Well-Posedness of the Nonlinear Schrödinger Equation

Abstract

The Nonlinear Schrödinger equation (NLSE) is a prototypical example of nonlinear partial differential equation. It is commonly used to describe propagation of light in nonlinear optical fibers and is of great importance in quantum mechanics. In this Capstone Project, we provide a complete proof of well-posedness, that is existence of a unique solution of the NLSE using one of the major mathematical techniques: the Banach fixed-point theorem. Both local and global results for inital data in L2(R) are obtained. Moreover, we briefly discuss possible extensions of the topic in terms of different function spaces, general nonlinearities and higher dimension.