Abstract

The recent observations of quantum droplet in ultra-cold atomic gases have opened up new avenues of fundamental research. The competition between mean-field and beyond mean-field interactions, in ultra-cold dilute alkali gases, is believed to be instrumental in stabilizing the droplets. This new understanding has motivated us to investigate the analytical solutions of a trapped cubic-quartic nonlinear Schrödinger equation (CQNLSE). The quartic contribution in the NLSE is derived from the beyond mean-field formalism of Bose–Einstein condensate (BEC). To the best of our knowledge, a comprehensive analytical description of CQNLSE is non-existent. Here, we study the existence of the analytical solutions which are of the cnoidal type for CQNLSE. The external trapping plays a significant role in the stabilization of the system. In the limiting case, the cnoidal wave solutions lead to the localized solution of bright solution and delocalized kink-antikink pair. The nonexistence of the sinusoidal mode in the current scheme is also revealed in our analysis.

Graphical abstract

中文翻译：

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关于解决正弦陷阱中三次三次非线性薛定ding方程的问题

摘要

最近对超冷原子气体中的量子滴的观察为基础研究开辟了新途径。在超冷的稀碱气体中，平均场相互作用和超出平均场相互作用之间的竞争被认为有助于稳定液滴。这种新的认识激发了我们去研究一个被困的三次四方非线性薛定ding方程（CQNLSE）的解析解。NLSE中的四次贡献来自Bose-Einstein凝结水（BEC）的超出平均场形式主义。据我们所知，不存在对CQNLSE的全面分析描述。在这里，我们研究CQNLSE的正弦类型的解析解的存在。外部陷阱在系统稳定中起着重要作用。在极限情况下，环形波解导致亮解和离结扭结-反扭结对的局部解。我们的分析还揭示了当前方案中不存在正弦波模式。

图形概要