Solitons in Bose-Einstein Condensates with time-dependent atomic   scattering length in an expulsive parabolic and complex potential
  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. 
Biao Li, Xiao-Fei Zhang, Yu-Qi Li, Yong Chen, W. M. Liu