GEOMETRY AND ANALYSIS FOR WAVE FIELDS

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In this research project, various space-time behavior of solutions to nonlinear dispersive equations, such as nonlinear Schrodinger equations (NLS) and KdV type equations, nonlinear hyperbolic equations, such as nonlinear wave and Klein-Gordon equations, and coupled systems of those equations, such as nonlinear field equations. The main results are the following :(1)Asymptotic completeness in the energy space H^1(R^3) for NLS with repulsive case has been proved.(2)A unified treatment for small data scattering for nonlinear field equations has been given in terms of critical and subcritical setting.(3)Existence and uniqueness of self-similar solutions for nonlinear wave equations have been proved in the framework of weak Lebesgue spaces.