It is argued that power-law spatial correlation at short distances is a generic property of spatiotemporal chaos exhibited by active dynamical elements coupled nonlocally. While this fact was suggested earlier from some numerical analysis of coupled limit cycles, further evidence is provided here from an analysis for chaotic Rössler oscillators and logistic maps. A theory is presented to explain why such short-range nonanalyticity of correlation with parameter-dependent exponent is so universal when the coupling is nonlocal.

A new equation for self-focusing in the presence of small beam nonparaxiality is derived. Analysis of this equation shows that nonparaxiality remains small as the beam propagates. Nevertheless, nonparaxiality arrests self-focusing when the beam width becomes comparable to its wavelength. A series of focusing-defocusing cycles of decreasing magnitude follows, ending with a final defocusing stage.

Stokes phenomenon is investigated in the complex semiclassical theory of chaotic tunneling, and a rule for locating contributing complex paths out of a complicated set of candidates is proposed. The proposal is based upon discovery of a tree structure hidden in entangled tunneling paths: The candidates are ordered along the tree, and the principle of exponential dominance (PED) is extended to prune noncontributing paths. This rule enables construction of semiclassical tunneling wave functions systematically. A phenomenon for which PED plays a crucial role is also demonstrated.

The topological pressure introduced by Ruelle and similar quantities describe dynamical multifractal properties of dynamical systems. These are important characteristics of mesoscopic systems in the classical regime. The original definitions of these quantities are based on the symbolic description of the dynamics. It is hard or nearly impossible to find a symbolic description and generating partition to a general dynamical system, therefore these quantities are often not accessible for further studies. Here we present a new method by which the symbolic description can be omitted. We apply the method for a mixing and an intermittent system.

We report on the first experimental observation of so-called nondegenerate photorefractive parametric amplification. We show that due to this effect it is possible for a weakly modulated photoinduced grating to be parametrically amplified via nonlinear interaction with a strongly modulated photoinduced grating.

The Bohigas-Giannoni-Schmit conjecture stating that the statistical spectral properties of systems which are chaotic in their classical limit coincide with random matrix theory (RMT) is proved. A new semiclassical field theory for individual chaotic systems is constructed in the framework of a nonlinear $\sigma$ model. The low lying modes are shown to be associated with the Perron-Frobenius (PF) spectrum of the underlying irreversible classical dynamics. It is shown that the existence of a gap in the PF spectrum results in RMT behavior. Moreover, our formalism offers a way of calculating system specific corrections beyond RMT.

We give experimental evidence for a new bifurcation structure that arises when smooth dynamical systems cross a boundary. Our experiment concerns a driven impacting leaf-spring oscillator with a very precise control of the driving amplitude. The results are in surprisingly good agreement with the predictions of a simple nonlinear mapping that is valid near grazing impact (i.e., impact with zero velocity). The agreement is surprising because a multitude of vibration modes of the spring is excited upon impact whereas the mapping is two dimensional. These findings point to the universality of the observed bifurcation structure.

We consider the higher order nonlinear Schrödinger (HNLS) equation describing nonlinear wave propagation in light guides with all higher order effects such as higher order dispersion, Kerr dispersion, and stimulated inelastic scattering. Using the Painlevé analysis, we derive all parametric conditions for soliton-type pulse propagation in HNLS fiber system. We generalize the Ablowitz-Kaup-Newell-Segur method to the $3\times 3$ eigenvalue problem, and constructed the Lax pair for the integrable case. The one soliton solution is generated from a Bäcklund transformation and an exact N-soliton solution is explicitly obtained from the Hirota bilinearization. The significance of the soliton solution is discussed.

We present an experimental study of highly localized, solitonlike structures that propagate on the two-dimensional surface of highly dissipative fluids. Like the well-known Faraday instability, these highly dissipative structures are driven by means of the spatially uniform, vertical acceleration of a thin fluid layer. These structures, harmonically coupled to the external driving frequency, are observed above a critical ratio of the viscous boundary layer height to the depth of the fluid layer for a wide range of fluid viscosities and system parameters.

It is shown that the description of anomalous scaling in turbulent systems requires the simultaneous use of two normalization scales. This phenomenon stems from the existence of two independent (infinite) sets of anomalous scaling exponents that appear in leading order, one set due to infrared anomalies and the other due to ultraviolet anomalies. To expose this clearly we introduce here a set of local fields whose correlation functions depend simultaneously on the two sets of exponents. Thus the Kolmogorov picture of “inertial range” scaling is shown to fail because of anomalies that are sensitive to the two ends of this range.

To obtain the turbulent velocity profiles in Couette and Poiseuille flows between parallel walls, we use statistical mechanics of point vortices. Our theory does not contain phenomenological constants. The only parameter taken from experiments is the total kinetic energy of the flow. We solve numerically the equation for averaged flow and show that the theoretical predictions are in good agreement with the experimental data.

We study how the interaction with an external incoherent environment induces a crossover from quantum to classical behavior for a particle whose classical motion is chaotic. Posing the problem in the semiclassical regime for weak bath coupling, we find that noise rather than dissipation is primarily responsible for the dephasing that leads to the quantum inhibition of mixing. However, the noise is also shown to inhibit the mixing by enhancing paths that interfere coherently.

We introduce the dynamic nonlinear optical skin effect in which a pulse incident on a saturable absorbing interface is self-reflected from a moving absorption front. The motion of the front causes the self-reflected wave to be redshifted by the Doppler effect, which in turn serves as an experimentally observable signature for the front propagation.

We suggest and implement a method for observing Manakov spatial solitons in crystals. We show experimentally that the propagation behavior of mutually trapped, orthogonally polarized beams in AlGaAs planar waveguides is identical to that of a fundamental (single polarization) soliton beam with the same total power.

A minimal model is constructed for two-dimensional fracture propagation. The heterogeneous process zone is presumed to suppress stress relaxation rate, leading to non-quasistatic behavior. Using the Yoffe solution, I construct and solve a dynamical equation for the tip stress. I dicuss a generic tip-velocity response to local stress and find that noise-free propagation is either at steady state or oscillatory, depening only on one material parameter. Noise gives rise to intermittency and quasiperiodicity. The theory explains the velocity oscillations and the complicated behavior seen in polymeric and amorphous brittle materials. I suggest experimental verifications and new connections between velocity measurements and material properties.

We consider a passive scalar convected by a multiscale random velocity field with short yet finite temporal correlations. Taking Kraichnan's limit of a white Gaussian velocity as a zero approximation we develop the perturbation theory with respect to a small correlation time and small non-Gaussianity of the velocity. We derive the renormalization (due to temporal correlations and non-Gaussianity) of the operator of turbulent diffusion. That allows us to calculate the respective corrections to the anomalous scaling exponents of the scalar field and show that they continuously depend on velocity correlation time and the degree of non-Gaussianity. The scalar exponents are thus nonuniversal as was predicted by Shraiman and Siggia on a phenomenological ground.

High-resolution direct numerical simulations of 3D Navier-Stokes turbulence with normal viscosity and hyperviscosity are carried out. It is found that the inertial-range statistics, both the scalings and the probability density functions, are independent of the dissipation mechanism, while the near-dissipation-range fluctuations show significant structural differences. Nevertheless, the relative scalings expressing the dependence of the moments at different orders are universal, and show unambiguous departure from the Kolmogorov 1941 description, including the $2/3$ law for the kinetic energy. Implications for numerical modeling of turbulence are discussed.

Electromagnetic radiation during the quasiclassical tunneling motion of a charge through a smooth potential barrier is considered. A general formula for the radiation spectral density per tunneling particle is derived, which is essentially classical (i.e., does not contain the Planck constant), and is given by a simple modification of the convenient result for the classical overbarrier motion.

Conversion of energy from a heat pulse to acoustic stress is theoretically and experimentally studied in detail. The heat pulse was generated by laser radiation delivered via an optical fiber into an absorbing liquid. The experimental results indicate that tensile stress and cavitation are induced in front of the fiber tip at a distance far below the optical penetration depth of the laser radiation. The occurrence of tensile stress in the acoustic near field of a submerged fiber is explained by acoustic diffraction of the thermoelastic expansion wave. Good agreement between experimental results and theoretical calculations based on a three-dimensional model was found.

We study the local diffusion rate $D$ in a Chirikov standard map near the critical golden curve. Numerical simulations confirm the predicted exponent $\alpha =5\mathrm{}$ for the power law decay of $D$ as approaching the golden curve via principal resonances with period $q_n$ $\left(D\sim {1/q}_n^{\alpha }\right)$. The universal self-similar structure of diffusion between principal resonances is demonstrated, and it is shown that other resonances may also play an important role.