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

Abstract

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 Nth 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)=ωR̃∕ωZ, where t is the scaled time, ωR̃ and ωZ 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.