Higher-Order Modes of Modulation Instability in Bose–Einstein Condensates with a Time-Dependent Three-Dimensional Parabolic Potential

摘要

By the similarity reduction and Darboux transformation, we derive higher-order modes of three-dimensional Bose– Einstein condensate modulation instability in the nonautonomous Gross–Pitaevskii equation and manipulate them by regulating the time-dependent potential and gain. Firstly, by the similarity reduction, the (3+1)-dimensional nonautonomous Gross–Pitaevskii equation reduces to a (1+1)-dimensional standard nonlinear Schr?dinger equation with constant coefficients. Then, considering the Akhmediev breather solution as the first-order modulation instability solution of the higher-order modes of Bose–Einstein condensate modulation instability, we achieve the Nth-order (N = 2, 3, 4, and 5) modulation instability solutions by the Darboux transformation. Finally, we verify the stable higherorder modes of Bose–Einstein condensate modulation instability and manipulate them by direct numerical simulation. The obtained results may raise the possibility of related experiments and potential applications in Bose–Einstein condensates and other related fields.