Integrable Dynamics of Knotted Vortex Filaments

9705005 Calini这个项目关注的是完全可积的偏微分方程和由涡丝演化模型产生的结闭曲线的拓扑性质之间的联系。首席研究员计划表明，在这种情况下，通过标准代表可以有效地解决有关结类型，打结和解结机制，结形成稳定性和结分类的问题。将使用的两个主要工具是具有周期边界条件的相关可积方程（其中包括连续海森堡模型，聚焦非线性薛定谔和正弦-戈登方程）的周期理论，以及Backlund变换理论。PI将使用多相解的构造方法来生成大类别的结代表，并将从理论上和计算上研究其结类型与相关Floquet谱之间的精确关系。Backlund变换将用于研究拓扑变化的可能机制，构建表示不同结类型之间转换的自相交曲线，并产生由具有特殊性质的曲线（如恒定扭转曲线）实现的结的示例。在物理世界的许多现象中，都存在着以打结和连接环的形式出现的复杂结构：龙卷风、等离子体环和恒星大气中的磁拱显示出复杂的漩涡形成；线粒体DNA是一种盘绕的、经常打结的分子；细菌链被发现形成缠结的环；在某些稳定的化学介质混合物中可以观察到连接和结。本项目关注的是结构已被充分理解的进化方程与涡丝动力学和DNA建模研究中出现的结环特性之间的联系。这项研究的主要目标是开发必要的数学工具，以有效地解决有关打结和解结机制的问题，结形成的稳定性以及结和链接的分类。这类问题在许多应用领域正变得越来越重要：例如，了解太阳冕中复杂的涡旋结构可以提供有关太阳磁场活动如何影响地球气候的信息，而拓扑变化，如环路的产生、打结和解结似乎是DNA分子复制机制的核心，这是癌症研究的一个基本问题。

9705005 Calini This project concerns the connection between completely integrable partial differential equations and the topological properties of knotted closed curves which arise from models of vortex filament evolution. The principal investigator plans to show that questions regarding knot types, mechanisms for knotting and unknotting, stability of knot formations and classification of knots by means of standard representatives can be effectively addressed in this context. The two main tools that will be used are the periodic theory of relevant integrable equations (among which are the Continuous Heisenberg Model, the focusing Nonlinear Schroedinger and the sine- Gordon equations) with periodic boundary conditions, and the theory of Backlund transformations. The PI will use constructive methods for multiphase solutions to generate large classes of knot representatives and will study a precise relation between their knot type and the associated Floquet spectrum both theoretically and computationally. Backlund transformations will be used to investigate possible mechanisms for topological changes, to construct self-intersecting curves that represent transitions between different knot types and to produce examples of knots that are realized by curves with special properties (such as curves of constant torsion). Complex structures in the form of knotted and linked loops are present in many phenomena of the physical world: tornadoes, plasma loops and magnetic arches in stellar atmospheres display complicated vortex formations; mitochondrial DNA is a coiled, often knotted, molecule; bacteria strands are found to form tangled loops; links and knots are observed in certain stable mixtures of chemical media. This project concerns the connection between evolution equations whose structure is well-understood and the properties of knotted loops which arise in the study of vortex filament dynamics and DNA modeling. The principal goal of this inve stigation is the development of the mathematical tools necessary to effectively address questions regarding the mechanisms for knotting and unknotting, the stability of knot formations and the classification of knots and links. Such issues are gaining great importance in a number of applied fields: for example, an understanding of the complex vortex structures in the solar crown can provide information on how the sun's magnetic activity affects the earth's climate, while topological changes such as loop creation, knotting and unknotting appear to be at the heart of the replication mechanism of the DNA molecule, a fundamental question in cancer research.