AI 中文总结
本文通过非标准Hirota双线性方法、行列式公式和左唯一性策略，对自旋-1玻色-爱因斯坦凝聚体在实数域上的解进行了首次完全分类，给出了新的显式解，并为自旋布居、相对相位和磁织构提供了实验可检验的预测。
AI 中文摘要
自旋-1玻色-爱因斯坦凝聚体为探索量子磁性和相干自旋动力学提供了基础平台。在粒子数和磁化强度守恒的条件下，其三个塞曼分量通过自旋交换相互作用耦合，从而产生丰富的磁相和非线性自旋结构；因此，对稳态进行严格的分类有助于阐明可能的自旋构型及其物理起源。本文作为我们先前工作\cite{LLWZ23,LLWZ24,LLWZ26}的延续，通过应用非标准Hirota双线性方法、行列式公式和一般左唯一性策略，研究了自旋-1玻色-爱因斯坦凝聚体在$\mathbb{R}$中解的分类，我们给出了一些新的显式解，并提供了解的完全分类。与经典Hirota方法相比，非标准Hirota方法需要引入辅助函数来构造Hirota双线性形式，这本质上对波函数增加了额外的约束，从而能够处理更复杂的方程。据我们所知，这是首次对自旋-1玻色-爱因斯坦凝聚体解的完全分类进行系统研究。我们的分类结果为自旋布居、相对相位和磁织构提供了实验可检验的预测。
英文摘要
Spin-1 Bose-Einstein condensates provide a fundamental platform for exploring quantum magnetism and coherent spin dynamics. Their three Zeeman components are coupled by spin-exchange interactions under conserved particle number and magnetization, giving rise to rich magnetic phases and nonlinear spin structures; a rigorous classification of stationary states therefore clarifies the possible spin configurations and their physical origins. In this paper, as a continuation of our previous work \cite{LLWZ23,LLWZ24,LLWZ26}, we study the classification of solution for the spin-1 Bose-Einstein condensates in $\mathbb{R}$, by applying the non-standard Hirota's bilinear method, the determinant formula and the general left-uniqueness strategy, we present some novel explicit solution and give the complete classification of the solutions. In comparison with the classical Hirota's method, the non-standard Hirota's method needs to introduce auxiliary functions to construct Hirota's bilinear form, which essentially adds additional constraint on the wave functions, so that it possible to deal with more complex equations. As far as we know, this is the first systematic study the complete classification of solutions to the spin-1 Bose-Einstein condensates. Our classification results yields experimentally testable predictions for spin populations, relative phases, and magnetic textures
Comments42 pages
