Multiple Higher-Order Pole Solutions in Spinor Bose–Einstein Condensates

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2024-04-01 DOI:10.1007/s00332-024-10024-8

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Abstract

In this study, multiple higher-order pole solutions of spinor Bose–Einstein condensates are explored by means of the inverse scattering transform, which are associated with different higher-order pole pairs of the transmission coefficient and give solutions to the spin-1 Gross–Pitaevskii equation. First, a direct scattering problem is introduced to map the initial data to the scattering data, which includes discrete spectrums, reflection coefficients, and a polynomial that replaces the normalized constants. In order to analyze symmetries and discrete spectra in the direct scattering problem, a generalized cross product is defined in four-dimensional vector Space. The inverse scattering problem is then characterized in terms of the \(4\times 4\) matrix Riemann–Hilbert problem that is subject to the residual conditions of these higher-order poles. In the reflectionless case, the Riemann–Hilbert problem can be converted into a linear algebraic system, which has a unique solution and allows us to explicitly obtain multiple higher-order pole solutions to the spin-1 Gross–Pitaevskii equation.

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旋子玻色-爱因斯坦凝聚态中的多重高阶极解

摘要本研究通过反散射变换探讨了自旋玻色-爱因斯坦凝聚体的多个高阶极点解，这些解与透射系数的不同高阶极点对相关联，并给出了自旋-1 格罗斯-皮塔耶夫斯基方程的解。首先，引入直接散射问题，将初始数据映射到散射数据，其中包括离散谱、反射系数和替代归一化常数的多项式。为了分析直接散射问题中的对称性和离散光谱，在四维向量空间中定义了广义交叉积。然后，反向散射问题用 \(4\times 4\) 矩阵黎曼-希尔伯特问题来描述，该问题受制于这些高阶极点的残差条件。在无反射情况下，黎曼-希尔伯特问题可以转换成一个线性代数系统，它有一个唯一的解，并允许我们明确地得到自旋-1 格罗斯-皮塔耶夫斯基方程的多个高阶极点解。

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