MULTISCALE METHODS AND ANALYSIS FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH WAVE OPERATOR

Abstract

The Schrödinger equation is a fundamental partial differential equation for the wave function of particles in quantum mechanics. The nonlinear Schrödinger equation with wave operator (NLSW) arises from the nonrelativistic limit of Klein-Gordon equation. Depending on the parameters, the solution can be highly oscillatory. In this thesis, different multiscale methods based on nested Picard iteration to solve NLSW and related equations are proposed, and uniform error estimates are established. Long-time dynamics of NLSW with weak nonlinearities are also studied.