KdV曲线与KdV孤立子

The recent introduction of the notion of geometric KdV flow is a very remarkable and interesting work, which successfully generalizes the classical KdV equation on general manifolds. The KdV curve and KdV soliton are two kinds of special global solution to the KdV flow equation..As a stable solution of the KdV flow, KdV curve gives a special kind of geometric curves. These curves can be regarded as a third-order analog to the geodesics and has interesting geometric properties. On the other hand, we introduce the notion of KdV solitons. The KdV soliton is defined as a global solution to the KdV flow equation with certain self-symmetry in the time direction, which generalizes the celebrated notion of solitons of the classical KdV equations. .We will study the geometric aspects of KdV curves and KdV solitons, and construct specific examples on manifolds with constant curvature. Then we will try to apply the Floer theory to construct a new kind of homology and establish a connection between the number of KdV curves and topology of the manifold..The KdV curve and KdV soliton have strong and extensive physics background. They are not only interesting because they are closely related to some famous problems in differential geometry, but also important since they provide potential support for the further research of the KdV flow.

KdV流的引入是最近一个十分引人关注的重要工作，它成功地把经典KdV方程推广到一般流形上。本项目主要研究KdV流方程的两类特殊整体解以及相应的几何问题。KdV流的稳定解定义了一类特殊的几何曲线，我们称之为KdV曲线。这类曲线可以视为测地线的三阶近似，有重要的几何意义。另一方面，我们引入了KdV孤立子的概念。这是一类具有自对称性的KdV流整体解，它推广了经典KdV方程孤立子解的重要思想。我们将研究KdV曲线的几何性质，并在常曲率流形上构造出具体的KdV曲线和KdV孤立子的例子。更重要的是我们将应用Floer同调理论在KdV曲线和KdV孤立子的数目和流形的拓扑之间建立联系，得到KdV曲线的多重存在性结果。KdV曲线和KdV孤立子不仅有着强烈的物理背景，有趣的历史渊源，以及自身重要的几何意义，并将有力推广Floer理论的应用，且对一般KdV流的深入研究提供重要依据

该项目主要研究KdV孤立子以及相关几何问题，主要分为三个部分。.1.几何孤立子。KdV流源自流体力学中对涡丝运动的研究，KdV孤立子是其一类具有特殊几何和物理意义的解，了解孤立子解对了解涡丝运动和流形结构有着重要的意义。我们把首先提出KdV孤立子的概念，并把它推广到一类哈密尔顿系统中，进而利用这个观点解释了涡丝运动方程的孤立子和磁测地线的密切关系。.2.反平均曲率流。反平均曲率流是涡丝运动方程的自然几何推广，也是一类重要的几何薛定谔型方程。我们证明了2维反平均曲率流的局部存在性，这是该方向第一个高维的存在性结果。.3.杨-米尔斯-希格斯场的收敛性。YMH场是量子理论的重要基石，也是数学中定义辛流形不变量的基础。我们深入研究了YMH场的收敛性，这一结果统一了以往许多调和映照和全纯曲线的收敛及爆破结论，并为在辛几何中相关应用铺平了道路。