标题： 霍普夫孤子作为广义折叠映射的分析 显示英文标题

标题： Analysis of Hopf solitons as generalized fold maps

摘要： 霍普夫指标是一种拓扑不变量，用于量化原像纤维的链接，是霍普尼结构和稳定性的基础。 在本工作中，我们提出了一种新的数学框架，用于建模具有高霍普夫指标的霍普尼，借鉴了奇点理论的语言和可微映射的拓扑学。 我们方法的核心是一个阶数为$n$的广义霍普夫映射，其结构通过折痕映射及其史坦因分解来捕捉。 我们证明，这种理论构造不仅与最近对高霍普夫指标霍普尼的实验观察高度一致，还提供了与不同点相关的纤维对可能配置的精确分类。 因此，我们的结果建立起了奇异映射的几何学与材料及其他物理系统中霍普尼复杂场配置的实验观测拓扑学之间的稳固桥梁。 显示英文摘要

摘要： The Hopf index, a topological invariant that quantifies the linking of preimage fibers, is fundamental to the structure and stability of hopfions. In this work, we propose a new mathematical framework for modeling hopfions with high Hopf index, drawing on the language of singularity theory and the topology of differentiable maps. At the core of our approach is the notion of a generalized Hopf map of order $n$, whose structure is captured via fold maps and their Stein factorizations. We demonstrate that this theoretical construction not only aligns closely with recent experimental observations of high-Hopf-index hopfions, but also offers a precise classification of the possible configurations of fiber pairs associated to distinct points. Our results thus establish a robust bridge between the geometry of singular maps and the experimentally observed topology of complex field configurations of hopfions in materials and other physical systems.