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Chinese Physics Letters

Small Amplitude Solitons in Bose–Einstein Condensates with External Perturbation

Fengjiao Wang, Xiaohong Yan, Denglong Wang

Published Jan 1, 2008• 13 citations
Publisher

Abstract

By developing a small amplitude soliton approximation method, we study analytically weak nonlinear excitations in cigar-shaped condensates with repulsive interatomic interaction under consideration of external perturbation potential. It is shown that matter wave solitons may exist and travel over a long distance without attenuation and change in shape by properly adjusting the strength of interatomic interaction to compensate for the effect of external perturbation potential.

DOI
10.1088/0256-307x/25/1/002

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Chinese Physics Letters•Jul 21, 2006•7 citations
By developing a small amplitude soliton approximation method, we study analytically weak nonlinear excitations in cigar-shaped condensates with repulsive interatomic interaction under consideration of external perturbation potential. It is shown that matter wave solitons may exist and travel over a long distance without attenuation and change in shape by properly adjusting the strength of interatomic interaction to compensate for the effect of external perturbation potential.
Abstract We analyze the formation and dynamics of W-shaped solitons in an inhomogeneous cigar-shaped Bose–Einstein condensate with repulsive interatomic interactions in the presence of spatially and temporally nonuniform nonlinearity and external potential. The condensate is assumed to have a large number of particles; hence it may be approximated by an effective one-dimensional equation, which is derived from the full three-dimensional Gross–Pitaevskii equation. New classes of W-shaped soliton solutions, which may propagate in the condensate system with the same hump value, are found for the first time by means of the similarity transformation technique. The functional form of these localized nonlinear structures is different from the usual ones reported in the setting of optical fibers. The dynamic behaviors of the W-shaped soliton pulses under different nonlinearity and external potential management that are of particular interest for applications in Bose–Einstein condensates are analyzed.
The dynamics of the weak nonlinear matter solitary waves in two-component Bose–Einstein condensates (BEC) with cigar-shaped external potential are investigated analytically by a perturbation method. In the small amplitude limit, the two-components can be decoupled and the dynamics of solitary waves are governed by a variable-coefficient Korteweg–de Vries (KdV) equation. The reduction to the KdV equation may be useful to understand the dynamics of nonlinear matter waves in two-component BEC. The analytical expressions for the evolution of soliton, emitted radiation profiles and soliton oscillation frequency are also obtained.
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 Gross-Pitaevskii equation for a Bose-Einstein condensate confined in an elongated cigar-shaped trap is reduced to an effective system of nonlinear equations depending on only one space coordinate along the trap axis. The radial distribution of the condensate density and its radial velocity are approximated by Gaussian functions with real and imaginary exponents, respectively, with parameters depending on the axial coordinate and time. The effective one-dimensional system is applied to a description of the ground state of the condensate, to dark and bright solitons, to the sound and radial compression waves propagating in a dense condensate, and to weakly nonlinear waves in repulsive condensate. In the low-density limit our results reproduce the known formulas. In the high-density case our description of solitons goes beyond the standard approach based on the nonlinear Schr\"odinger equation. The dispersion relations for the sound and radial compression waves are obtained in a wide region of values of the condensate density. The Korteweg--de Vries equation for weakly nonlinear waves is derived and the existence of bright solitons on a constant background is predicted for a dense enough condensate with a repulsive interaction between the atoms.
To analytically investigate the matter-wave solitons of Bose–Einstein condensates (BECs) in time-dependent complex potential, we consider a cubic-quintic Gross–Pitaevskii (GP) equation with distributed coefficients and a dissipative term. By introducing a suitable ansatz, we establish the criterion of the modulational instability (MI) of the system and present an explicit expression for the growth rate of a purely growing MI. Effects of the parabolic background potential, as well as of the linear potential, the gain/loss parameter, and the two- and three-body interatomic interactions on the MI are investigated. We show how the feeding/loss parameter can be well used to control the instability of the system. The analytical resolution of the considered GP equation leads to exact bright, dark and kink solitary wave solutions which are used to investigate analytically the dynamics of matter-wave solitons in BECs under consideration. These analytical investigations show that the amplitude and the motion of bright, dark and kink solitary waves depend on the strengths of the two- and three-body interatomic interactions, as well as on the strengths of the external trapping potential and the parameter of the gain/loss of atoms in the condensate.
We discuss the possible nonlinear waves of atomic matter wave in a Bose–Einstein condensate. One and two of two-dimensional (2D) dark solitons in the Bose–Einstein condensed system are investigated. A rich dynamics is studied for the interactions between two solitons. The interaction profiles of two solitons are greatly different if the angle between them are different. If the angle is small enough, the maximum amplitude during the interaction between two solitons is even less than that of a single soliton. However, if the angle is large enough, the maximum amplitude of two solitons can gradually attend to the sum of two soliton amplitudes.