Dynamics of analytical three-dimensional solutions in Bose-Einstein condensates with time-dependent gain and potential

Using the F-expansion method we systematically present exact solutions of the three-dimensional nonlinear generalized Gross-Pitaevskii equation, with time-varying gain or loss, in both attractive and expulsive harmonic confinement regimes. This approach allows us to obtain solitons for a large variety of solutions depending on the time-varying potential and the gain or loss profiles. The dynamics of these matter waves, including quasibreathing solitons, double-quasibreathing solitons, and three-quasibreathing solitons, is discussed. The explicit functions that describe the evolution of the amplitude, width, and trajectory of the soliton's wave center are presented exactly. It is demonstrated that an arbitrary additional time-dependent gain function can be added to the model to control the amplitude and width of the soliton and the nonlinearity without affecting the motion of the solitons' wave center. Additionally, a number of exact traveling waves, including the Faraday pattern formation, have been found. The obtained results may raise the possibility of relative experiments and potential applications.