RUI: Topology and Stability of Integrable Vortex Filament Motion

我们研究了涡丝方程（VFE）的周期解，这是一个模拟理想流体中涡线自诱导动力学的非线性偏微分方程。VFE与三次聚焦非线性薛定谔（NLS）方程密切相关，后者是一种典型孤子方程，是各种情况下（从深水水波到非线性光学介质）非线性波传播的模型。我们的研究有几个目标，包括：更深入地了解与NLS方程（周期孤子类似物）有限间隙解类相关的闭涡细丝的拓扑性质；将周期NLS解的线性稳定性与相应涡丝的线性稳定性联系起来；并研究了保持结能泛函的涡旋细丝流的扰动。我们的工作涉及流体中丝状结构动力学的规范模型（例如在空气中传播的烟圈，或由强大的水下电流产生的强烈涡度回路）。尽管它很简单，但这个模型有丰富的解类，实现了许多有趣的拓扑特征：我们发现了越来越复杂的结涡环，并观察到拓扑变化，如链交叉和自交。许多数学领域汇集在一起，以解决诸如打结丝状结构的拓扑性质、稳定性和能量学等问题。这项工作还将继续整合研究和教育，为本科生提供旨在更好地为研究生课程做准备的研究经验，并支持研究生在主要的本科院校参与高级研究。

We study periodic solutions of the Vortex Filament Equation (VFE), a nonlinear partial differential equation that models the self-induced dynamics of a vortex line in an ideal fluid. The VFE is closely related to the cubic focusing Nonlinear Schroedinger (NLS) equation, a canonical soliton equation arising as model of nonlinear wave propagation in a variety of situations (from water waves in deep water to nonlinear optical media). Our research has several goals including: a deeper understanding of the topological properties of closed vortex filaments associated to the class of finite-gap solutions of the NLS equation (the periodic analogues of solitons); relating the linear stability of periodic NLS solutions to the linear stability of the corresponding vortex filaments; and studying perturbations of the vortex filament flow that preserve knot energy functionals.Our work concerns a canonical model of the dynamics of filamentary structures in fluids (such as smoke rings traveling through the air, or loops of intense vorticity generated by a strong underwater current). Despite its simplicity, this model has a rich class of solutions that realize many interesting topological features: we find knotted vortex loops of increasing complexity, and observe topological changes such as strand crossing and self-intersections. Many areas of mathematics are brought together to address questions such as the topological properties of knotted filamentary structures, their stability, and their energetics. This work also continues to integrate research and education, by providing undergraduate students with research experiences aimed at better preparing them for graduate programs, and by supporting graduate students involved in advanced research at a primarily undergraduate institution.