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«Узбекский физический журнал»

Dynamics of a bouncing matter-wave soliton under periodically modulated atomic interactions

Kh. Kh. Ismailov

Published Feb 5, 2022• 0 citations
Publisher

Abstract

We consider, by analytical and numerical means, the dynamics of a matter-wave soliton bouncing on the reflecting surface (atomic mirror) under the influence of gravity, when twobody interactions between atoms are periodically modulated in time. The analytical description is based on the variational approach. The frequencies of small amplitude oscillations of the soliton's center-of-mass and width are obtained. Analytical predictions are compared to the results of numerical simulations of the Gross-Pitaevskii equation and qualitative agreement between them is found.

DOI
10.52304/.v24i1.299

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We consider, by analytical and numerical means, the dynamics of a matter-wave soliton bouncing on the reflecting surface (atomic mirror) under the influence of gravity, when twobody interactions between atoms are periodically modulated in time. The analytical description is based on the variational approach. The frequencies of small amplitude oscillations of the soliton's center-of-mass and width are obtained. Analytical predictions are compared to the results of numerical simulations of the Gross-Pitaevskii equation and qualitative agreement between them is found.
The 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 to the results of numerical simulations of the Gross-Pitaevskii equation and qualitative agreement between them is found.
The dynamics of a two-soliton molecule bouncing on the reflecting atomic mirror under the effect of gravity have been studied by analytical and numerical methods. The analytical description is based on the variational approximation. In numerical simulations, we observe the resonance oscillations of the two-soliton’s center-of-mass position and width, induced by modulated atomic mirror. Theoretical predictions are verified by numerical simulations of the nonlocal Gross–Pitaevskii equation (GPE) and qualitative agreement between them is found. Hamiltonian dynamic system for a dipolar Bose–Einstein condensates (BECs) has been studied.
We investigate the properties of modulational instability in the Salerno equation in quasi-one dimension in Bose-Einstein condensate (BEC). We analyzed the regions of modulational instability of nonlinear plane waves and determine the conditions of its existence in BEC. The existence and stability of strongly localized modes in discrete media is investigated with the framework of the Salerno model by using a linear analysis method. The regions of stability and instability are determined. Also the existence of localized modes for different values of parameters is shown numerically by homoclinic orbits intersection method. The response of Bose–Einstein condensates is studied with strong dipole–dipole atomic interactions to periodically varying perturbation. The dynamics is governed by the Gross–Pitaevskii equation with an additional nonlinear term, corresponding to nonlocal dipolar interactions. A mathematical model, based on the variational approximation, has been developed and applied to study parametric excitation of the condensates due to coefficient of nonlocal nonlinearity varying periodically. The model predicts the waveform of solitons in dipolar condensates and describes their small amplitude dynamics quite accurately. Theoretical predictions are verified by numerical simulations of the nonlocal Gross–Pitaevskii equation and a good agreement between them is found. The results can lead to better understanding of the properties of ultra-cold quantum gases, such as 52Cr, 164Dy and 168Er, where the long-range dipolar atomic interactions dominate the usual contact interactions. 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 iv 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 a qualitative agreement between them is found.
We study the motion of bright matter-wave solitons in nonlinear potentials, produced by periodic or random spatial variations of the atomic scattering length. We obtain analytical results for the soliton motion, the radiation of matter wave, and the radiative soliton decay in such configurations of the Bose-Einstein condensate. The stable regimes of propagation are analyzed. The results are in remarkable agreement with the numerical simulations of the Gross-Pitaevskii equation with periodic or random spatial variations of the mean field interactions.
Abstract Using the time-dependent variational approach and the numerical solution of the Gross–Pitaevskii (GP) equation, the dynamics of a matter-wave soliton are investigated in a two-dimensional constant external force field. Firstly, the time-dependent variational approach is used to study the movements of the center-of-mass position in a simple harmonic trap and a two-dimensional weakly transmissive reflective barrier. Because the mathematical model for describing soliton dynamics in simple harmonic traps is simple, the analytic dynamical results of a soliton wave packet center-of-mass position is given directly. Since the model describing the weakly transmissive reflective barrier is difficult to solve analytically, we give the Newton’s equation of the center-of-mass position of motion, and then use the numerical solution method to study the movement of the center-of-mass in the case of several parameters. To study the motion of the soliton in the harmonic potential trap in the two-dimensional external force field under rotation , we use the split-step finite difference method to solve the two-dimensional GP equation; the motion of the center-of-mass position is discussed under the conditions of several parameter combinations. The results show that the rotation has a very important influence on the movement of the center-of-mass. When there is no rotation effect, the center-of-mass position acts as a simple harmonic motion under the combined action of the external force field and the potential well. When there is a rotation effect, the center-of-mass position is almost static.
The dynamics of the soliton in a self-attractive Bose–Einstein condensate under the gravity are investigated. First, we apply the inverse scattering method, which gives rise to equation of motion for the center-of-mass coordinate of the soliton. We analyze the amplitude-frequency characteristic for nonlinear resonance. Applying the Krylov–Bogoliubov method for the small parameters the dynamics of soliton on the phase plane are considered. Hamiltonian chaos under the action of the gravity on the Poincaré map are studied.