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Journal of Northwest Normal University

Head-on collisions and phase shifts of solitons in Bose-Einstein condensate

Duan Wen-Shan

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Head-on collisions between two solitary waves(e.i.solitons) in a disk-shaped Bose-Einstein condensate(BEC) are investigated by using the extended PLK method.Two dark solitons in the system can be described by two KdV equations,their phase shifts are obtained.Based on the comparisons between a disk-shaped BEC and a cigar-shaped BEC,it is found that the values of phase shifts after head-on collision in the two cases are approximately equivalent when the wave number tends to 1.

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Head-on collisions between two solitary waves(e.i.solitons) in a disk-shaped Bose-Einstein condensate(BEC) are investigated by using the extended PLK method.Two dark solitons in the system can be described by two KdV equations,their phase shifts are obtained.Based on the comparisons between a disk-shaped BEC and a cigar-shaped BEC,it is found that the values of phase shifts after head-on collision in the two cases are approximately equivalent when the wave number tends to 1.
The ring dark solitons and their head-on collisions in a Bose–Einstein condensates with thin disc-shaped potential are studied. It is shown that the system admits a solution with two concentric ring solitons, one moving inwards and the other moving outwards, which in small-amplitude limit, are described by the two cylindrical KdV equations in the respective reference frames. By using the extended Poincare–Lighthill–Kuo perturbation method, the analytical phase shifts following the head-on collisions between two ring dark solitary waves are derived. It is shown that the phase shifts decrease with the radial coordinate r according to the r−1/3 law and depend on the initial soliton amplitude and radius.
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Systems of solitary waves in the one-dimensional Gross-Pitaevskii equation, which models a trapped atomic Bose-Einstein condensate, are investigated theoretically. To analyze the soliton nature of these solitary waves, a particle analogy for the solitary waves is formulated. Exact soliton solutions exist in the absence of an external trapping potential, which behave in a particlelike manner, and we find the particle analogy we employ to be a good model also when a harmonic trapping potential is present up to a gradual shift in the trajectories when the harmonic trap period is short compared with the collision time of the solitons. We find that the collision time of the solitons is dependent on the relative phase of the solitons as they collide. In the case of two solitons, the particle model is integrable, and the dynamics are completely regular. In the case of a system of two solitary waves of equal norm, the solitons are shown to retain their phase difference for repeated collisions. This phase preservation can be used to find regimes where there is agreement between the wave and particle models. This also implies that soliton regimes may be found in three-dimensional geometries where solitary waves can be made to repeatedly collide out of phase, stabilizing the condensate against collapse. The extension to three particles supports both regular and chaotic regimes. The trajectory shift observed for two solitons carries over to the case of three solitons. This shift aside, the agreement between the particle model and the wave dynamics remains good, even in chaotic regimes.
Interactions of solitary waves in a cylindrically confined Bose-Einstein condensate are investigated by simulating their head-on collisions. Slow vortex rings and fast solitons are found to collide elastically contrary to the situation in the three-dimensional homogeneous Bose gas. Strongly inelastic collisions are absent for low density condensates but occur at higher densities for intermediate velocities. The scattering behavior is rationalized by use of dispersion diagrams. During inelastic collisions, spherical shell-like structures of low density are formed and they eventually decay into depletion droplets with solitary-wave features. The relation to similar shells observed in a recent experiment by Ginsberg et al. [Phys. Rev. Lett. 94, 040403 (2005)] is discussed.