This work deals with a class of square matrix nonlinear Schrödinger (MNLS) systems whose reductions include two equations that model hyperfine spin \(F = 1\) spinor Bose–Einstein condensates in the focusing and defocusing dispersion regimes, and two novel (mixed sign) equations that were recently shown to be integrable. Our main goal is to discuss the bright soliton solutions and their interactions for the focusing MNLS and for the two mixed sign systems within the framework of the inverse scattering transform. The nature of the solitons and their interactions depend on whether the associated norming constants (polarization matrices) are rank-one matrices (giving rise to ferromagnetic solitons) or full rank (corresponding to polar solitons). By computing the long-time asymptotics of the 2-soliton solutions, we determine how the polarization matrix of each soliton changes because of the interaction. Explicit formulas for the soliton interactions are given for all possible types of interacting solitons, namely ferromagnetic–ferromagnetic, polar–polar, and polar–ferromagnetic soliton interactions, and for all three inequivalent reductions of the MNLS systems that admit regular bright soliton solutions. We also present bound states, representing 2 solitons travelling with the same velocity, for all three systems.

中文翻译：

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某些方阵非线性Schrödinger系统中的孤子相互作用

这项工作涉及一类方阵非线性Schrödinger（MNLS）系统，其减少量包括两个模型，这些方程对超精细自旋\（F = 1 \）进行建模旋量玻色－爱因斯坦在聚焦和散焦色散状态下凝结，两个新的（混合符号）方程最近被证明是可积分的。我们的主要目标是讨论逆散射变换框架内聚焦MNLS和两个混合符号系统的亮孤子解及其相互作用。孤子的性质及其相互作用取决于相关的规范常数（极化矩阵）是秩一矩阵（产生铁磁孤子）还是满秩（对应于极性孤子）。通过计算2孤子解的长时间渐近性，我们确定了每个孤子的极化矩阵由于相互作用而如何变化。针对所有可能的相互作用孤子类型，给出了孤子相互作用的显式公式，分别是铁磁-铁磁，极性-极性和极性-铁磁孤子相互作用，以及对于允许规则的明亮孤子解的MNLS系统的所有三个不等价减少。我们还给出了所有三个系统的束缚态，表示两个孤子以相同的速度传播。