On global properties of solutions of some nonlinear Schrodinger-type equations

Abstract

The Schrodinger equation, an equation central to quantum mechanics, is a dispersive equation which means, very roughly speaking, that its solutions have a wave-like nature, and spread out over time. In this thesis, we will consider global behaviour of solutions of two nonlinear variations of the Schrodinger equation. In particular, we consider the nonlinear magnetic Schrodinger equation for u : R3 × R→ C, iut = (i∇+A)u+ V u+ g(u), u(x, 0) = u0(x), where A : R3 → R3 is the magnetic potential, V : R3 → R is the electric potential, and g = ±|u|2u is the nonlinear term. We show that under suitable assumptions on the electric and magnetic potentials, if the initial data is small enough in H1, then the solution of the above equation decomposes uniquely into a standing wave part, which converges as t→∞, and a dispersive part, which scatters. We also consider the Schrodinger map equation ~ut = ~u×∆~u for ~u : R2 × R → S2. We obtain a global well-posedness result for this equation with radially symmetric initial data without any size restriction on the initial data. Our technique involves translating the Schrodinger map equation into a cubic, non-local Schrodinger equation via the generalized Hasimoto transform. There, we also show global well-posedness for the nonlocal Schrodinger equation with radially-symmetric initial data in the critical space L2(R2), using the framework of Kenig-Merle and Killip-Tao-Visan.