Modern Mathematical Research for Equations in Mathematical Physics

研究了数学物理中许多重要的微分方程的解及其解空间的结构。主要结果如下。(1)流体动力学方程：考虑沿斜面向下的粘性不可压缩流体的自由面问题。得到了一类全非线性方程初值问题全局及时解存在的充分条件。我们还考虑了自由表面问题的近似非线性方程，并证明了一组行周期波的存在。它表现为零解的一个分支。然后对分岔分支的延拓进行了数值计算，发现了周期加倍和向环面分岔等情况。(2)计算机辅助证明或验证数值计算：在Fortran 77EX上为富士通大型计算机（M1800）开发了一套计算机内部分析软件，作为进行内部分析的必要工具。下一个目的是利用这种分析方法来跟踪上述周期解的分支。(3)量子力学方程：具有可穿透壁势的薛定谔方程的谱分析。三体系统总散射截面的半经典渐近性。(4)随机分析：Wiener泛函积分的一些渐近问题及其在热方程中的应用。Wiener空间上Sobolev空间的构造。(5)动力系统与分岔理论：二参数族下异斜轨道的分岔及其在动力系统混沌行为中的应用。

We investigated many important differential equations in mathematical physics about the structure of their solutions and of their solution spaces. Main results are the following.(1) Equation of Fluid Dynamics : We considered the free surface problem of viscous incompressible fluid down an inclined plane. We obtained some sufficient conditions to guarantee the existence of global in time solution for the initial value problem of the full nonlinear equations. We also considered an approximate nonlinear equation of the free surface problem and proved the occurrence of a family of traveling periodic waves. It appeared as a bifurcation from the zero solution. Then we computed numerically the continuation of the bifurcation branches and found the occurrence of period doubling and bifurcations to torus and so on. (2) Computer Aided Proof or Validated Numerical Computation : A software of Internal analysis by computer has been made for the big Fujitsu computer (M1800) on the Fortran 77EX, which is a necessary tool for this analysis. The next aim is to use this method for the analysis to trace the bifurcation branch of the above periodic solutions. (3) Equation of Quantum Mechanics : Spectral analysis of Schroedinger equation with penetrable wall potentials. Semi-classical asymptotics for total scattering cross sections of three body systems. (4) Stochastic analysis : Some asymptotic problems of Wiener functional integration and applications to heat equation. Construction of Sobolev space on the Wiener space. (5) Dynamical Systems and Bifurcation Theory : Bifurcation of heteroclinic orbit under the family of two parameters and application to chaotic behaviors of a dynamical system.