Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0909.2397 · International Journal of Modern Physics B

VARIATIONAL ANALYSIS OF FLAT-TOP SOLITONS IN BOSE–EINSTEIN CONDENSATES

Bakhtiyor B. Baizakov, A. BOUKETIR, A. MESSIKH, A. BENSEGHIR, B. A. PUMAROV

Published Jun 9, 2011• 28 citations
arXivHTMLPDF

Abstract

Static and dynamic properties of matter-wave solitons in dense Bose–Einstein condensates, where three-body interactions play a significant role, have been studied by a variational approximation (VA) and numerical simulations. For experimentally relevant parameters, matter-wave solitons may acquire a flat-top shape, which suggests employing a super-Gaussian trial function for VA. Comparison of the soliton profiles, predicted by VA and those found from numerical solution of the governing Gross–Pitaevskii equation shows good agreement, thereby validating the proposed approach.

DOI
10.1142/s0217979211101521

Related papers

Statics and dynamics solitons in Bose-Einstein condensates

A. Benseghir, Bakhram Umarov, Bakhtiyor B. Baizakov +4 more

Static and dynamic properties of matter-wave solitons in dense Bose-Einstein condensates, where three-body interactions play a significant role, have been studied by a variational approximation (VA) and numerical simulations. For experimentally relevant parameters, matter-wave solitons may acquire a flat-top shape, which suggests employing a superGaussian trial function for VA. Comparison of the soliton profiles, predicted by VA and those found from numerical solution of the governing Gross-Pitaevskii equation shows good agreement, thereby validating the proposed approach.

•Jan 1, 2009•0 citations

The stability of matter-wave solitons in 2D linear and nonlinear optical lattices

Chen Hai-Jun, Xiangfu Li

By means of variational solution and direct numerical simulation of the Gross-Pitaevskii equation (GPE), we have studied the stability of matter-wave solitons in two-dimensional (2D) Bose-Einstein condensations (BECs), with 2D linear and nonlinear optical lattices (OLs). Using the static variational approach and Vakhitov-Kolokolov criterion necessary for stability, we obtain the stability condition for solitons in different combinations of OL's parameters. We show that the 2D linear and nonlinear optical lattices allow us to stabilize 2D solitons for both attractive and repulsive interactions. We also study the time-evolution problems of 2D BECs, using the time-dependent variational approach and numerical solution of GPE for 2D linear and nonlinear OLs. Very good agreement between the results corresponding to both treatments is observed.

Acta Physica Sinica•Jan 1, 2013•5 citations

Variational Monte Carlo study of soliton excitations in hard-sphere Bose gases

R. Rota, S. Giorgini

By using a full many-body approach, we calculate the excitation energy, the effective mass, and the density profile of soliton states in a three-dimensional Bose gas of hard spheres at zero temperature. The many-body wave function used to describe the soliton contains a one-body term, derived from the solution of the Gross-Pitaevskii equation, and a two-body Jastrow term, which accounts for the repulsive correlations between atoms. We optimize the parameters in the many-body wave function via a variational Monte Carlo procedure, calculating the grand-canonical energy and the canonical momentum of the system in a moving reference frame where the soliton is stationary. As the density of the gas is increased, significant deviations from the mean-field predictions are found for the excitation energy and the density profile of both dark and gray solitons. In particular, the soliton effective mass ${m}^{*}$ and the mass $m\mathrm{\ensuremath{\Delta}}N$ of missing particles in the region of the density depression are smaller than the result from the Gross-Pitaevskii equation, their ratio, however, being well reproduced by this theory up to large values of the gas parameter. We also calculate the profile of the condensate density around the soliton notch, finding good agreement with the prediction of the local-density approximation.

arXiv:1506.02970•Physical Review A•Oct 30, 2015•0 citations

Dynamics of Bose-Einstein condensates: Variational solutions of the Gross-Pitaevskii equations

Víctor M. Pérez‐García, Humberto Michinel, J. I. Cirac +2 more

A variational technique is applied to solve the time-dependent nonlinear Schr\"odinger equation (Gross-Pitaevskii equation) with the goal to model the dynamics of dilute ultracold atom clouds in the Bose-Einstein condensed phase. We derive analytical predictions for the collapse, equilibrium widths, and evolution laws of the condensate parameters and find them to be in very good agreement with our numerical simulations of the nonlinear Schr\"odinger equation. It is found that not only the number of particles, but also both the initial width of the condensate and the effect of different perturbations to the condensate may play a crucial role in the collapse dynamics. The results are applicable when the shape of the condensate is sufficiently simple.

Physical Review A•Aug 1, 1997•406 citations

Matter-wave bright solitons in bose-einstein condensate / Abdelaziz Benseghir

A. Benseghir

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.

•Jan 1, 2015•0 citations

Reexamination of dynamical stabilization of matter-wave solitons

A. P. Itin, Tōru Morishita, Shinichi Watanabe

We consider dynamical stabilization of Bose-Einstein condensates by time-dependent modulation of the scattering length. The problem has been studied before by several methods: Gaussian variational approximation, the method of moments, the method of modulated Townes soliton, and the direct averaging of the Gross-Pitaevskii equation. We summarize these methods and find that the numerically obtained stabilized solution has a different configuration than that assumed by the theoretical methods (in particular a phase of the wave function is not quadratic with $r$). We show that there is presently no clear evidence for stabilization in a strict sense, because in the numerical experiments only metastable (slowly decaying) solutions have been obtained. In other words, neither numerical nor mathematical evidence for a new kind of soliton solutions has been revealed so far. The existence of the metastable solutions is nevertheless an interesting and complicated phenomenon on its own. We try some non-Gaussian variational trial functions to obtain better predictions for the critical nonlinearity ${g}_{cr}$ for metastabilization but other dynamical properties of the solutions remain difficult to predict.

arXiv:cond-mat/0506472•Physical Review A•Sep 20, 2006•36 citations
Static and dynamic properties of matter-wave solitons in dense Bose–Einstein condensates, where three-body interactions play a significant role, have been studied by a variational approximation (VA) and numerical simulations. For experimentally relevant parameters, matter-wave solitons may acquire a flat-top shape, which suggests employing a super-Gaussian trial function for VA. Comparison of the soliton profiles, predicted by VA and those found from numerical solution of the governing Gross–Pitaevskii equation shows good agreement, thereby validating the proposed approach.
Static and dynamic properties of matter-wave solitons in dense Bose-Einstein condensates, where three-body interactions play a significant role, have been studied by a variational approximation (VA) and numerical simulations. For experimentally relevant parameters, matter-wave solitons may acquire a flat-top shape, which suggests employing a superGaussian trial function for VA. Comparison of the soliton profiles, predicted by VA and those found from numerical solution of the governing Gross-Pitaevskii equation shows good agreement, thereby validating the proposed approach.
By means of variational solution and direct numerical simulation of the Gross-Pitaevskii equation (GPE), we have studied the stability of matter-wave solitons in two-dimensional (2D) Bose-Einstein condensations (BECs), with 2D linear and nonlinear optical lattices (OLs). Using the static variational approach and Vakhitov-Kolokolov criterion necessary for stability, we obtain the stability condition for solitons in different combinations of OL's parameters. We show that the 2D linear and nonlinear optical lattices allow us to stabilize 2D solitons for both attractive and repulsive interactions. We also study the time-evolution problems of 2D BECs, using the time-dependent variational approach and numerical solution of GPE for 2D linear and nonlinear OLs. Very good agreement between the results corresponding to both treatments is observed.
By using a full many-body approach, we calculate the excitation energy, the effective mass, and the density profile of soliton states in a three-dimensional Bose gas of hard spheres at zero temperature. The many-body wave function used to describe the soliton contains a one-body term, derived from the solution of the Gross-Pitaevskii equation, and a two-body Jastrow term, which accounts for the repulsive correlations between atoms. We optimize the parameters in the many-body wave function via a variational Monte Carlo procedure, calculating the grand-canonical energy and the canonical momentum of the system in a moving reference frame where the soliton is stationary. As the density of the gas is increased, significant deviations from the mean-field predictions are found for the excitation energy and the density profile of both dark and gray solitons. In particular, the soliton effective mass m∗{m}^{*}m∗ and the mass $m\mathrm{\ensuremath{\Delta}}N$ of missing particles in the region of the density depression are smaller than the result from the Gross-Pitaevskii equation, their ratio, however, being well reproduced by this theory up to large values of the gas parameter. We also calculate the profile of the condensate density around the soliton notch, finding good agreement with the prediction of the local-density approximation.
A variational technique is applied to solve the time-dependent nonlinear Schr\"odinger equation (Gross-Pitaevskii equation) with the goal to model the dynamics of dilute ultracold atom clouds in the Bose-Einstein condensed phase. We derive analytical predictions for the collapse, equilibrium widths, and evolution laws of the condensate parameters and find them to be in very good agreement with our numerical simulations of the nonlinear Schr\"odinger equation. It is found that not only the number of particles, but also both the initial width of the condensate and the effect of different perturbations to the condensate may play a crucial role in the collapse dynamics. The results are applicable when the shape of the condensate is sufficiently simple.
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 consider dynamical stabilization of Bose-Einstein condensates by time-dependent modulation of the scattering length. The problem has been studied before by several methods: Gaussian variational approximation, the method of moments, the method of modulated Townes soliton, and the direct averaging of the Gross-Pitaevskii equation. We summarize these methods and find that the numerically obtained stabilized solution has a different configuration than that assumed by the theoretical methods (in particular a phase of the wave function is not quadratic with rrr). We show that there is presently no clear evidence for stabilization in a strict sense, because in the numerical experiments only metastable (slowly decaying) solutions have been obtained. In other words, neither numerical nor mathematical evidence for a new kind of soliton solutions has been revealed so far. The existence of the metastable solutions is nevertheless an interesting and complicated phenomenon on its own. We try some non-Gaussian variational trial functions to obtain better predictions for the critical nonlinearity gcr{g}_{cr}gcr​ for metastabilization but other dynamical properties of the solutions remain difficult to predict.