The Geometric Background of biHamiltonian Systems

项目负责人：Gloria mari - beffa在本项目中，PI提出研究非线性双哈密顿偏微分方程的几何实现的存在性，以及这种存在性对双哈密顿系统和实现流形的意义。她特别打算描述可能实现的流形，并将它们的几何形状与它们实现的系统类型联系起来。在过去，PI研究了在实现流形时如何通过曲线几何生成双哈密顿结构。她还研究了KdV型方程在厄米对称流形中的几何实现，将微分不变量的射影特征与KdV型方程联系起来。在这个项目中，她建议深化这种关系，并将其扩展为抛物流形。在与伊斯特伍德的合作中，她将使用经典微分几何中的工具来寻找流动中的投影结构。解决这个微分几何问题将很可能创造出kdv型抛物流形的几何实现。她还将研究其他几何中类似联系的可能性，例如ofSchrödinger， mKdV和正弦戈登流到黎曼几何。最后，在与Calini和Ivey的联合工作中，pi将研究孤立子方程解的几何和拓扑性质，特别是与有限间隙解（周期情况）相对应的解。这项研究将试图把这些溶液的性质与其光谱参数联系起来。设置合适的几何背景通常是解决问题的基本步骤。我们希望保持不变的特性和规律是由几何结构决定的：相对论（用洛伦兹几何设置空间和时间之间的相互作用）和计算机成像（在需要保留3D透视时使用射影几何）是最著名的例子。但现在许多工程师和物理学家，包括一些从事数据收集的团体，认为微分几何的基本知识是基础。几何思维是司空见惯的，因为通常为问题找到正确的几何选择是解决问题的第一步——这完全取决于你如何（或用哪种几何眼睛）看待它。双哈密顿非线性方程结构丰富，常用于模拟不同类型的现象。它们丰富的结构使我们能够找到大量关于它们建模和预测行为的系统的信息。最著名的完全可积系统是双哈密顿系统，它们的解预测了流体的行为，从浅水中的水波到飞机翼尖后的尾涡。原则上，这些现象不存在于任何给定的几何背景中。当我们找到一个完全可积系统的几何实现时，我们通过两种不同的方式获得信息：1)我们知道这些现象的行为可以在特定的几何中可视化，我们学习如何可视化它。特别地，同样的现象可以用不止一种几何来描述；2)我们了解到，一些几何图形是以前不知道存在的现象的宿主。例如，通过将射影几何和中心等仿射几何联系起来，我们可以发现表现得像孤立波的星形曲线的演变。我们对这两个显然不相关的学科之间的关系理解得越好，我们就越能将我们广泛的几何知识，它们的联系和性质，转移到对完全可积系统的理解上。反之亦然。

AbstractAward: DMS-0804541Principal Investigator: Gloria Mari-BeffaIn this project the PI proposes to research the existence ofgeometric realizations for nonlinear biHamiltonian PDEs and theimplications of this existence for the biHamiltonian system andfor the realizing manifold alike. In particular she intends todescribe possible realizing manifolds and to link their geometryto the type of system they realize. In the past the PI hasstudied how biHamiltonian structures are generated by thegeometry of curves in realizing manifolds. She also studiedgeometric realizations of equations of KdV type in HermitianSymmetric manifolds, linking the projective character ofdifferential invariants to equations of KdV-type. In this projectshe proposes to deepen this relation and to expand it toparabolic manifolds. In joint work with M. Eastwood she will beusing tools in classical differential geometry to find projectivestructures on flows. Resolving this differential geometry problemwill very likely create geometric realizations of KdV-type inparabolic manifolds. She will also investigate the possibility ofa similar connection in other geometries, for example that ofSchrödinger, mKdV and sine-Gordon flows to Riemanniangeometry. Finally, in joint work with Calini and Ivey, the PIwill study geometric and topological properties of solutions torealizations of soliton equations, in particular thosecorresponding to finite-gap solutions (periodic case). The studywill try to link properties of these solutions to their spectralparameter.Setting an appropriate geometric background is often afundamental step in the resolution of a problem. A choice ofgeometry establishes the properties and laws we wish to keepunchanged: Relativity (with Lorentzian geometry setting theinteraction between space and time) and computer imaging (usingprojective geometry when 3D perspective needs to be preserved)are some of the best-known examples. But nowadays many engineersand physicists, including some groups working on data collection,consider a basic knowledge of differential geometry to befundamental. Geometric thought is commonplace, as often findingthe right choice of geometry for a problem is an initial step inits resolution - it all depends on how (or with which geometriceyes) you look at it -. BiHamiltonian nonlinear equations arevery rich in structure and they are often used to model differenttypes of phenomena. Their rich structure allows us to find agreat deal of information about the system they model and topredict behavior. The best-known completely integrable systemsare bi-Hamiltonian, and their solutions predict the behavior offluids, from water waves in shallow water to the trailingvortices behind the wing tips of an airplane. These phenomena donot, in principle, exist within any given geometricbackground. When we find geometric realizations for a completelyintegrable system we gain information in two different ways: 1)We learn that the behavior of these phenomena can be visualizedwithin a certain geometry, and we learn how to visualize it. Inparticular, the same phenomena can be described in more than onegeometry; 2) We learn that some geometries are hosts to phenomenathat were not known to exist in that context before. For example,by linking projective and centro-equi-affine geometries, we canfind evolutions of star-shaped curves that behave like solitarywaves. The better we understand the relation between these twoapparently unrelated subjects, the more we can transfer ourextensive geometric knowledge, their connections and properties,to the understanding of completely integrable systems. Andvice-versa.