Generation of matter-wave solitons of the Gross-Pitaevskii equation with a time-dependent complicated potential

We examine the generation of bright matter-wave solitons in the Gross-Pitaevskii equation describing Bose-Einstein condensates with a time-dependent complex potential, which is composed of a repulsive parabolic background potential and a gravitational field. By performing a modified lens-type transformation, an explicit expression for the growth rate of a purely growing modulational instability is presented and analyzed. We point out the effects of the gravitational field, as well as of the parameter related to the feeding or loss of atoms in the condensate, on the instability growth rate. It is evident from numerical simulations that the feeding with atoms and the magnetic trap have opposite effects on the dynamics of the system. It is shown that the feeding or loss parameter can be well used to control the instability domain. Our study shows that the gravitational field changes the condensate trail of the soliton trains during the propagation. We also perform a numerical analysis to solve the Gross-Pitaevskii equation with a time-dependent complicated potential. The numerical results on the effect of both the gravitational field and the parameter of feeding or loss of atoms in the condensate agree well with predictions of the linear stability analysis. Another result of the present work is the modification of the background wave function in the Thomas-Fermi approximation during the numerical simulations.