Caustics and the Indefinite Signature Schrödinger Equation, Linear and Nonlinear

The evolution of surface waves in deep water is given by a Schrodinger-like equation. In deep water surface water waves evolve under the nonlinear equation 2i u =1/4(uxx - 2uyy) + q |u|2u Where x, y are coordinates in R2, q is a constant. The techniques for the Schrodinger equation can be used in the study of the evolution of (1.1.19), although the behavior is often quite different. This thesis will focus on three main areas. First, it will concentrate on the behavior the linear Schroedinger equation iut + delta u = 0, in particular, on the asymptotic behavior of eitdelta u0 as $t arrow 0. This has a connection to the asymptotic behavior of the pointwise Fourier inversion SRf as R arrow infinity. Secondly, this thesis will address the behavior of the indefinite signature Schroedinger equation iut + Lu = 0, where [equasion follows] as well as i ut + Lu = F(u), where F is a nonlinearity. Finally, some supercritical local existence results will be obtained for a power-type nonlinearity, iut + delta u = |u|alphau u(0,x) = chiB(0;1), and a global existence result for [equation follows]