椭圆余弦势阱中三-四次非线性薛定谔方程的求解
On Solving Cubic-Quartic Nonlinear Schrödinger Equation in a Cnoidal Trap
摘要：
最近对超冷原子气体中量子液滴的观测为基础研究开辟了新的途径。在超冷的稀碱气体中，平均场和超平均场相互作用之间的竞争被认为是有助于稳定液滴的。这些新的认识促使我们去研究一个被囚禁的四次非线性薛定谔方程（CQNLSE）的解析解。NLSE中的四次贡献由玻色-爱因斯坦凝聚体（BEC）的超平均场形式导出。据我们所知，CQNLSE的全面分析描述是不存在的。在这里，我们研究CQNLSE解析解的存在性，这是椭圆余弦型。外捕集对系统的稳定起着重要作用。在极限情况下，椭圆余弦波解导致亮解的定域解和离域扭折-反扭折对。在我们的分析中还揭示了在目前的计划正弦模式的不存在。
英文摘要：
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, are believed to be instrumental in stabilizing the droplets. These 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.
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