Integrable Dynamics of Knotted Vortex Filaments

小行星9705005 这个项目涉及完全可积的部分 微分方程和纽结闭的拓扑性质 这些曲线来自涡丝演化模型。校长 研究者计划表明，关于线结类型、机制 用于打结和解开，结形成的稳定性和分类 可以有效地解决通过标准代表的结 在这方面。将使用的两个主要工具是周期理论 相关的可积方程（其中包括连续 Heisenberg模型、聚焦非线性薛定谔方程和正弦-戈登方程），以及 贝克兰德变换。PI将使用建设性方法， 多相解决方案，以生成大类结代表， 将研究它们的结类型和相关的 Floquet频谱的理论和计算。贝克兰德 转换将被用来调查可能的机制， 拓扑变化，以构建表示 不同结类型之间的过渡，并生成结的示例 由具有特殊属性的曲线（例如 恒定扭转）。 结和连接环形式的复杂结构存在于 物理世界的许多现象：龙卷风，等离子体环， 恒星大气中的磁拱显示出复杂的涡旋 线粒体DNA是一种卷曲的，经常打结的分子; 发现细菌链形成缠结的环;链接和结， 在某些稳定的化学介质混合物中观察到。这个项目 关于演化方程之间的联系，其结构是 以及研究中出现的打结环的性质 涡丝动力学和DNA建模的研究其主要目标是 研究是开发必要的数学工具， 有效解决打结机制的问题， 解结、结形成的稳定性和结的分类 和链接。这些问题在许多应用领域中越来越重要。 领域：例如，了解复杂的旋涡结构， 太阳冕可以提供太阳磁场活动的信息 影响地球的气候，而拓扑变化，如环的创建， 打结和解开似乎是复制的核心 DNA分子的作用机制，癌症的基本问题 research.

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.