Applied analysis of differential equations in math.sci.

点击翻译按钮获取中文摘要

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.