Modern Analysis for the Equations of Mathematical Science

主要研究数学科学中微分方程组解的结构和解空间的结构。(1)研究了双曲型偏微分方程的适定性，特别是具有三重特征根的方程的有效双曲性。(2)随机微分方程解的研究，特别是用由高斯核构造热核的Levi-sum方法在wiener混沌中展开Donsker's delta函数。(3)对可解晶格模型状态空间结构的研究。用量子群表示的六顶点模型相关函数的显式表达式(4)动力系统分岔和混沌的研究。由具有三重零特征值的简并向量场的微扰产生几何奇异吸引子。同斜轨或异斜轨的余维二分岔，特别是无限多的同斜双轨。(5)流体动力学方程的分岔问题。用计算机辅助证明自由曲面问题的平稳分岔和Hopf分岔的研究和证明

Main theme is the investigation of the structure of the solution and the structure of the solution space for the system of the differential equations in the mathematical science.(1) Investigation for the characterization of the wellposedness of the hyperbolic partial differential equation, specially the effective hyperbolicity for the equation which has triple chracteristic roots. Propagation of singularities along the bicharacteristics(2) Investigations of the solutions of stochastic differential equations, especially the expansion of the Donsker's delta function by the wiener chaos using the Levi-sum method which constructs the heat kernel from the Gauss kernel. Approximation scheme for the heat kernel using the Donsker's delta function(3) Investigation for the structure of the space of states for solvable lattice models. Explicit formula for the correlation function for the six-vertex model using the representation of quantum groups(4) Investigation of bifurcations and chaos in the Dynamical systems. Appearance of the geometric strange attractors by small perturbation of the degenerate vector field which has triple zero characteristic values. Codimension two bifurcations of homoclinic or heteroclinic orbits, specially infinitely many numbers of homoclinic doubling bif.(5) Bifurcation problems for the equations of fluid dynamics. Investigations and proof of the stationary and Hopf bifurcations for the free surface problems using the computer assisted proof