Applied analysis of differential equations in math.sci.

本研究的目的是研究数学科学中常微分方程组和偏微分方程组的解的结构以及解空间的结构。重点分析了结构随参数变化的规律。1.量子力学中的方程组：（1）研究薛定谔算符和泡利算符的微扰对本征值分布的影响;（2）由Dirac算符的散射算符重构势的逆散射问题。2.流体动力学方程组：由自由表面的Boussinesq方程组产生的热对流。热传导状态的稳定性分析。通过计算机辅助证明，研究了线性化系统的特征值随Rayleigh数和Marangoni数的大变化而发生的移动，并证明了在这些参数的特定值下，热传导态的相应不稳定性。由此证明了系统的平稳分岔，并对系统的Hopf分岔进行了研究。解曲线和分支分支的全局结构分析是下一个研究课题，需要发展新的分析方法。对偏微分方程组进行了计算机辅助证明分析的第一步，并给出了一个判据，保证了该方法解核的存在性，该解核对应于特定的参数值。3.格点模型与量子群：研究了Yang-Baxter方程的椭圆函数解，用统一的方法建立了椭圆型量子群理论。4.动力系统：对于余维为3的同宿轨道，发现了无数的同宿加倍分支。研究了双参数向量场出现这种分岔现象的条件。

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.