Vinca Institute of Nuclear Sciences, PO Box 522, 11001 Belgrade, Serbia.
Chaos. 2009 Dec;19(4):043105. doi: 10.1063/1.3248269.
The Bose-Einstein condensate (BEC), confined in a combination of the cigar-shaped trap and axial optical lattice, is studied in the framework of two models described by two versions of the one-dimensional (1D) discrete nonpolynomial Schrodinger equation (NPSE). Both models are derived from the three-dimensional Gross-Pitaevskii equation (3D GPE). To produce "model 1" (which was derived in recent works), the 3D GPE is first reduced to the 1D continual NPSE, which is subsequently discretized. "Model 2," which was not considered before, is derived by first discretizing the 3D GPE, which is followed by the reduction in the dimension. The two models seem very different; in particular, model 1 is represented by a single discrete equation for the 1D wave function, while model 2 includes an additional equation for the transverse width. Nevertheless, numerical analyses show similar behaviors of fundamental unstaggered solitons in both systems, as concerns their existence region and stability limits. Both models admit the collapse of the localized modes, reproducing the fundamental property of the self-attractive BEC confined in tight traps. Thus, we conclude that the fundamental properties of discrete solitons predicted for the strongly trapped self-attracting BEC are reliable, as the two distinct models produce them in a nearly identical form. However, a difference between the models is found too, as strongly pinned (very narrow) discrete solitons, which were previously found in model 1, are not generated by model 2-in fact, in agreement with the continual 1D NPSE, which does not have such solutions either. In that respect, the newly derived model provides for a more accurate approximation for the trapped BEC.
玻色-爱因斯坦凝聚体（BEC）被限制在雪茄形状的陷阱和轴向光晶格的组合中，在两个版本的一维（1D）离散非线性薛定谔方程（NPSE）描述的框架中进行研究。这两个模型都源自三维 Gross-Pitaevskii 方程（3D GPE）。为了产生“模型 1”（最近的工作中推导出的），首先将 3D GPE 简化为 1D 连续 NPSE，然后对其进行离散化。“模型 2”以前没有考虑过，它是通过首先对 3D GPE 进行离散化，然后在维度上进行降维得到的。这两个模型看起来非常不同；特别是，模型 1 由一个用于一维波函数的离散方程表示，而模型 2 包括一个用于横向宽度的附加方程。然而，数值分析表明，在这两个系统中，基本非交错孤子的行为非常相似，就其存在区域和稳定性极限而言。这两个模型都允许局域模式的崩塌，再现了被限制在紧密陷阱中的自吸引 BEC 的基本特性。因此，我们得出结论，对于被强限制的自吸引 BEC 预测的离散孤子的基本特性是可靠的，因为这两个不同的模型以几乎相同的形式产生它们。然而，我们也发现了模型之间的差异，因为以前在模型 1 中发现的强固定（非常窄）离散孤子，在模型 2 中不会产生——事实上，与没有此类解决方案的连续 1D NPSE 一致。在这方面，新推导出的模型为被限制的 BEC 提供了更精确的近似。
