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中许多重要的微分方程的解结构和解空间。主要结果如下：(1)流体动力学方程：考虑了粘性不可压缩流体沿斜面的自由表面问题。我们得到了保证全非线性方程初值问题全局时间解存在的充分条件。我们还考虑了自由表面问题的近似非线性方程，并证明了一系列行波的出现。它表现为与零解的分歧。然后我们对分岔分支的延拓进行了数值计算，发现了倍周期和环面分岔等现象的发生。 (2)计算机辅助证明或验证数值计算：在富士通大型计算机(M1800)的Fortran 77EX上制作了计算机内部分析软件，是本次分析的必备工具。下一个目标是利用该方法进行分析，追踪上述周期解的分岔分支。 (3) 量子力学方程：具有穿壁势的薛定谔方程的谱分析。三个体系统的总散射截面的半经典渐进。 (4) 随机分析：维纳泛函积分的一些渐近问题及其在热方程中的应用。在维纳空间上构造索博列夫空间。 (5) 动力系统和分岔理论：二参数族下异宿轨道的分岔及其在动力系统混沌行为中的应用。

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。