Modern Mathematical Research for Equations in Mathematical Physics

数学物理方程的现代数学研究

• 批准号: 02452007
• 负责人: NISHIDA Takaaki
• 金额: $ 4.8万
• 依托单位: Kyoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (B)
• 财政年份: 1990
• 资助国家: 日本
• 起止时间: 1990 至 1991
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-02452007/
• 关键词: Equation of Quantum Mechanics; Spectral Theory; Stochastic Analysis; Wiener Functional; Equation of Fluid Dynamics; Free Surface Problems; Validated Numerical Computation; Dynamical Systems and Chaos; Wiener汎関数; 無限次元リ-群の表現; 可解格子模型; シュレディンガ-方程式

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.

我们研究了数学物理学中有关其解决方案及其溶液空间结构的许多重要微分方程。主要结果是以下。（1）流体动力学方程：我们认为倾斜平面下粘性不可压缩的流体的自由表面问题。我们获得了一些足够的条件，以确保整体解决方案的存在，以解决整个非线性方程的初始值问题。我们还考虑了自由表面问题的近似非线性方程，并证明了旅行周期性波的发生。它是从零溶液中作为分叉的。然后，我们以数值计算分叉分支的延续，发现周期倍增和分叉对圆环等等。 （2）计算机辅助证明或经过验证的数值计算：通过Fortran 77EX上的Big Fujitsu计算机（M1800）进行了计算机内部分析软件，这是该分析的必要工具。下一个目的是将此方法用于分析以追踪上述周期性解决方案的分叉分支。 （3）量子力学方程：具有可渗透壁电位的Schroedinger方程的光谱分析。三个身体系统的总散射横截面的半古典渐近学。 （4）随机分析：Wiener功能整合的某些渐近问题和对热方程式的应用。在维也纳空间上建造Sobolev空间。 （5）动力学系统和分叉理论：两个参数的家族中的杂斜轨道的分叉，并应用于动态系统的混乱行为。

项目成果

Ikebe,Teruo(同): "Spectral and scattering theory for the Schrodinger operators with penetrable wall potentials" J.Math.Kyoto University. 31. 219-258 (1991)

Ikebe, Teruo（同）：“具有穿透壁势的薛定谔算子的光谱和散射理论”J.Math.Kyoto University（京都大学） 31. 219-258 (1991)。

Shinzo Watanabe: "Donsker's delta function in the Malliavin calculus" Stochastic Analysis,Libre Amicorum for Moshe Zakai,ed.by MayorーWolf,E.etc.495-502 (1991)

Shinzo Watanabe：“Malliavin 演算中的 Donsker δ 函数”随机分析，Moshe Zakai 的 Libre Amicorum，Mayor-Wolf 编辑，E.etc.495-502 (1991)

Shigekawa, I.: ""Sobolev spaces over the Wiener space based on an Ornstein-Uhlenbeck operator"" J. Math. Kyoto University.

Shigekawa, I.：“基于 Ornstein-Uhlenbeck 算子的维纳空间上的 Sobolev 空间”J. Math。

Michio Jimbo: "Combinatorics of representations of Ug(sl/n,(1)) at g=0" to appear in Comm.Math.Phys.

Michio Jimbo：“在 g=0 时 Ug(sl/n,(1)) 表示的组合”出现在 Comm.Math.Phys 中。

Kokubu Hiroshi: "Heteroclinic bifurcations associatel with different saddle indices" Proc.Intern.Conf.on Dynamical Sys.& Related Topics(Nagoya 1990)ed.by Shinaina World Sci.Publ.236-261 (1991)

Kokubu Hiroshi：“异斜分岔与不同鞍指数相关”Proc.Intern.Conf.on Dynamical Sys。