Analytical three-dimensional bright solitons and soliton pairs in Bose-Einstein condensates with time-space modulation

We provide analytical three-dimensional bright multisoliton solutions to the $(3+1)$-dimensional Gross-Pitaevskii equation with time- and space-dependent potential, time-dependent nonlinearity, and gain or loss. The zigzag propagation trace and the breathing behavior of solitons are observed. Different shapes of bright solitons and fascinating interactions between two solitons can be achieved with different parameters. The obtained results may raise the possibility of relative experiments and potential applications.