there is an infinity of multisoliton branches. Thus, the Thirring solitons are structurally unstable. In another parameter region (far from the Thirring limit), solitons exist everywhere.

We propose a new class of models for the mean motion of ideal incompressible fluids in three dimensions, including stratification and rotation. In these models, the amplitude of the rapid fluctuations introduces a length scale, $\alpha$, below which wave activity is filtered by both linear and nonlinear dispersion. This filtering enhances the stability and regularity of the new fluid models without compromising either their large scale behavior, or their conservation laws. These models also describe geodesic motion on the volume-preserving diffeomorphism group for a metric containing the $H^1$ norm of the fluid velocity.

The occurrence of a finite time singularity in the incompressible Euler equations in three dimensions is studied numerically using the technique of adaptive mesh refinement. As opposed to earlier treatments, a prescribed accuracy is guaranteed over the entire integration domain. A singularity in the vorticity could be traced down to five levels of refinement which corresponds to a resolution of ${2048}^3$ mesh points in a nonadaptive treatment. The growth of vorticity fits a power law behavior proportional to $1/(T^{*}-t)$ where $T^{*}$ denotes the time when the singularity occurs.

Turbulence in the heat flow of helium II manifests itself as a tangle of quantized vortex lines. It has been known for some time that there are two different forms of turbulence, called the $T-1$ and the $T-2$ turbulent states. Experiments show that when a critical velocity is exceeded there is a large increase of the superfluid vortex line density and a transition occurs from $T-1$ to $T-2$. Until now, the nature of the two turbulent states and of the critical velocity has been a mystery. We present a model which solves this problem by addressing for the first time the issue of the stability of the normal fluid. The computed transition is found to be in good agreement with the observations.

Using experimental and numerical systems, we show that it is possible to maintain excellent synchronization between a drive and a response system even when there is a large (50%) parameter mismatch between them, and they are coupled only through a scalar signal. By optimizing the coupling consistent with a stability constraint, we show that a consequence of the optimized coupling is that the synchronization is maintained even in the presence of bifurcations in the drive system, despite the condition that the response parameters are held constant.

Kinetic lattice methods are a very attractive representation of nonlinear macroscopic systems because of their inherent parallelizability on multiple processors and their avoidance of the nonlinear convective terms. By uncoupling the velocity lattice from the spatial grid, one can employ higher order (non-space-filling) isotropic lattices—lattices which greatly enhance the stable parameter regions, particularly in thermal problems. In particular, the superiority of the octagonal lattice over previous models used in 2D (hexagonal or square) and 3D (projected face-centered hypercube) is shown.

Grid turbulence in a flowing soap film is investigated with ambient pressure $P$ as one of the control parameters. Reducing $P$ significantly decreases air drag thus minimizing three-dimensional effects on the flowing film. It was found that while the total kinetic energy $\frac{1}{2}〈v^2〉=\int E\left(k\right)\mathrm{dk}$ of velocity fluctuations increased noticeably as $P$ was reduced, the energy spectrum $E\left(k\right)$ for wave numbers greater than the injection scale was remarkably insensitive to the pressure change, and obeyed a power law $E\left(k\right)\sim k^{-3.3\mathrm{}\pm 0.3}$.

We show theoretically and experimentally that a basic pattern formation mechanism in hydrodynamics, the Faraday instability, can be observed in optics (for modulated class $B$ lasers with inhomogeneous broadening). As for the Faraday problem, stationary waves are excited parametrically and wave numbers are selected by the dispersion curve of the medium. The mechanism is evidenced by a multiple-scale analysis of the physical laser model, and is confirmed experimentally using a Nd-doped fiber laser.

We present a new method for the computation of Lyapunov exponents utilizing representations of orthogonal matrices applied to decompositions of $M$ or $M\tilde{M}$ where $M$ is the tangent map. This method uses a minimal set of variables, does not require renormalization or reorthogonalization, can be used to efficiently compute partial Lyapunov spectra, and does not break down when the Lyapunov spectrum is degenerate.

This work focuses on the ability of any feedback approach to control a given chaotic system. The concept of reachability for nonlinear systems is presented in a differential geometric framework. This concept is used to examine the ability of any perturbation signal to redirect the system's trajectory flow within its phase space. The structure of the controlling input and its global relation to the underlying dynamics of the system are crucial for the ability to effectively control the system within the phase space. The concepts are illustrated on a well known example: the Lorenz system.