RUI: Stability analysis for soliton solutions of the Vortex Filament Equation and beyond

该项目的主要目标是研究涡旋细丝(涡度集中的细长细丝)的动力学。研究的重点是描述涡丝运动的涡丝方程(VFE)，假设涡丝在不可压缩的无粘流体中的运动是由其自身的诱导引起的。VFE通过Hasimoto映射与非线性薛定谔方程相联系。该项目将利用这一关系来研究VFE的特解(Hasimoto孤子)的稳定性，并将分析扩展到包含了最初被理想化的VFE模型忽略的物理效应的模型。Hasimoto孤子的形式是沿着涡旋丝传播的局域环路。在实验室和数值实验中都观察到了湍流中涡旋的孤立波。虽然最常见的涉及涡旋细丝的现象是水龙卷和龙卷风，但对涡旋细丝的研究在科学的其他方面有着引人入胜的应用，例如在超流和超导方面。在具体的应用中，Hasimoto孤子出现在例如超导、龙卷风、孤子在流体中的粒子传输和激光-物质相互作用的背景下。尽管文献中对它们进行了大量的研究，但对它们的稳定性知之甚少。这个项目的主要目的是研究解的稳定性，这是至关重要的，因为只有稳定的解才能在实验中实现和观察。该项目涉及本科生和研究生的广泛参与。在做研究的过程中，学生将学习进行这项研究所需的复杂的数值和数学方法。

The principal objective of the project is the study of dynamics of vortex filaments (slender filaments where vorticity is concentrated). The research is focused on the vortex filament equation (VFE) that describes the motion of the vortex filament, assuming that the motion of a vortex filament in an incompressible, inviscid fluid is due to its own induction. The VFE is related to the nonlinear Schroedinger equation via a Hasimoto map. The project will take advantage of this relation to investigate stability of special solutions (Hasimoto solitons) for VFE, and to extend the analysis to the models that incorporate physical effects initially neglected by the idealized VFE model. Hasimoto solitons take the form of localized loops traveling along the vortex filament. Solitary waves of vortices in turbulent fluids have been observed both in laboratory and numerical experiments. While the most commonly encountered phenomena involving vortex filaments are waterspouts and tornadoes, the study of vortex filaments has fascinating applications to other aspects of science such as in superfluidity and superconductivity. In specific applications, Hasimoto solitons occur, for example, in the context of superconductivity, of tornadoes, of particle transport in a fluid by solitons, and of laser-matter interaction. Although they have been studied abundantly in the literature not much is known about their stability. The main purpose of this project is studying solutions' stability properties, which is of fundamental importance, since only stable solutions can be realized and observed experimentally. The project involves a broad participation of undergraduate and graduate students. In the process of doing research the students will learn sophisticated numerical and mathematical methods needed to carry this research through.