In this paper,we analyze Gross-Pitaevskii equation which describes the dynamics of a bright soliton in trapped atomic Bose-Einstein condensates,and obtain the exact bright soliton solution of Gross-Pitaevskii equation in spherically-symmetric non-harmonic trap.
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.