Ukrainian Journal of Physical Optics

Bright and Dark Solitons in a (2+1)-Dimensional Spin-1 Bose-Einstein Condensates

Yan Li, Q Chen, Houria Triki, Feng Liu, Youwen Sun, S Xu, Qin Zhou

Published Jan 1, 2024• 27 citations

Abstract

The three-coupled Gross-Pitaevskii equation with a harmonic potential in a (2+1)-dimensional spin-1 Bose-Einstein condensate is studied in this paper.Using the Hirota bilinear method, the bright and dark soliton solutions of the system with attractive and repulsive interactions are obtained, and the amplitudes and velocities of those solitons are also given.In addition, we analyze the influence of harmonic potential on the dynamics of solitons and discuss the interaction between solitons through asymptotic analysis.As a result, we find that the amplitude and velocity of solitons are related to the harmonic potential, and the interaction between the two solitons is elastic.

DOI: 10.3116/16091833/ukr.j.phys.opt.2024.s1060

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Ukrainian Journal of Physical Optics

Bright and Dark Solitons in a (2+1)-Dimensional Spin-1 Bose-Einstein Condensates

Yan Li, Q Chen, Houria Triki, Feng Liu, Youwen Sun, S Xu, Qin Zhou

Published Jan 1, 2024• 27 citations

Publisher

Abstract

DOI: 10.3116/16091833/ukr.j.phys.opt.2024.s1060

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g>0

). By using an adiabatic perturbation theory in the parameter

{g}_{12}/g

, we show that, contrary to the case of two solitons in scalar condensates, the interactions between solitons are attractive when the interparticle interactions between condensates are repulsive

{g}_{12}>0

. As a result, the relative motion of dark solitons with equal chemical potential $\ensuremath{\mu}$ is well approximated by harmonic oscillations of angular frequency ${w}_{r}=(\ensuremath{\mu}/\ensuremath{\hbar})\sqrt{(8/15){g}_{12}/g}$. We also show that, in finite systems, the resonance of this anomalous excitation mode with the spin-density mode of lowest energy gives rise to alternating dynamical instability and stability fringes as a function of the perturbative parameter. In the presence of harmonic trapping (with angular frequency $\mathrm{\ensuremath{\Omega}}$) the solitons are driven by the superposition of two harmonic motions at a frequency given by ${w}^{2}={(\mathrm{\ensuremath{\Omega}}/\sqrt{2})}^{2}+{w}_{r}^{2}$. When

{g}_{12}<0

, these two oscillators compete to give rise to an overall effective potential that can be either single well or double well through a pitchfork bifurcation. All our theoretical results are compared with numerical solutions of the Gross-Pitaevskii equation for the dynamics and the Bogoliubov equations for the linear stability. A good agreement is found between them.

We consider vector solitons of mixed bright-dark types in quasi-one-dimensional spinor

(F=1)

Bose-Einstein condensates. Using a multiscale expansion technique, we reduce the corresponding nonintegrable system of three coupled Gross-Pitaevskii equations (GPEs) to an integrable Yajima-Oikawa system. In this way, we obtain approximate solutions for small-amplitude vector solitons of dark-dark-bright and bright-bright-dark types, in terms of the ${m}_{F}=+1,\ensuremath{-}1,0$ spinor components, respectively. By means of numerical simulations of the full GPE system, we demonstrate that these states indeed feature soliton properties, i.e., they propagate undistorted and undergo quasielastic collisions. It is also shown that in the presence of a parabolic trap the bright component(s) is (are) guided by the dark one(s) and, as a result, the small-amplitude vector soliton as a whole performs quasiharmonic oscillations. The oscillation frequency is found as a function of the spin-dependent interaction strength for both small-amplitude and large-amplitude solitons.