微分几何中若干物理发展方程的研究

Research on nonlinear evolution equations in physics and differential geometry has always been the concern of mathematicians. Recently, the introduction of the notion of geometric KdV flow has successfully generalized this classical physics equation on Kahler manifolds respectively. On this basis, the main research goal of this project is to 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 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. On the other hand, we will further explore the generalization of KdV equations on Riemannian manifolds and show the global existence results. Meanwhile, we will keep exploring the construction of conservation laws. All these works will helps us to impove the theory of KdV flow.

物理中的非线性发展方程在微分几何中的研究一直备受数学家们关注。近期，KdV几何流的引入成功把KdV方程推广到Kahler流形上，并得到了一定的全局存在性结果。 以此为基础，本项目主要研究物理中几类典型发展方程在Kahler流形上统一的几何推广，我们拟给出一个新的几何模型来统一描述包括mKdV方程，导数非线性薛定谔方程等，深入研究新几何模型的整体解的存在性将对物理方程自身的适定性研究有重要意义；另一方面，本项目将研究KdV方程在黎曼流形上的几何推广并探索其整体存在性结果，这将进一步完善KdV几何流理论。

本项目集中研究讨论了物理中几类典型发展方程在Kahler流形上统一的几何推广，给出了一个新的几何模型—“薛定谔-Airy流”来统一描述包括KdV方程，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，研究了该模型在一定条件下的整体解的存在性；另一方面，本项目深入研究了KdV方程在黎曼流形上的几何推广，给出了一类新的几何流模型，该模型同时又可以看做薛定谔-Airy流一定条件时的特例，我们讨论了与新模型相关的守恒律的构造，进一步完善了KdV几何流理论。