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Rogue waves for a \((2+1)\)-dimensional Gross-Pitaevskii equation with time-varying trapping potential in the Bose-Einstein condensate. (English) Zbl 1443.81040
Summary: 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, \(\alpha(t)\), and of \(\Omega(t)=\omega_{\widetilde{R}}/\omega_Z\), where \(t\) is the scaled time, \(\omega_{\widetilde{R}}\) and \(\omega_Z\) are the confinement frequencies in the radial and axial directions: When \(\Omega(t)=0\), the first-order rogue wave exhibits as an eye-shaped distribution; When \(\Omega(t)\) is a periodic function, the rogue wave periodically raises; When \(\Omega(t)\) is an exponential function, the rogue wave appears on the exponentially increasing background; When \(\alpha(t)\) decreases, background and amplitude of the rogue wave both increase. The second-order rogue waves reach the maxima only once or three times when \(\Omega(t)\) is a constant. With \(\Omega(t)\) being the exponential function, the backgrounds of the second-order rogue waves exponentially increase with \(t\) increasing.
81Q80 Special quantum systems, such as solvable systems
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q82 PDEs in connection with statistical mechanics
78A60 Lasers, masers, optical bistability, nonlinear optics
Full Text: DOI
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