Rogue waves for a (2+1)-dimensional Gross–Pitaevskii equation with time-varying trapping potential in the Bose–Einstein condensate

The Bose–Einstein condensates (BECs) are seen during the studies in atomic optics, cavity opto-mechanics, cavity quantum electrodynamics, black-hole astrophysics, laser optics and atomtronics. Under investigation in this paper is a (2+1)-dimensional Gross–Pitaevskii equation with time-varying trapping potential which describes the dynamics of a (2+1)-dimensional BEC. Based on the Kadomtsev–Petviashvili hierarchy reduction, we construct the bilinear forms and the N th order rogue-wave solutions in terms of the Gramian. With the help of the analytic and graphic analysis, we exhibit the first- and second-order rogue waves under the influence of the strength of the interatomic interaction, α ( t ) , and of Ω ( t ) = ω R ∕ ω Z , where t is the scaled time, ω R and ω Z are the confinement frequencies in the radial and axial directions: When Ω ( t ) = 0 , the first-order rogue wave exhibits as an eye-shaped distribution; When Ω ( t ) is a periodic function, the rogue wave periodically raises; When Ω ( t ) is an exponential function, the rogue wave appears on the exponentially increasing background; When α ( t ) decreases, background and amplitude of the rogue wave both increase. The second-order rogue waves reach the maxima only once or three times when Ω ( t ) is a constant. With Ω ( t ) being the exponential function, the backgrounds of the second-order rogue waves exponentially increase with t increasing.