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Annalen der Physik

Investigation of Bose–Einstein Condensation Dynamics by Homotopy Perturbation and Bernoulli Sub‐Equation Function Methods

Devrim Özlem Görmüş, Hüseyin Şirin, Abdullah Engin Çalık

Published Feb 1, 2026• 0 citations
Publisher

Abstract

ABSTRACT In this study, the dynamics of a quasi 1D Bose–Einstein condensate is investigated with the help of homotopy perturbation and Bernoulli sub‐equation function methods, and their effectiveness is compared. The quasi‐1D condensate, composed of interacting atoms at low temperatures and low densities, is modeled by the 1D Gross–Pitaevskii equation, which provides highly accurate results under the assumptions of mean‐field theory. In this context, the equation is considered both in the absence of a trap potential and in the presence of the Pöschl–Teller trap potential, which holds special significance in quantum mechanics. For the first time, the analysis of the 1D Gross–Pitaevski equation with the Pöschl–Teller trap potential is carried out using the Bernoulli sub‐equation function and homotopy perturbation methods.

DOI
10.1002/andp.202500311

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ABSTRACT In this study, the dynamics of a quasi 1D Bose–Einstein condensate is investigated with the help of homotopy perturbation and Bernoulli sub‐equation function methods, and their effectiveness is compared. The quasi‐1D condensate, composed of interacting atoms at low temperatures and low densities, is modeled by the 1D Gross–Pitaevskii equation, which provides highly accurate results under the assumptions of mean‐field theory. In this context, the equation is considered both in the absence of a trap potential and in the presence of the Pöschl–Teller trap potential, which holds special significance in quantum mechanics. For the first time, the analysis of the 1D Gross–Pitaevski equation with the Pöschl–Teller trap potential is carried out using the Bernoulli sub‐equation function and homotopy perturbation methods.
We present for the first time an approach about Bose-Einstein condensates made up of atoms with attractive interatomic interactions confined to the Pöschl-Teller hyperbolic potential. In this paper, we consider a Bose-Einstein condensate confined in a cigar-shaped, and it was modeled by the mean field equation known as the Gross-Pitaevskii equation. An analytical (variational method) and numerical (two-step Crank-Nicolson) approach is proposed to study the proposed model of interatomic interaction. The solutions of the one-dimensional Gross-Pitaevskii equation obtained in this paper confirmed, from a theoretical point of view, the possibility of the Pöschl-Teller potential to confine Bose-Einstein condensates. The chemical potential as a function of the depth of the Pöschl-Teller potential showed a behavior very similar to the cases of Bose-Einstein condensates and superfluid Fermi gases in optical lattices and optical superlattices. The results presented in this paper can open the way for several applications in atomic and molecular physics, solid state physics, condensed matter physics, and material sciences.
In this study, after we have briefly introduced the standard Gross-Pitaevskii equation, we have suggested fractional Gross-Pitaevskii equations to investigate the time-dependent ground state dynamics of the Bose-Einstein condensation of weakly interacting bosonic particle system which can includes non-Markovian processes or non-Gaussian distributions and long-range interactions. Only we focused the time-fractional Gross-Pitaevskii equation and have obtained solutions of the standard Gross-Pitaevskii and time-fractional Gross-Pitaevskii equations for attractive and repulsive interactions in the case external trap potentials V(x)=0V(x)=0V(x)=0 and optical lattice potential V(x)=±sin⁡2xV(x) =\pm\sin^{2}xV(x)=±sin2x by using Homotopy Perturbation Method. We have found that the Homotopy Perturbation Method solutions of the Gross-Pitaevskii equation for these potentials and interactions are the same analytical results of it. Furthermore we have also found that solutions of the time-fractional Gross-Pitaevskii equation for these potentials and interactions can be given in terms of Mittag-Leffler function. The solutions of the time-fractional Gross-Pitaevskii equation provide that the time evolution of the ground state dynamics of Bose-Einstein condensation of bosonic particles deviates exponential form, and evolutes with time as stretched exponentially.
We try to find the analytical solutions to the time-independent Gross-Pitaevskii equation, a nonlinear Schrödinger equation used in the simulation of Bose—Einstein condensates trapped in a harmonic potential. Both the homotopy analysis method and the Galerkin spectral method are applied. We investigate the one-dimensional case and obtain the approximate analytical solutions successfully. Comparison between the analytical solutions and the numerical solutions has been made. The results indicate that they agree very well with each other when the atomic interaction is not too strong.
Abstract In this paper, we theoretically investigate the dynamics and stability of matter‐wave solitons (MWS) in a cigar‐shaped Bose–Einstein condensate (BEC) consisting of atoms with attractive interatomic interactions trapped by Pöschl–Teller (PT) potential. In this scenario, we propose the idea of the BEC being dragged by PT potential at a constant speed. The BEC was modeled by the quasi‐one‐dimensional Gross–Piteaevskii Equation (1D‐GPE). In addition, an analytical (time‐dependent variational method) and numerical (split‐step Crank–Nicolson method) approach has been proposed to investigate the dynamic properties of solitons during the temporal evolution of this system. Both analytical and numerical results demonstrated that MWSs dragged by PT potential can be stable. From a general point of view, the results obtained in this paper can motivate both theoretical and experimental investigations on the dynamics and stability of solitons in dilute and ultracold unitary Bose gases trapped by PT potential, nonlinear optical properties in PT quantum wells, and propagation of solitons in nonlinear media.
The homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions for the Gross-Pitaevskii equations, a set of nonlinear Schrödinger equation used in simulation of spin-1 Bose-Einstein condensates trapped in a harmonic potential. We investigate the one-dimensional case and get the approximate analytical solutions successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are in agreement well with each other when the atomic interaction is weakly. We also find a class of exact solutions for the stationary states of the spin-1 system with harmonic potential for a special case.