物理中的几何流与几何孤立子

Research on nonlinear evolution equations in physics and differential geometry has always been the concern of mathematicians. The results on Schrodinger flow and geometric KdV flow have successfully generalized the two classical physics equations on Kahler manifolds respectively. On this basis, the main research goal of this project is to discuss the global existence of KdV flow without curvature conditions and we will give a unified geometric interpretation for several classical evolutions equations on Kahler manifolds. More precisely, we would introduce a class of new geometric flows to unify some physical and mechanical models such as KdV equations, complex-valued mKdV equations, derivative nonlinear Schrodinger equations, Airy equations, Da Rios equation and the motion equations of voetex filament etc. The global existence results of this new flow will be of great significance to the research on the well-posedeness for these physics equations. Moreover, we will further explore the generalization of KdV equations on Riemannian manifolds and show the global existence results and we will keep exploring the construction of conservation laws. On the other hand, we introduce the notion of geometric solitons which are defined as global sotions to geometric flows with certain self-symmetry in the time direction. Geometric solitons could be thought as geometric generalizations of solitons of Schrodinger equation and KdV equation.We will study the geometric aspects and the existence of geometric solitons which will help us understand KdV flow and Schrodinger flow much more comprehensive.Meanwhile, we will seek specific examples on several classic manifolds. All these works will helps us to impove the theory of KdV geometric flows.

对物理与微分几何中非线性发展方程的研究一直备受数学家们关注。薛定谔流、KdV几何流则分别成功把两类经典物理方程推广到了Kahler流形上，并得到了一定曲率条件下的整体存在性。以此为基础，本项目主要研究一般情形下KdV流的整体存在性；探索物理中几类典型发展方程在Kahler流形上统一的几何描述，我们拟给出一个新的几何模型来统一复向量值修正KdV方程，薛定谔方程，导数非线性薛定谔方程，Airy方程，Da Rios方程，涡丝运动方程等物理方程，对新几何流的研究尤其关于整体解的存在性研究将对这些物理方程自身的适定性研究有重要意义；本项目还将研究KdV方程在黎曼流形上的几何推广并探索其守恒率的构造和整体存在性结果。另一方面，我们给出几何孤立子的概念，这是一类具有对称性的几何流的整体解，可以推广薛定谔方程、KdV方程的孤立子解。我们将深入研究几何孤立子的几何性质和存在性, 并探索典型流形上具体的例子。

对物理与微分几何中非线性发展方程的研究一直备受数学家们关注。薛定谔流、KdV几何流则分别成功把两类经典物理方程推广到了Kahler流形上，并得到了一定曲率条件下的整体存在性。在此基础上，本项目研究探索了物理中几类经典方程在流形上统一的几何描述，我们提出了薛定谔-Airy几何流的定义来统一复向量值mKdV方程，薛定谔方程，导数非线性薛定谔方程，Airy方程，涡丝运动方程等物理方程；同时我们还在完备黎曼流形上定义了一类新的几何流，它既与推广的三维Landau-Lifshit方程有密切关系，同时也可以看做上述薛定谔-Airy流的在目标流形带有常全纯截面曲率时的特例，有其特殊的意义。这两类几何流问题我们均研究讨论了他们的局部存在性、正则性情况，并在一定条件下得到了整体存在性的结果。另一方面，我们提出了几何孤立子的概念，这是一类具有对称性的几何流的整体解，可以推广薛定谔方程、KdV方程的孤立子解，并在几何层面可以看到孤立子解本质上与底空间及目标空间中的等距变换有关。此外我们还给出了一些典型曲面上的与薛定谔流、KdV流有关的几何孤立子的具体的例子。最后，本项目的后续研究中我们将继续深入研究KdV方程在黎曼流形上的几何描述以及相关的几何孤立子问题。