Controllable nonautonomous localized waves and dynamics for a quasi-1D Gross–Pitaevskii equation in Bose–Einstein condensations with attractive interaction

This paper investigates dynamical behaviors and controllability of some nonautonomous localized waves based on the Gross–Pitaevskii equation with attractive interatomic interactions. Our approach is a relation constructed between the Gross–Pitaevskii equation and the standard nonlinear Schrödinger equation through a new self-similarity transformation which is to convert the exact solutions of the latter to the former’s. Subsequently, one can obtain the nonautonomous breather solutions and higher-order rogue wave solutions of the Gross–Pitaevskii equation. It has been shown that the nonautonomous localized waves can be controlled by the parameters within the self-similarity transformation, rather than relying solely on the nonlinear intensity, spectral parameters, and external potential. The control mechanism can induce an unusual number of loosely bound higher-order rogue waves. The asymptotic analysis of unusual loosely bound rogue waves shows that their essence is energy transfer among rogue waves. Numerical simulations test the dynamical stability of obtained localized wave solutions, which indicate that modifying the parameters in the self-similarity transformation can improve the stability of unstable localized waves and prolong their lifespan. We numerically confirm that the rogue wave controlled by the self-similarity transformation can be reproduced from a chaotic initial background field, hence anticipating the feasibility of its experimental observation, and propose an experimental method for observing these phenomena in Bose–Einstein condensates. The method presented in this paper can help to induce and observe new stable localized waves in some physical systems.