Applied analysis of differential equations in math.sci.

math.sci 中微分方程的应用分析。

基本信息

批准号：
08404007
负责人：
NISHIDA Takaaki
金额：
$ 6.78万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
1996
资助国家：
日本
起止时间：
1996 至 1997
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08404007/
关键词：
Nonlinear Part.Diff.Eq's Structure of Solution Space Theory of Dynamical Systems Bifurcation Th'y Equations in Quantum Mechanics Operator Theory Quantum Group of Elliptic Type Fluid Dynamical Equations 量子力学の方程式系可解格子模型解の特異性の伝播表現論 /

项目摘要

The purpose of this research is to investigate the structure of not only solutions but also solution spaces for the system of ordinary and partial differential equations in mathematical sciences. Especially the great emphasis is put on the analysis of the change of structures depending on the parameters. 1.System of equations in quantum mechanics : (1) Research on the change of distribution of eigenvalues by the perturbation of the Schrodinger operators and Pauli operators, (2) Inverse scattering problems to reconstruct the potential from the scattering operator for Diracoperator. 2.Systems of equations in fluid dynamics : Heat convection by Boussinesq equations with free surface. The stability analysis of the heat conduction state. The movement of eigenvalues of the linearized system depending on the large change of Rayleigh number and/or Marangoni number are investigated by computer assisted proof and the corresponding instability of the heat conduction state is proved at the specific values of those parameters. The stationary bifurcation can be proved from it. The Hopf bifurcation is under investigations. The analysis on the global structure of solution curves and bifurcation branches is the next subject, for which the new analytical method should be developed. The first step for the analysis by the computer assisted proof for the system of partial differential equations is just taken and a criterion is proposed to guarantee the existence of solution coreesponding specific parameter value by the method. 3.Lattice model and quantum group : The solutions of elliptic function for Yang-Baxter equation are investigated and the theory of quantum groups of elliptic type is established in an unified method. 4.Dynamical systems : Infinitely many homoclinic doubling bifurcations are found for homoclinic orbits with some degeneracy of codimension 3. The conditions of vector fields with two parameters which have this bifurcation phenomena are investigated.

这项研究的目的是研究数学科学中普通和部分微分方程系统的解决方案的结构，还要研究解决方案空间。特别是重点放在对结构变化的分析上，具体取决于参数。 1.量子力学中的方程系统：（1）研究特征值分布的研究通过schrodinger操作员和保利操作员的扰动而变化，（2）逆散射问题以从散射操作员的散射范围中重建越野摩托车的潜力。 2.流体动力学方程式的系统：由自由表面的布斯西尼克方程进行的热对流。热传导状态的稳定性分析。线性化系统的特征值的运动取决于雷利数和/或马龙诺尼数的较大变化，并通过计算机辅助证明进行了研究，并以这些参数的特定值证明了热传导状态的相应不稳定性。可以从中证明固定分叉。 HOPF分叉正在调查中。对解决方案曲线和分叉分支的全局结构的分析是下一个主题，应为其开发新的分析方法。仅采用了计算机辅助证明部分偏微分方程系统进行分析的第一步，并提出了标准以确保通过该方法核心对应特定参数值的解决方案的存在。 3.局部模型和量子组：研究了杨 - 巴克斯特方程的椭圆函数的解，并以统一的方法建立了椭圆类型的量子群的理论。 4.动力系统：发现了具有共同敏感的一定脱位的同质轨道的无限层层次加倍分叉。