Modern Analysis for the Equations of Mathematical Science

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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