arXiv subjects

F. Kh. Abdullaev

Publications and source records attributed to F. Kh. Abdullaev.

At least 19 recordsLinked to original sources

arXiv2026

Nonlinearity management of matter-wave vector solitons of Bose-Einstein condensates in two dimensions

The evolution of matter-wave vector solitons in two dimensions under nonlinearity management is studied. The averaged over strong and rapid modulations in time of the inter-species interactions vector Gross-Pitaevskii equation is derived. The averaging gives the appearance of the effective nonlinear quantum pressure depending on the population of the other component. Using this system of equations, the dynamical stabilization of the two-dimensional vector matter-wave solitons under the management of the mean-field nonlinearity is investigated. It is shown that the nonlinear quantum pressure arrests the collapse in the averaged system, giving rise to stable vector solitons. Full numerical simulations of the original, time-dependent system of Gross-Pitaevskii equations confirm that the management strongly prolongs the lifetime of these solitons compared to the unmanaged case.

cond-mat.quant-gas↗

arXiv2023

Nonlinearity managed vector solitons

The evolution of vector solitons under nonlinearity management is studied. The averaged over strong and rapid modulations in time of the inter-species interactions vector Gross-Pitaevskii equation (GPE) is derived. The averaging gives the appearance of the effective nonlinear quantum pressure depending on the population of the other component. Using this system of equations, the existence and stability of the vector solitons under the action of the strong nonlinearity management (NM) is investigated. Using a variational approach the parameters of NM vector solitons are found. The numerical simulations of the full time-dependent coupled GPE confirms the theoretical predictions.

nlin.PS↗

arXiv2023

Dynamics of imbalanced quasi-one-dimensional binary Bose-Einstein condensate in external potentials

In the framework of coupled 1D Gross-Pitaevskii equations, we explore the dynamics of a binary Bose-Einstein condensate where the intra-component interaction is repulsive, while the inter-component one is attractive. The existence regimes of stable self-trapped localized states in the form of symbiotic solitons have been analyzed. Imbalanced mixtures, where the number of atoms in one component exceeds the number of atoms in the other component, are considered in parabolic potential and box-like trap. When all the intra-species and inter-species interactions are repulsive, we numerically find a new type of symbiotic solitons resembling dark-bright solitons. A variational approach has been developed which allows us to find the stationary state of the system and frequency of small amplitude dynamics near the equilibrium. It is shown that the strength of inter-component coupling can be retrieved from the frequency of the localized state's vibrations.

cond-mat.quant-gas↗

arXiv2022

Compacton existence and spin-orbit density dependence in Bose-Einstein condensates

We demonstrate the existence of compactons matter waves in binary mixtures of Bose-Einstein condensates (BEC) trapped in deep optical lattices (OL) subjected to equal contributions of intra-species Rashba and Dresselhaus spin-orbit coupling (SOC) under periodic time modulations of the intra-species scattering length. We show that these modulations lead to the rescaling of the SOC parameters that involve the density imbalance of the two components. This gives rise to a density-dependent SOC parameters strongly influence the existence and stability of compacton matter waves. The stability of SOC-compactons is investigated both by linear stability analysis and by time integrations of the coupled Gross-Pitaevskii equations. We find that SOC restricts the parameter ranges for stable stationary SOC-compacton existence but, on the other side, it gives a more stringent signature of their occurrence. In particular, SOC-compactons should appear when the intra-species interactions and the number of atoms in the two components are perfectly balanced (or close to being balanced for metastable cases). The possibility to use SOC-compactons as a tool for indirect measurements of the number of atoms and/or the intra-species interactions, is also suggested.

cond-mat.quant-gas↗

arXiv2022

Management of solitons in medium with competing cubic and quadratic nonlinearities

Management of solitons in media with competing quadratic and cubic nonlinearities is investigated. Two schemes, using rapid modulations of a mismatch parameter, and of the Kerr nonlinearity parameter are studied. For both cases, the averaged in time wave equations are derived. In the case of mismatch management, the region of the parameters where stabilization is possible is found. In the case of Kerr nonlinearity management, it is shown that the effective $χ^{(2)}$ nonlinearity depends on the intensity imbalance between fundamental (FH) and second (SH) harmonics. Predictions obtained from the averaged equations are confirmed by numerical simulations of the full PDE's.

nlin.PS↗

arXiv2021

Oscillations of a quasi-one-dimensional dipolar supersolid

The properties of a supersolid state (SS) in quasi-one-dimensional dipolar Bose-Einstein condensate is studied, considering two possible mechanisms of realization - due to repulsive three-body atomic interactions and quantum fluctuations in the framework of the Lee-Huang-Yang (LHY) theory. The proposed theoretical model, based on minimization of the energy functional, allows evaluating the amplitude of the SS for an arbitrary set of parameters in the governing Gross-Pitaevskii equation (GPE). To explore the dynamics of the SS first, we numerically construct its ground state in different settings, including periodic boundary conditions, box-like trap and parabolic potential, then impose a perturbation. In oscillations of the perturbed supersolid we observe the key manifestation of SS, namely the free flow of the superfluid fraction through the crystalline component of the system. Two distinct oscillation frequencies of the supersolid associated with the superfluid fraction and crystalline components of the wave function are identified from numerical simulations of the GPE.

cond-mat.quant-gas↗

arXiv2021

Confinement of matter-wave solitons on top of a pedestal-shaped potential

Reflection of wave packets from downward potential steps and attractive potentials, known as a quantum reflection, has been explored for bright matter-wave solitons with the main emphasis on the possibility to trap them on top of a pedestal-shaped potential. In numerical simulations, we observed that moving solitons return from the borders of the potential and remain trapped for a sufficiently long time. The shuttle motion of the soliton is accompanied by shedding some amount of matter at each reflection from the borders of the trap, thus reducing its norm. The one- and two- soliton configurations are considered. A discontinuous jump of trajectories of colliding solitons has been discussed. The time-shift observed in a step-like decay of the moving soliton's norm in the two-soliton configuration is linked to the trajectory jump phenomenon. The obtained results can be of interest for the design of new soliton experiments with Bose-Einstein condensates.

cond-mat.quant-gas↗

arXiv2020

Bosonic impurity in a one-dimensional quantum droplet in the Bose-Bose mixture

We study an impurity immersed in the mixture of Bose ultracold gases in the regime where a quantum droplet exists. The quasi-one-dimensional geometry is considered. We find an effective attractive potential that acts by the quantum droplet onto the impurity. The bound states of the impurity in this potential are investigated. These impurity bound states can provide potential probes for the presence of quantum fluctuations effects on the droplet properties. In the case of strong impurity-BEC coupling, we study the properties of the nonlinear local modes on the impurity induced by the quantum fluctuations.

cond-mat.quant-gas↗

arXiv2019

Generation of density waves in dipolar quantum gases by time-periodic modulation of atomic interactions

We study the emergence of density waves in dipolar Bose-Einstein condensates (BEC) when the strength of dipole-dipole atomic interactions is periodically varied in time. The proposed theoretical model, based on the evolution of small perturbations of the background density, allows to compute the growth rate of instability (gain factor) for arbitrary set of input parameters, thus to identify the regions of instability against density waves. We find that among other modes of the system the roton mode is most effectively excited due to the contribution of sub-harmonics of the excitation frequency. The frequency of temporal oscillations of emerging density waves coincides with the half of the driving frequency, this being the hallmark of the parametric resonance, is characteristic to Faraday waves. The possibility to create density waves in dipolar BECs, which can persist after the emergence, has been demonstrated. The existence of a stationary spatially periodic solution of the nonlocal Gross-Pitaevskii equation has been discussed. The effect of three-body atomic interactions, which is relevant to condensates with increased density, upon the properties of emerging waves has been analyzed too. Significant modification of the condensate's excitation spectrum owing to three-body effects is shown.

cond-mat.quant-gas↗

arXiv2019

Collective dynamics of Fermi-Bose mixtures with an oscillating scattering length

Collective oscillations of superfluid mixtures of ultra cold fermionic and bosonic atoms are investigated while varying the fermion-boson scattering length. We study the dynamics with respect to excited center of mass modes and breathing modes in the mixture. Parametric resonances are also analyzed when the scattering length varies periodically in time, by comparing partial differential equation (PDE) models and ordinary differential equation (ODE) models for the dynamics. An application to the recent experiment with fermionic $^{6}$Li and bosonic $^{7}$Li atoms, which approximately have the same masses, is discussed.

cond-mat.quant-gas↗

arXiv2017

Binary matter-wave compactons induced by inter-species scattering length modulations

Binary mixtures of quasi one-dimensional Bose-Einstein condensates (BEC) trapped in deep optical lattices (OL) in the presence of periodic time modulations of the inter-species scattering length, are investigated. We adopt a mean field description and use the tight binding approximation and the averaging method to derive averaged model equations in the form of two coupled discrete nonlinear Schrödinger equations (DNLSE) with tunneling constants that nonlinearly depend on the inter-species coupling. We show that for strong and rapid modulations of the inter-species scattering length, the averaged system admits exact compacton solutions, e.g. solutions that have no tails and are fully localized on a compact which are achieved when the densities at the compact edges are in correspondence with zeros of the Bessel function (zero tunneling condition). Deviations from exact conditions give rise to the formation of quasi-compactons, e.g. non exact excitations which look as compactons for any practical purpose, for which the zero tunneling condition is achieved dynamically thanks to an effective nonlinear dispersive coupling induced by the scattering length modulation. Stability properties of compactons and quasi-compactons are investigated by linear analysis and by numerical integrations of the averaged system, respectively, and results compared with those from the original (unaveraged) system. In particular, the occurrence od delocalizing transitions with existence of thresholds in the mean inter-species scattering length is explicitly demonstrated. Under proper management conditions, stationary compactons and quasi-compactons are quite stable and robust excitations that can survive on very long time scale. A parameter design and a possible experimental setting for observation of these excitations are briefly discussed.

cond-mat.quant-gas↗

arXiv2017

Optical solitons in periodically managed PT-symmetric media

The dynamics of light beams in the nonlinear optical media with periodically modulated in the longitudinal direction parity-time distribution of the complex refractive index is investigated. The possibility of dynamical stabilization of PT-symmetric solitons is demonstrated.

nlin.PS↗

arXiv2017

Solitons in a ${\cal PT}$- symmetric $χ^{(2)}$ coupler

We consider the existence and stability of solitons in a $χ^{(2)}$ coupler. Both the fundamental and second harmonics undergo gain in one of the coupler cores and are absorbed in the other one. The gain and losses are balanced creating a parity-time (${\cal PT}$) symmetric configuration. We present two types of families of ${\cal PT}$-symmetric solitons, having equal and different profiles of the fundamental and second harmonics. It is shown that gain and losses can stabilize solitons. Interaction of stable solitons is shown. In the cascading limit the model is reduced to the ${\cal PT}$-symmetric coupler with effective Kerr-type nonlinearity and balanced nonlinear gain and losses.

physics.optics↗

arXiv2016

Tunable spin-orbit coupled Bose-Einstein condensates in deep optical lattices

Binary mixtures of Bose-Einstein condensates trapped in deep optical lattices and subjected to equal contributions of Rashba and Dresselhaus spin-orbit coupling (SOC), are investigated in the presence of a periodic time modulation of the Zeeman field. SOC tunability is explicitly demonstrated by adopting a mean-field tight-binding model for the BEC mixture and by performing an averaging approach in the strong modulation limit. In this case, the system can be reduced to an unmodulated vector discrete nonlinear Schrödinger equation with a rescaled SOC tunning parameter $α$, which depends only on the ratio between amplitude and frequency of the applied Zeeman field. The dependence of the spectrum of the linear system on $α$ has been analytically characterized. In particular, we show that extremal curves (ground and highest excited states) of the linear spectrum are continuous piecewise functions (together with their derivatives) of $α$, which consist of a finite number of decreasing band lobes joined by constant lines. This structure also remains in presence of not too large nonlinearities. Most important, the interactions introduce a number of localized states in the band-gaps that undergo change of properties as they collide with band lobes. The stability of ground states in the presence of the modulating field has been demonstrated by real time evolutions of the original (un-averaged) system. Localization properties of the ground state induced by the SOC tuning, and a parameter design for possible experimental observation have also been discussed.

cond-mat.quant-gas↗

arXiv2015

Multidimensional Discrete Compactons in Nonlinear Schrödinger Lattices with Strong Nonlinearity Management

The existence of multidimensional lattice compactons in the discrete nonlinear Schrödinger equation in the presence of fast periodic time modulations of the nonlinearity is demonstrated. By averaging over the period of the fast modulations, a new effective averaged dynamical equation arises with coupling constants involving Bessel functions of the first and zeroth kind. These terms allow one to solve, at this averaged level, for exact discrete compacton solution configurations in the corresponding stationary equation. We focus on seven types of compacton solutions: single site and vortex solutions are found to be always stable in the parametric regimes we examined. Other solutions such as double site in- and out-of-phase, four site symmetric and anti-symmetric, and a five site compacton solution are found to have regions of stability and instability in two-dimensional parametric planes, involving variations of the strength of the coupling and of the nonlinearity. We also explore the time evolution of the solutions and compare the dynamics according to the averaged with those of the original dynamical equations without the averaging. Possible observation of compactons in the BEC loaded in a deep two-dimensional optical lattice with interactions modulated periodically in time is discussed.

nlin.PS↗

arXiv2015

Some case example exact solutions for quadratically nonlinear optical media with $\mathcal{PT}$-symmetric potentials

In the present paper we consider an optical system with a $χ^{(2)}$-type nonlinearity and unspecified $\mathcal{PT}$-symmetric potential functions. Considering this as an inverse problem and positing a family of exact solutions in terms of cnoidal functions, we solve for the resulting potential functions in a way that ensures the potentials obey the requirements of $\mathcal{PT}$-symmetry. We then focus on case examples of soliton and periodic solutions for which we present a stability analysis as a function of their amplitude parameters. Finally, we numerically explore the nonlinear dynamics of the associated waveforms to identify the outcome of the relevant dynamical instabilities of localized and extended states.

nlin.PS↗

arXiv2014

Compacton matter waves in binary Bose gases under strong nonlinear management

The existence of compacton matter waves in binary mixtures of quasi one-dimensional Bose-Einstein condensates in deep optical lattices and in the presence of nonlinearity management, is first demonstrated. For this, we derive an averaged vector discrete nonlinear Schrödinger equation (DNLSE) and show that compacton solutions of different types can exist as stable excitations. Stability properties are studied by linear analysis and by direct numerical integrations of the DNLSE system and their dependence on the inter- and intra-species scattering lengths, investigated. We show that under proper management conditions, compactons can be very robust excitations that can emerge spontaneously from generic initial conditions. A possible experimental setting for compacton observation is also discussed.

cond-mat.quant-gas↗

arXiv2014

Matter wave soliton bouncer

Dynamics of a matter wave soliton bouncing on the reflecting surface (atomic mirror) under the effect of gravity has been studied by analytical and numerical means. The analytical description is based on the variational approach. Resonant oscillations of the soliton's center of mass and width, induced by appropriate modulation of the atomic scattering length and the slope of the linear potential are analyzed. In numerical experiments we observe the Fermi type acceleration of the soliton when the vertical position of the reflecting surface is periodically varied in time. Analytical predictions are compared with the results of numerical simulations of the Gross-Pitaevskii equation and qualitative agreement between them is found.

cond-mat.quant-gas↗