Studies on the rogue waves of a (3+1)-dimensional Gross-Pitaevskii equation in the Bose-Einstein condensation

On the basis of the application in the Bose-Einstein condensation, we investigate a (3 + 1)-dimensional Gross-Pitaevskii equation with distributed time-dependent coefficients. With the aid of the Kadomtsev-Petviashvili hierarchy reduction method, we construct the Nth-order rogue-wave solutions in terms of the Gram determinant by introducing appropriate constraints. Using different coefficients for the diffraction beta(t) and gain/loss gamma(t), we demonstrate the behaviors of the first- and second-order rogue waves by analytical and graphical means. We find that only if gamma(t) = 3 beta(t), the rogue waves appear on the constant backgrounds; otherwise, the heights of the backgrounds change as time goes on. With the different choices of beta(t) and gamma(t), the long-live, rapid-reducing and periodic rogue waves are discussed. The separated and aggregated second-order rogue waves are also shown on the constant and periodical backgrounds.