Discrete instability in nonlinear lattices

The discrete multiscale analysis for boundary value problems in nonlinear discrete systems leads to a first order discrete modulational instability above a threshold amplitude for wave numbers beyond the zero of group velocity dispersion. Applied to the electrical lattice [Phys. Rev. E, 51 (1995) 6127 ], this acurately explains the experimental instability at wave numbers beyond 1.25 . The theory is also briefly discussed for sine-Gordon and Toda lattices.Comment: 1 figure, revtex, published: Phys. Rev. Lett. 83 (1999) 232

Two dimensional modulational instability in photorefractive media

We study theoretically and experimentally the modulational instability of broad optical beams in photorefractive nonlinear media. We demonstrate the impact of the anisotropy of the nonlinearity on the growth rate of periodic perturbations. Our findings are confirmed by experimental measurements in a strontium barium niobate photorefractive crystal.Comment: 8 figure

Modulational instability in nonlocal Kerr-type media with random parameters

Modulational instability of continuous waves in nonlocal focusing and defocusing Kerr media with stochastically varying diffraction (dispersion) and nonlinearity coefficients is studied both analytically and numerically. It is shown that nonlocality with the sign-definite Fourier images of the medium response functions suppresses considerably the growth rate peak and bandwidth of instability caused by stochasticity. Contrary, nonlocality can enhance modulational instability growth for a response function with negative-sign bands.Comment: 6 pages, 12 figures, revTeX, to appear in Phys. Rev.

Pattern generation by dissipative parametric instability

Nonlinear instabilities are responsible for spontaneous pattern formation in a vast number of natural and engineered systems, ranging from biology to galaxy buildup. We propose a new instability mechanism leading to pattern formation in spatially extended nonlinear systems, which is based on a periodic antiphase modulation of spectrally dependent losses arranged in a zigzag way: an effective filtering is imposed at symmetrically located wave numbers k and -k in alternating order. The properties of the dissipative parametric instability differ from the features of both key classical concepts of modulation instabilities, i.e., the Benjamin-Feir instability and the Faraday instabiltyity. We demonstrate how the dissipative parametric instability can lead to the formation of stable patterns in one- and two-dimensional systems. The proposed instability mechanism is generic and can naturally occur or can be implemented in various physical systems

Fusion in the entwined category of Yetter--Drinfeld modules of a rank-1 Nichols algebra

We rederive a popular nonsemisimple fusion algebra in the braided context, from a Nichols algebra. Together with the decomposition that we find for the product of simple Yetter-Drinfeld modules, this strongly suggests that the relevant Nichols algebra furnishes an equivalence with the triplet W-algebra in the (p,1) logarithmic models of conformal field theory. For this, the category of Yetter-Drinfeld modules is to be regarded as an \textit{entwined} category (the one with monodromy, but not with braiding).Comment: 36 pages, amsart++, times, xy. V3: references added, an unnecessary assumption removed, plus some minor change

Spatial-temporal dynamics of the terahertz field generated by femtosecond filament

We present the study on spatial distribution of the maximum of terahertz field amplitude in time domain when generated by a femtosecond filament. It is shown that as a result of the propagation of the terahertz field forms a spherical wave front, on the edge of which the maximum of amplitude has a temporary delay in contrary to its central part

Investigation of interaction femtosecond laser pulses with skin and eyes mathematical model

We present a mathematical model of linear and nonlinear processes that takes place under the action of femtosecond laser radiation on the cutaneous covering. The study is carried out and the analytical solution of the set of equations describing the dynamics of the electron and atomic subsystems and investigated the processes of linear and nonlinear interaction of femtosecond laser pulses in the vitreous of the human eye, revealed the dependence of the pulse duration on the retina of the duration of the input pulse and found the value of the radiation power density, in which there is a self-focusing is obtained. The results of the work can be used to determine the maximum acceptable energy, generated by femtosecond laser systems, and to develop Russian laser safety standards for femtosecond laser systems

Modulational instability and nonlocality management in coupled NLS system

The modulational instability of two interacting waves in a nonlocal Kerr-type medium is considered analytically and numerically. For a generic choice of wave amplitudes, we give a complete description of stable/unstable regimes for zero group-velocity mismatch. It is shown that nonlocality suppresses considerably the growth rate and bandwidth of instability. For nonzero group-velocity mismatch we perform a geometrical analysis of a nonlocality management which can provide stability of waves otherwise unstable in a local medium.Comment: 15 pages, 12 figures, to be published in Physica Script

Strong Collapse Turbulence in Quintic Nonlinear Schr\"odinger Equation

We consider the quintic one dimensional nonlinear Schr\"odinger equation with forcing and both linear and nonlinear dissipation. Quintic nonlinearity results in multiple collapse events randomly distributed in space and time forming forced turbulence. Without dissipation each of these collapses produces finite time singularity but dissipative terms prevents actual formation of singularity. In statistical steady state of the developed turbulence the spatial correlation function has a universal form with the correlation length determined by the modulational instability scale. The amplitude fluctuations at that scale are nearly-Gaussian while the large amplitude tail of probability density function (PDF) is strongly non-Gaussian with power-like behavior. The small amplitude nearly-Gaussian fluctuations seed formation of large collapse events. The universal spatio-temporal form of these events together with the PDF for their maximum amplitudes define the power-like tail of PDF for large amplitude fluctuations, i.e., the intermittency of strong turbulence.Comment: 14 pages, 17 figure