标题： 到一维离散非多项式薛定谔方程的两条路径 显示英文标题

标题： Two routes to the one-dimensional discrete nonpolynomial Schrödinger equation

摘要： 在由香肠形陷阱和轴向光晶格组合限制的玻色-爱因斯坦凝聚体（BEC）在两个模型的框架下被研究，这两个模型由一维（1D）离散非多项式薛定谔方程（NPSE）的两个版本描述。 这两个模型都来源于三维 Gross-Pitaevskii 方程（3D GPE）。 为了生成“模型 1”（该模型在最近的研究中被推导出），首先将 3D GPE 约化为一维连续 NPSE，随后对其进行离散化。 “模型 2”之前未被考虑，其推导方法是首先对 3D GPE 进行离散化，然后进行维度约化。 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 包含一个额外的方程用于横向宽度。 然而，数值分析显示两种系统中基本非交错孤子的行为相似，包括它们的存在区域和稳定性极限。 两个模型都允许局域模式的坍缩，再现了在紧密陷阱中自吸引 BEC 的基本特性。 因此，我们得出结论，对于强束缚自吸引 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这些特性。 然而，也发现了模型之间的差异，即先前在模型 1 中发现的强烈钉扎（非常狭窄）的离散孤子并未被模型 2 生成——事实上，这与连续的一维 NPSE 一致，后者也没有这样的解。 在这方面，新推导的模型为束缚 BEC 提供了更精确的近似。 显示英文摘要

摘要： 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 Schr\"odinger 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", that was not considered before, is derived by first discretizing the 3D GPE, which is followed by the reduction of 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.