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arXiv:cond-mat/0512169 · Physical Review E

Class of solitary wave solutions of the one-dimensional Gross-Pitaevskii equation

Rajneesh Atre, Prasanta Kumar Panigrahi, G. S. Agarwal

Published May 19, 2006• 177 citations
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Abstract

We present a large family of exact solitary wave solutions of the one-dimensional Gross-Pitaevskii equation, with time-varying scattering length and gain or loss, in both expulsive and regular parabolic confinement regimes. The consistency condition governing the soliton profiles is shown to map onto a linear Schrödinger eigenvalue problem, thereby enabling one to find analytically the effect of a wide variety of temporal variations in the control parameters, which are experimentally realizable. Corresponding to each solvable quantum mechanical system, one can identify a soliton configuration. These include soliton trains in close analogy to experimental observations of Streckeret al. [Nature (London) 417, 150 (2002)], spatiotemporal dynamics, solitons undergoing rapid amplification, collapse and revival of condensates, and analytical expression of two-soliton bound states, to name a few.

DOI
10.1103/physreve.73.056611

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We present a large family of exact solitary wave solutions of the one-dimensional Gross-Pitaevskii equation, with time-varying scattering length and gain or loss, in both expulsive and regular parabolic confinement regimes. The consistency condition governing the soliton profiles is shown to map onto a linear Schrödinger eigenvalue problem, thereby enabling one to find analytically the effect of a wide variety of temporal variations in the control parameters, which are experimentally realizable. Corresponding to each solvable quantum mechanical system, one can identify a soliton configuration. These include soliton trains in close analogy to experimental observations of Streckeret al. [Nature (London) 417, 150 (2002)], spatiotemporal dynamics, solitons undergoing rapid amplification, collapse and revival of condensates, and analytical expression of two-soliton bound states, to name a few.
The evolution of traveling and solitary waves in Bose-Einstein condensates (BECs) with a time-dependent scattering length in an attractive/repulsive parabolic potential is studied. The homogeneous balance principle and the F-expansion technique are used to solve the one-dimensional Gross-Pitaevskii equation with time-varying coefficients. We obtained three classes of new exact traveling wave and localized solutions. Our results demonstrate that the BEC solitary wave solutions can be manipulated and controlled by the time-dependent scattering length.
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.
We present three families of one-soliton solutions for (2+1)-dimensional Gross—Pitaevskii equation with both time-dependent scattering length and gain or loss in a harmonic trap. Then we investigate the dynamics of these solitons in Bose—Einstein condensates (BECs) by some selected control functions. Our results show that the intensities of these solitons first increase rapidly to the condensation peak, then decay very slowly to the background; thus the lifetime of a bright soliton, a train of bright solitons and a dark soliton in BECs can be all greatly extended. Our results offer a useful method for observing matter-wave solitons in BECs in future experiments.
We present three families of one-soliton solutions for (2+l)-dimensional Gross-Pitaevskii equation withboth time-dependent scattering length and gain or loss in a harmonic trap.Then we investigate the dynamics of thesesolitons in Bose-Einstein condensates (BECs) by some selected control functions.Our results show that the intensitiesof these solitons first increase rapidly to the condensation peak,then decay very slowly to the background;thus thelifetime of a bright soliton,a train of bright solitons and a dark soliton in BECs can be all greatly extended.Our resultsoffer a useful method for observing matter-wave solitons in BECs in future experiments.
We present an analytical study on the dynamics of bright and dark solitons in Bose–Einstein condensates with time-varying atomic scattering length in a time-varying external parabolic potential. A set of exact soliton solutions of the one-dimensional Gross–Pitaevskii equation are obtained, including fundamental bright solitons, higher-order bright solitons, and dark solitons. The results show that the soliton's parameters (amplitude, width, and period) can be changed in a controllable manner by changing the scattering length and external potential. This may be helpful to design experiments.
In this paper, we consider a Gross-Pitaevskii (GP) equation with a time-dependent nonlinearity and a spatiotemporal complex linear term which describes the dynamics of matter-wave solitons in Bose-Einstein condensates (BECs) with time-dependent interatomic interactions in a parabolic potential in the presence of feeding or loss of atoms. We establish the integrability conditions under which analytical solutions describing the modulational instability and the propagation of both bright and dark solitary waves on a continuous wave background are obtained. The obtained integrability conditions also appear as the conditions under which the solitary waves of the BECs can be managed by controlling the functional gain or loss parameter. For specific BECs, the dynamics of bright and dark solitons are investigated analytically through the found exact solutions of the GP equation. Our results show that under the integrability conditions, the gain or loss parameter of the GP equation can be used to manage the motion of both bright and dark solitons. We show that for BECs with loss (gain) of atoms, the bright and dark solitons during their propagation have a compression (broadening) in their width. Furthermore, under a safe range of parameters and under the integrability conditions, it is possible to squeeze a bright soliton of BECs with loss of atoms into the assumed peak matter density, which can provide an experimental tool for investigating the range of validity of the 1D GP equation. Our results also reveal that under the conditions of the solitary wave management, neither the injection or the ejection of atoms from the condensate affects the soliton peak during its propagation.