F. Kh. Abdullaev

Stability of trapped Bose-Einstein condensates

In three-dimensional trapped Bose-Einstein condensate (BEC), described by the time-dependent Gross-Pitaevskii-Ginzburg equation, we study the effect of initial conditions on stability using a Gaussian variational approach and exact numerical simulations. We also discuss the validity of the criterion for stability suggested by Vakhitov and Kolokolov. The maximum initial chirp (initial focusing defocusing of cloud) that can lead a stable condensate to collapse even before the number of atoms reaches its critical limit is obtained for several specific cases. When we consider two- and three-body nonlinear terms, with negative cubic and positive quintic terms, we have the conditions for the existence of two phases in the condensate. In this case, the magnitude of the oscillations between the two phases are studied considering sufficient large initial chirps. The occurrence of collapse in a BEC with repulsive two-body interaction is also shown to be possible.

Coherent atomic oscillations and resonances between coupled Bose-Einstein condensates with time-dependent trapping potential

We study the quantum coherent tunneling between two Bose-Einstein condensates separated through an oscillating trap potential. The cases of slow and rapid varying in the time trap potential are considered. In the case of a slowly varying trap, we study the nonlinear resonances and chaos in the oscillations of the relative atomic population. Using the Melnikov function approach, we find the conditions for chaotic macroscopic quantum-tunneling phenomena to exist. Criteria for the onset of chaos are also given. We find the values of frequency and modulation amplitude which lead to chaos on oscillations in the relative population, for any given damping and the nonlinear atomic interaction. In the case of a rapidly varying trap, we use the multiscale expansion method in the parameter «51/V, where V is the frequency of modulations, and we derive the averaged system of equations for the modes. The analysis of this system shows that new macroscopic quantum self-trapping regions, in comparison with the constant trap case, exist.

Modulational instability in optical fibers with variable dispersion

Abstract We study modulational instability in optical fibres whose dispersion varies with the propagation distance. We distinguish between two cases, viz. periodic and random dispersion variation. In the former case it is found that due to the parametric resonance between the dispersion modulation and the modes of the linearized system new domains of modulational instability arise for anomalous as well as normal average dispersion. In the latter case stochastic parametric resonances again lead to the occurrence of modulational instability for normal dispersion. If the dispersion is anomalous the region of modulational instability increases and the respective gain decreases in comparison with a fibre with constant dispersion.

Dissipative periodic waves, solitons, and breathers of the nonlinear Schrödinger equation with complex potentials.

Exact solutions for the generalized nonlinear Schrödinger equation with inhomogeneous complex linear and nonlinear potentials are found. We have found localized and periodic solutions for a wide class of localized and periodic modulations in the space of complex potentials and nonlinearity coefficients. Examples of stable and unstable solutions are given. We also demonstrated numerically the existence of stable dissipative breathers in the presence of an additional parabolic trap.

Modulational instability in optical fibers near the zero dispersion point

Modulational instability (MI) of electromagnetic waves in an optical fiber near the zero dispersion point is investigated both analytical and numerical. The effect of fourth order dispersion is taken into account and a new region with MI is found. The possibility of MI is shown for the case of positive second and fourth order group dispersion. For both positive and negative fourth order dispersion a recurrence phenomenon is observed analogous to the Fermi-Pasta-Ulam problem.

Disintegration of a soliton in a dispersion-managed optical communication line with random parameters.

The propagation of dispersion-managed solitons in optical fiber links with a random dispersion map has been studied. Two types of randomness are considered:random dispersion magnitudes and random lengths of the spans. By numerical simulations, disintegration of a soliton propagating in such an optical communication line is shown to occur. It is observed that the stability of the soliton propagation is affected more by modulations of the dispersion magnitudes of the spans than by modulations of the span lengths. Results of numerical simulations of the soliton breakup distance confirm theoretical predictions in the averaged dynamics limit.

Zeno effect and switching of solitons in nonlinear couplers

The Zeno effect is investigated for soliton type pulses in a nonlinear directional coupler with dissipation. The effect consists in increase of the coupler transparency with increase of the dissipative losses in one of the arms. It is shown that localized dissipation can lead to switching of solitons between the arms. Power losses accompanying the switching can be fully compensated by using a combination of dissipative and active (in particular, parity-time-symmetric) segments.

Localized modes in chi((2)) media with PT-symmetric localized potential

We study the existence and stability of solitons in the quadratic nonlinear media with spatially localized parity-time- (PT)-symmetric modulation of the linear refractive index. Families of stable one- and two-hump solitons are found. The properties of nonlinear modes bifurcating from a linear limit of small fundamental harmonic fields are investigated. It is shown that the fundamental branch, bifurcating from the linear mode of the fundamental harmonic is limited in power. The power maximum decreases with the strength of the imaginary part of the refractive index. The modes bifurcating from the linear mode of the second harmonic can exist even above the PT-symmetry-breaking threshold. We found that the fundamental branch bifurcating from the linear limit can undergo a secondary bifurcation colliding with a branch of two-hump soliton solutions. The stability intervals for different values of the propagation constant and gain or loss gradient are obtained. The examples of dynamics and excitations of solitons obtained by numerical simulations are also given.

Localized modes of binary mixtures of Bose-Einstein condensates in nonlinear optical lattices

The properties of the localized states of a two-component Bose-Einstein condensate confined in a nonlinear periodic potential (nonlinear optical lattice) are investigated. We discuss the existence of different types of solitons and study their stability by means of analytical and numerical approaches. The symmetry properties of the localized states with respect to nonlinear optical lattices are also investigated. We show that nonlinear optical lattices allow the existence of bright soliton modes with equal symmetry in both components and bright localized modes of mixed symmetry type, as well as dark-bright bound states and bright modes on periodic backgrounds. In spite of the quasi-one-dimensional nature of the problem, the fundamental symmetric localized modes undergo a delocalizing transition when the strength of the nonlinear optical lattice is varied. This transition is associated with the existence of an unstable solution, which exhibits a shrinking (decaying) behavior for slightly overcritical (undercritical) variations in the number of atoms.

Generalized neighbor-interaction models induced by nonlinear lattices

It is shown that the tight-binding approximation of the nonlinear Schrödinger equation with a periodic linear potential and periodic in space nonlinearity coefficient gives rise to a number of nonlinear lattices with complex, both linear and nonlinear, neighbor interactions. The obtained lattices present nonstandard possibilities, among which we mention a quasilinear regime, where the pulse dynamics obeys essentially the linear Schrödinger equation. We analyze the properties of such models both in connection to their modulational stability, as well as in regard to the existence and stability of their localized solitary wave solutions.