  We present two families of analytical solutions of the one-dimensional
nonlinear Schr\"{o}dinger equation which describe the dynamics of bright and
dark solitons in Bose-Einstein condensates (BECs) with the time-dependent
interatomic interaction in an expulsive parabolic and complex potential. We
also demonstrate that the lifetime of both a bright soliton and a dark soliton
in BECs can be extended by reducing both the ratio of the axial oscillation
frequency to radial oscillation frequency and the loss of atoms. It is
interested that a train of bright solitons may be excited with a strong enough
background. An experimental protocol is further designed for observing this
phenomenon.
