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Communications in Theoretical Physics

Dynamics of Analytical Matter-Wave Solutions in Three-Dimensional Bose—Einstein Condensates with Two- and Three-Body Interactions

Hai-Qin Jin, Jun-Rong He, Jian-Chu Liang, Zebin Cai, Lin Yi

Published Feb 1, 2014• 2 citations
Publisher

Abstract

Using the F-expansion method we present analytical matter-wave solutions to Bose—Einstein condensates with two- and three-body interactions through the generalized three-dimensional Gross—Pitaevskii equation with time-dependent coefficients, for the periodically time-varying interactions and quadratic potential strength. Such solutions exist under certain conditions, and impose constraints on the functions describing potential strength, nonlinearities, and gain (loss). Various shapes of analytical matter-wave solutions which have important applications of physical interest are studied in details.

DOI
10.1088/0253-6102/61/2/11

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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.
Bose-Einstein condensates with time varying two- and three-body interatomic interactions, confined in a linear potential and exchanging atoms with the thermal cloud are investigated. Using the extended tanh-function method with an auxiliary equation, i.e., the Lenard equation, many exact solutions describing the dynamics of matter-wave condensates are derived. An important issue is the time management of the cubic and the quintic nonlinearities by tuning the rate of exchange of atoms between the condensate and the thermal background. In addition, adjusting the strength of the linear potential, the rate of exchange of atoms, and many other free parameters allow one to control many features of the condensate such as its height, width, position, velocity, acceleration, and its direction, respectively. Full numerical solutions corroborate the analytical predictions.
We present two families analytical solutions to the extended Gross-Pitaevskii equation (GPE), which describe the dynamics of bright and dark solitons in Bose-Einstein condensations(BECs) with varying coefficients of two and three-body interactions in a parabolic and complex potential. The properties of the dynamical evolutions of the bright and dark matter-wave solitons are investigated. It is interesting that the matter-wave can be controlled by applying the external magnetic field and a background.
The matter-wave solutions of Bose-Einstein condensates with three-body interaction are examined through the one-dimensional Gross-Pitaevskii equation. By using a modified lens-type transformation and a further extension of the tanh-function method we obtain the exact analytical solutions which describe the propagation of kink-shaped solitons, anti-kink-shaped solitons, and other families of solitary waves. We realize that the shape of a kink solitary wave depends on both the scattering length and the parameter of atomic exchange with the substrate. The stability of the solitary waves is examined using analytical and numerical methods. Our results can also be applied to nonlinear optics in the presence of cubic-quintic media.
We investigate the profiles and stability of the localized nonlinear matter waves in a Bose–Einstein condensate with spatially inhomogeneous two- and three-body interactions in various external potentials both analytically and numerically. Several families of localized stationary solutions of the one-dimensional mean-field Gross–Pitaevskii equation with cubic–quintic nonlinearities are obtained by similarity transformation. It is shown that the whole Bose–Einstein condensate can have an infinite number of localized nonlinear matter waves determined by certain non-negative integers. The linear and dynamical stabilities of the localized stationary solutions are studied numerically, and some stable localized nonlinear matter waves are found. We demonstrate that the attractive inhomogeneous two- and three-body interactions support linearly and dynamically stable localized nonlinear matter waves, but attractive two-body interaction along with repulsive three-body interaction destroys the linear stability. Moreover, an anharmonic potential is proposed to stabilize the localized nonlinear matter waves in the Bose–Einstein condensate with spatially inhomogeneous interactions.