Modern Mathematical Research for Equations in Mathematical Physics

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We investigated many important differential equations in mathematical physics about the structure of their solutions and of their solution spaces. Main results are the following.(1) Equation of Fluid Dynamics : We considered the free surface problem of viscous incompressible fluid down an inclined plane. We obtained some sufficient conditions to guarantee the existence of global in time solution for the initial value problem of the full nonlinear equations. We also considered an approximate nonlinear equation of the free surface problem and proved the occurrence of a family of traveling periodic waves. It appeared as a bifurcation from the zero solution. Then we computed numerically the continuation of the bifurcation branches and found the occurrence of period doubling and bifurcations to torus and so on. (2) Computer Aided Proof or Validated Numerical Computation : A software of Internal analysis by computer has been made for the big Fujitsu computer (M1800) on the Fortran 77EX, which is a necessary tool for this analysis. The next aim is to use this method for the analysis to trace the bifurcation branch of the above periodic solutions. (3) Equation of Quantum Mechanics : Spectral analysis of Schroedinger equation with penetrable wall potentials. Semi-classical asymptotics for total scattering cross sections of three body systems. (4) Stochastic analysis : Some asymptotic problems of Wiener functional integration and applications to heat equation. Construction of Sobolev space on the Wiener space. (5) Dynamical Systems and Bifurcation Theory : Bifurcation of heteroclinic orbit under the family of two parameters and application to chaotic behaviors of a dynamical system.