arXiv subjects

Fatkhulla Kh. Abdullaev

Publications and source records attributed to Fatkhulla Kh. Abdullaev.

At least 19 recordsLinked to original sources

arXiv2026

Front-based construction of quantum droplets, bubbles, and hole states in a Bose mixture

We study nonlinear holes, bubbles, kink fronts, and quantum-droplet-like states in an elongated Bose mixture governed by an extended Gross-Pitaevskii equation with attractive cubic mean-field interactions and repulsive quartic Lee-Huang-Yang contribution. We classify the constant-amplitude backgrounds: the Bogoliubov-de Gennes spectrum shows that the upper branch is modulationally stable, whereas the lower branch is unstable to long-wavelength perturbations. Grand-potential and pressure determine the vacuum-finite-density coexistence point and the unique background supporting stationary kink and antikink fronts. Symmetric holes on the stable background have a density deficit and integral width that diverge near coexistence as the hole separates into two nonlinear fronts. A multiplicative kink-antikink construction connects bell-shaped states to broad flat-top droplets and approaches the stationary quantum-droplet family as the front separation increases. An antikink-kink sum with equal signs generates bubble-like density depletions without a phase jump, whereas their difference produces dark-hole-like profiles with a $π$ phase difference between the asymptotic backgrounds. Bubble depth increases with front separation, distinguishing gray-like and dark-like profiles. Real-time simulations show that deep bubbles persist under weak random perturbations, demonstrating finite-time robustness without establishing spectral stability. In the asymmetric two-component model, localized relative-density modes can induce spinor-driven breakup on a modulationally stable background, while robust holes persist near the kink limit. Collisions of kink-antikink droplet-like states display phase-dependent merging, asymmetric particle transfer, and effective reflection. These results connect nonlinear fronts with bubbles, phase-jump holes, and self-bound droplets in beyond-mean-field Bose mixtures.

cond-mat.quant-gas↗

arXiv2026

Quantum scattering of droplets by wells and barriers in one-dimensional Bose-Bose mixtures

We investigate, both analytically and numerically, the scattering of quasi-one-dimensional quantum droplets from Pöschl-Teller potential wells and barriers. For attractive wells, we find a sharp transition between complete reflection and transmission at a critical incident velocity for both small and large flat-top droplets. The scattering interactions differ: small, soliton-like droplets form a spatially symmetric trapped mode at the critical velocity, showing their compressibility and coherence characteristics, while large droplets develop a spatially asymmetric trapped state, revealing incompressibility and internal structure. The critical velocity depends non-monotonically on atom number: it rises in the small, compressible-droplet regime, falls in the incompressible, flat-top regime, and turns at the crossover point. We also show that the reflectionless well generates a $π$-phase shift, strongly altering droplet-droplet collisions relative to free space. The persistence of a confined mode after collisions between trapped and incident droplets depends sensitively on their relative phase. For the repulsive barrier, we identify regimes of complete reflection, partial return, and full transmission, depending on incident velocity, barrier height, and particle number. Our predictions match direct numerical simulations in all cases.

cond-mat.quant-gas↗

arXiv2026

Josephson Dynamics of 2D Bose-Einstein Condensates in Dual-Core Trap: Homogeneous, Droplet-Droplet, and Vortex-Vortex Regimes

The dynamics of a two-dimensional Bose-Einstein condensate mixture, loaded into a dual-core trap, when beyond-mean-field effects are taken into account, are considered. The effects of quantum fluctuations are described by the Lee-Huang-Yang correction terms in the extended coupled Gross-Pitaevskii equations. The spatially uniform and inhomogeneous BEC cases are studied. In the first case, the parameter regimes associated with macroscopic quantum tunnelling, self-trapping, and revival-like localisation dynamics are found. The Josephson oscillation frequencies for both the zero-phase and the $π$-phase modes are derived. As the total atom number varies, the dynamics exhibit a nontrivial bifurcation structure: along the zero-phase branch, two pitchfork bifurcations generate bistability and hysteresis, while the $π$-phase branch shows a single pitchfork bifurcation. In the second case, the Josephson dynamics for quantum droplets and vortices are investigated. Predictions for the oscillation frequencies of the atomic population between quantum droplets are obtained and fully validated by direct numerical simulations of coupled extended GP equations. The existence of the Andreev-Bashkin nondissipative drag through simulations of droplet-droplet interactions is shown. The Josephson dynamics of vortex states are studied. Vortices with topological charge $S$ and sufficiently small particle number are typically unstable, breaking up into $S+1$ (occasionally $S+2$) fundamental fragments, with the breakup time increasing as the particle number grows. Unstable asymmetric vortices show splitting and/or crescent-like instability. For vortices with sufficiently large norms, long-time simulations confirm robust stability against small perturbations; in this regime, Josephson oscillations and Andreev-Bashkin-type entrainment for vortex states with charges $S=1, 2$, and $3$ are investigated.

cond-mat.quant-gas↗

arXiv2025

Modulational instability and discrete quantum droplets in a deep quasi-one-dimensional optical lattice

We study the properties of modulational instability and discrete breathers arising in a quasi-one-dimensional discrete Gross-Pitaevskii equation with Lee-Huang-Yang corrections. Conditions for modulation instability and instability regions of nonlinear plane waves are determined in parameter space. We analytically investigate the existence of different quantum droplet solutions, including intersite, onsite, front-like, flat-top and dark localized modes, using the Page method and variational approach. Their stability is checked using linear stability analyses and numerical simulations. The analytical predictions corroborated with the numerical simulations.

nlin.PS↗

arXiv2025

Beyond mean-field effects in Josephson oscillations and self-trapping of Bose-Einstein condensates in two-dimensional dual-core traps

We study a binary Bose gas in a symmetric dual-core, pancake-shaped trap, modelled by two linearly coupled two-dimensional Gross-Pitaevskii equations with Lee-Huang-Yang corrections. Two different cases are considered. First, we consider a spatially uniform condensate, where we identify the domains of parameters for macroscopic quantum tunnelling, self-trapping and localisation revivals. The analytical formulas for the Josephson frequencies in the zero- and $π$-phase modes are derived. As the total atom number varies, the system displays a rich bifurcation structure. In the zero-phase, two successive pitchfork bifurcations generate bistability and hysteresis, while the $π$-phase exhibits a single pitchfork bifurcation. The second case is when the quantum droplets are in a dual-core trap. Analytical predictions for the oscillation frequencies are derived via a variational approach for the coupled dynamics of quantum droplets, and direct numerical simulations validate the results. We identify critical values of the linear coupling that separate Josephson and self-trapped regimes as the particle number changes. We also found the Andreev-Bashkin superfluid drag effect in numerical simulations of the droplet-droplet interactions in the two-core geometry.

cond-mat.quant-gas↗

arXiv2024

Dynamics of quasi-one-dimensional quantum droplets in Bose-Bose mixtures

The properties of quasi-one-dimensional quantum droplets of Bose-Einstein condensates are investigated analytically and numerically, taking into account the contribution of quantum fluctuations. Through the development of a variational approach employing the super-Gaussian function, we identify stationary parameters for the quantum droplets. The frequency of breathing mode oscillations in these quantum droplets is estimated. Moreover, the study reveals that periodic modulation in time of the atomic scattering length induces resonance oscillations in quantum droplet parameters or the emission of linear waves, contingent on the amplitude of the external modulation. A similar analysis is conducted for the Lee-Huang-Yang fluid, confined in a parabolic potential. Theoretical predictions are corroborated through direct numerical simulations of the governing extended Gross-Pitaevskii equation. Additionally, we study the collision dynamics of quasi-one-dimensional quantum droplets.

cond-mat.quant-gas↗

arXiv2024

Beyond mean-field effects in dynamics of BEC in the double-well potential

The nonlinear dynamics of a quasi-1D BEC loaded in a double-well potential is studied. The beyond mean-field corrections to the energy in the form of the Lee-Huang-Yang term are taken into account. One-dimensional geometry is considered. The problem is described in the scalar approximation by the extended Gross-Pitaevskii (EGP) equation with the attractive quadratic nonlinearity, due to the Lee-Huang-Yang correction, and the effective cubic mean-field nonlinearity describing the residual intra- and inter-species interactions. To describe tunneling and localization phenomena, a two-mode model was obtained. The frequencies of the Josephson oscillations are found and confirmed by the full numerical simulations of the EGP equation. The parametric resonance in the Josephson oscillations, when the height of the barrier is periodically modulated, is studied. The predictions of the dimer model, including the case of the one-dimensional Lee-Huang-Yang superfluid, have been proven.

cond-mat.quant-gas↗

arXiv2024

Modulational instability in a quasi-one-dimensional Bose-Einstein condensates

In this work, we investigate the modulational instability of plane wave solutions within a modified Gross-Pitaevskii equation framework. The equation features cubic and quartic nonlinearity. It models the behaviour of quasi-one-dimensional Bose-Einstein condensates in symmetric Bose-Bose mixtures of ultra-dilute cold atoms. Our study demonstrates the pivotal role of the competition between mean-field attractions and quantum fluctuation-induced repulsions. This competition significantly affects the emergence and evolution of modulational instability. By employing linear stability analysis, we identify the essential conditions that lead to modulational instability. We find that the stability of plane wave solutions significantly depends on the interaction among system parameters. Further development of the instability leads to the fragmentation of the BEC into a chain of quantum droplets. We calculated the quantity of quantum droplets generated during the nonlinear phase of the instability. Our analytical results are corroborated by numerical simulations of the modified quasi-1D Gross-Pitaevskii equation. These simulations vividly depict the formation, interaction, and coalescence of droplets during the nonlinear phase of modulational instability. The investigation shows that linear stability analysis of the modified Gross-Pitaevskii equation, considering quantum fluctuations, precisely forecasts modulational instability phenomena across different domains of parameter spaces.

cond-mat.quant-gas↗

arXiv2023

Effects of quantum fluctuations on macroscopic quantum tunneling and self-trapping of BEC in a double well trap

We study the influence of quantum fluctuations on the macroscopic quantum tunneling and self-trapping of a two-component Bose-Einstein condensate in a double-well trap. Quantum fluctuations are described by the Lee-Huang-Yang term in the modified Gross-Pitaevskii equation. Employing the modified Gross-Pitaevskii equation in scalar approximation, we derive the dimer model using a two-mode approximation. The frequencies of Josephson oscillations and self-trapping conditions under quantum fluctuations are found analytically and proven by numerical simulations of the modified Gross-Pitaevskii equation. The tunneling and localization phenomena are investigated also for the case of the Lee-Huang-Yang fluid loaded in the double-well potential.

cond-mat.quant-gas↗

arXiv2022

Modulational instability and quantum droplets in a two-dimensional Bose-Einstein condensate

Modulational instability of a uniform two-dimensional binary Bose-Einstein condensate (BEC) in the presence of quantum fluctuations is studied. The analysis is based on the coupled Gross-Pitaevskii equations. It is shown that quantum fluctuations can induce instability when the BEC density is below a threshold. The dependence of the growth rate of modulations on the BEC parameters is found. It is observed that an asymmetry of the interaction parameters and/or initial densities of the components typically decreases the growth rate. Further development of the instability results in a break-up of the BEC into a set of quantum droplets. These droplets merge dynamically with each other so that the total number of droplets decreases rapidly. The rate of this decrease is evaluated numerically for different initial parameters.

cond-mat.quant-gas↗

arXiv2021

Discrete Solitons of the Ginzburg-Landau Equation

In this chapter we review recent results concerning localized and extended dissipative solutions of the discrete complex Ginzburg-Landau equation. In particular, we discuss discrete diffraction effects arising both from linear and nonlinear properties, the existence of self-localized dissipative solitons in the presence of cubic-quintic terms and modulational instability induced by saturable nonlinearities. Dynamical stability properties of localized and extended dissipative discrete solitons are also discussed.

nlin.PS↗

arXiv2020

The variational approximation for two-dimensional quantum droplets

The dynamics of a two-dimensional Bose-Einstein condensate in a presence of quantum fluctuations is studied. The properties of localized density distributions, quantum droplets (QDs), are analyzed by means of the variational approach. It is demonstrated that the super-Gaussian function gives a good approximation for profiles of fundamental QDs and droplets with non-zero vorticity. The dynamical equations for parameters of QDs are obtained. Fixed points of these equations determine the parameters of stationary QDs. The period of small oscillations of QDs near the stationary state is estimated. It is obtained that periodic modulations of the strength of quantum fluctuations can actuate different processes, including resonance oscillations of the QD parameters, an emission of waves and a splitting of QDs into smaller droplets.

nlin.PS↗

arXiv2020

Nonlinearity managed dissipative solitons

We study the dynamics of localized pulses in the complex cubic-quintic Ginzburg-Landau (GL) equation with strong nonlinearity management. The generalized complex GL equation, averaged over rapid modulations of the nonlinearity, is derived. We employ the variational approach to this averaged equation to derive a dynamical system for the dissipative soliton parameters. By using numerical simulations of this dynamical system and the full GP equation, we show that nonlinearity-managed dissipative solitons exist in the model. The parameter regions for stabilization of exploding soliton are obtained.

nlin.PS↗

arXiv2019

Stationary and dynamical properties of one-dimensional quantum droplets

The dynamics of quantum droplets in 1D is analyzed on the basis of the variational approach (VA). It is shown that the VA based on the super-Gaussian function gives a good approximation of stationary states. The period of small oscillations of the perturbed droplet is obtained. It is found numerically that oscillations are almost undamped for many periods. Based on the VA, an existence of stable localized states for different combinations of signs of nonlinearities is demonstrated.

nlin.PS↗

arXiv2018

Flat bands and dynamical localization of spin-orbit coupled Bose-Einstein condensates

Flat bands and dynamical localization of binary mixtures of Bose-Einstein condensates, with spin-orbit coupling subjected to a deep optical lattice which is shaking in time and to a periodic time modulation of the Zeeman field, are investigated. In contrast with usual dynamical localization in the absence of spin-orbit coupling, we find that to fully suppress the tunneling in the system the optical lattice shaking is not enough, and a proper tuning of the spin-orbit term, achievable via the Zeeman field modulation, is also required. This leads to a sequence of Zeeman parameter values where energy bands become flat, the tunneling in the system is suppressed, and the dynamical localization phenomenon occurs. Exact wave functions at the dynamical localization points show that the binary mixture localizes on a dimer with the two components occupying different sites. This type of localization occurs in exact form also for the ground state of the system at the dynamical localization points in the presence of nonlinearity and remains valid, although in approximate form, for a wide range of the Zeeman parameter around these points. The possibility of observing the above phenomena in real experiments is also briefly discussed.

cond-mat.quant-gas↗

arXiv2018

Solitons and Josephson-type oscillations in Bose-Einstein condensates with spin-orbit coupling and time-varying Raman frequency

The existence and dynamics of solitons in quasi-one-dimensional Bose-Einstein condensates (BEC) with spin-orbit coupling (SOC) and attractive two-body interactions are described for two coupled atomic pseudo-spin components with slowly and rapidly varying time-dependent Raman frequency. By varying the Raman frequency linearly in time, it was shown that ordinary nonlinear Schrödinger-type bright solitons can be converted to striped bright solitons and vice versa. The internal Josephson oscillations between atom-number of the coupled soliton components, and the corresponding center-of-mass motion, are studied for different parameter configurations. In this case, a mechanism to control the soliton parameters is proposed by considering parametric resonances, which can emerge when using time-varying Raman frequencies. Full numerical simulations confirm variational analysis predictions when applied to the region where regular solitons are expected. In the limit of high frequencies, the system is described by a time-averaged Gross-Pitaevskii formalism with renormalized nonlinear and SOC parameters and modified phase-dependent nonlinearities. By comparing full-numerical simulations with averaged results, we have also studied the lower limits for the frequency of Raman oscillations in order to obtain stable soliton solutions.

cond-mat.quant-gas↗

arXiv2015

Faraday waves in BEC with engineering three-body interactions

We consider Bose-Einstein condensates with two- and three-body interactions periodically varying in time. Two models of time-dependent three-body interactions, with quadratic and quartic dependence on the two-body atomic scattering length $a_s$, are studied. It is shown that parametric instabilities in the condensate leads to the generation of Faraday waves (FW), with wavelengths depending on the background scattering length, as well as on the frequency and amplitude of the modulations of $a_s$. In an experimental perspective, this opens a new possibility to tune the period of Faraday patterns by varying not only the frequency of modulations and background scattering length, but also through the amplitude of the modulations. The latter effect can be used to estimate the parameters of three-body interactions from the FW experimental results. Theoretical predictions are confirmed by numerical simulations of the corresponding extended Gross-Pitaevskii equation.

cond-mat.quant-gas↗

arXiv2015

Discrete solitons of spin-orbit coupled Bose-Einstein condensates in optical lattices

We study localized nonlinear excitations of a dilute Bose-Einstein condensate (BEC) with spin-orbit coupling in a deep optical lattice (OL). We use Wannier functions to derive a tight-binding model that includes the spin-orbit coupling (SOC) at the discrete level in the form of a generalized discrete nonlinear Schödinger equation. Spectral properties are investigated and the existence and stability of discrete solitons and breathers with different symmetry properties with respect to the OL is demonstrated. We show that the symmetry of the modes can be changed from on-site to inter-site and to asymmetric modes simply by changing the interspecies interaction. Asymmetric modes appear to be novel modes intrinsic of the SOC.

cond-mat.quant-gas↗