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New Journal of Physics

Modulational instability and bright solitary wave solution for Bose–Einstein condensates with time-dependent scattering length and harmonic potential

Lei Wu, Jie-Fang Zhang, Lu Li

Published Mar 21, 2007• 80 citations
Publisher

Abstract

We consider the one-dimensional Gross–Pitaevskii (GP) equation, which governs the dynamics of Bose–Einstein condensate (BEC) matter waves with time-dependent scattering length and a harmonic trapping potential. We present the integrable condition for the one-dimensional GP equation and obtain the exact analytical solution which describes the modulational instability and the propagation of a bright solitary wave on a continuous wave (cw) background. Moreover, by employing the adiabatic perturbation theory for a bright soliton, we obtain approximative bright solitary wave solutions under near-integrable conditions. Both the exact and approximative solutions show that the amplitude of a bright solitary wave with zero boundary condition depends on the scattering length while its motion depends on the external trapping potential.

DOI
10.1088/1367-2630/9/3/069

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We consider the one-dimensional Gross–Pitaevskii (GP) equation, which governs the dynamics of Bose–Einstein condensate (BEC) matter waves with time-dependent scattering length and a harmonic trapping potential. We present the integrable condition for the one-dimensional GP equation and obtain the exact analytical solution which describes the modulational instability and the propagation of a bright solitary wave on a continuous wave (cw) background. Moreover, by employing the adiabatic perturbation theory for a bright soliton, we obtain approximative bright solitary wave solutions under near-integrable conditions. Both the exact and approximative solutions show that the amplitude of a bright solitary wave with zero boundary condition depends on the scattering length while its motion depends on the external trapping potential.
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.
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.
The dynamics of a bright matter wave soliton in a quasi one-dimensional Bose–Einstein condensate (BEC) with a periodically rapidly varying time trap is considered. The governing equation is based on averaging the fast modulations of the Gross–Pitaevskii (GP) equation. This equation has the form of a GP equation with an effective potential of a more complicated structure than an unperturbed trap. In the case of an inverted (expulsive) quadratic trap corresponding to an unstable GP equation, the effective potential can be stable. For the bounded space trap potential it is showed that bifurcation exists, i.e. the single-well potential bifurcates to the triple-well effective potential. The stabilization of a BEC cloud on-site state in the temporary modulated optical lattice is found. This phenomenon is analogous to the Kapitza stabilization of an inverted pendulum. The analytical predictions of the averaged GP equation are confirmed by numerical simulations of the full GP equation with rapid perturbations.
We consider a cubic-quintic Gross–Pitaevskii equation which governs the dynamics of Bose–Einstein condensate matter waves with time-dependent scattering length and spatiotemporal complex potential. By introducing phase-imprint parameters in the system, we present the integrable condition for the equation and obtain the exact analytical solutions, which describe the propagation of a solitary wave. By applying specific time-modulated feeding/loss functional parameter, various types of magnetic trap strengths, and phase-imprint parameters, the dynamics of the solutions can be controlled. Solitary wave solutions with breathing and snaking behaviors are reported.
We investigate dynamical effects of a bright soliton in Bose–Einstein condensed (BEC) systems with local and smooth space variations of the two-body atomic scattering length. It includes a discussion about the possible observation of a new type of standing nonlinear atomic matter wave in cigar-type traps. A rich dynamics is observed in the interaction between the soliton and an inhomogeneity. By considering an analytical time-dependent variational approach and also full numerical simulation of one-dimensional and three-dimensional Gross–Pitaevskii equations, we study processes such as trapping, reflection and transmission of the bright matter soliton due to the impurity. We also derive conditions for the collapse of the bright solitary wave, considering a quasi-one-dimensional BEC with attractive local inhomogeneity.
A one-dimensional (1D) Gross-Pitaevskii (GP) equation which governs the dynamics of Bose-Einstein condensates (BECs) with time-dependent scattering length in a harmonic potential is considered. Through a modified phase imprint transformation into the macroscopic order parameter, we suitably generate a derivative cubic-quintic Gross-Pitaevskii (DCQ-GP) equation that describes the dynamics of condensates in the harmonic potential for both twoand three-body interatomic interactions in the presence of nonlinear dispersion. Employing a modified lens-type transformation, we reduce the DCQ-GP equation into a derivative cubic-quintic Ginzburg-Landau (DCQ-GL) equation that allows us to investigate the modulational instability (MI) of the system and study, through the analytical exact solutions of the considered 1D GP equation, the effects of the three-body interactions and nonlinear dispersion on the dynamics of the BECs. We observe that for BECs with repulsive two-body interatomic interactions, the attractive three-body interactions and the nonlinear dispersion have opposite effects on the condensate: (i) The attractive three-body interactions enhance the instability, while the nonlinear dispersion softens the instability of the system; (ii) the attractive three-body interactions contributes to the ejections of atoms from the condensates while the nonlinear dispersion contributes to the accumulation of atoms in the condensates; (iii) the attractive three-body interactions contributes to the broadening of the matter wave solitons, while the nonlinear dispersion contributes to the compression of the matter wave solitons.