Compact bright pulse in inhomogeneous and nonlinear medium: Case of the Bose–Einstein Condensate

Abstract

The extended one-dimensional Gross–Pitaevskii equation which describes the dynamics of localized wave in nonlocal nonlinear media, in particular the Bose–Einstein condensates (BECs) with time-dependent interatomic interactions in a parabolic potential in the presence of feeding or loss of atoms, is investigated. Through analytical methods invoking a modified lens-type transformation, an equivalent nonlocal GP equation with constant coefficients is derived as well as the integrability conditions. In the limit of zero transverse kinetic energy, we show that the nonlocal GP equation exhibits a stationary bright pulse with strictly localized support and with the width independent of the amplitude. However due to the property of conservation of the norm, the BEC amplitude will be a function of both the pulse width and the number of trap atoms. Similarly, the obtained integrability conditions also appear as the conditions under which the compact bright waves describing the BECs can be managed by controlling the parameters of the external potential. Thus, the number of trap atoms can be managed through the functional gain or loss which is not sensitive to the strength of the magnetic trap. It also appears that, in the presence of the nonlocal interaction, the shape of the BEC interpolates between the shape of the compact bright pulse and that of the NLS bright pulse, and the number of condensed atoms participating to the formation of the BEC increases with the strength of the nonlocal interactions.