RUI: Topology and Stability of Integrable Vortex Filament Motion

我们研究涡丝方程(VFE)的周期解，这是一个非线性偏微分方程，它模拟了理想流体中涡线的自诱导动力学。VFE与立方聚焦非线性薛定谔(NLS)方程密切相关，立方聚焦非线性薛定谔(NLS)方程是一种典型的孤子方程，是研究各种情况(从深水中的水波到非线性光学介质)中的非线性波传播的模型。我们的研究目标包括：更深入地理解与NLS方程(孤子的周期类)的有限间隙解相关的闭合涡丝的拓扑性质；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；以及研究保持纽结能量泛函的涡丝流动的微扰。我们的工作涉及流体中细丝结构动力学的正则模型(如流经空气的烟环，或由强大的水下水流产生的强涡度环)。尽管它很简单，但这个模型有一类丰富的解，实现了许多有趣的拓扑特征：我们发现了越来越复杂的打结涡环，并观察到了链交叉和自交等拓扑变化。数学的许多领域被聚集在一起，以解决诸如打结的丝状结构的拓扑性质、它们的稳定性和它们的能量学等问题。这项工作还继续将研究和教育结合在一起，为本科生提供研究经验，旨在为他们的研究生课程做好更好的准备，并通过支持研究生在以本科生为主的机构参与高级研究。

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.