On the Initial Value Problem for the One Dimensional Quasi-Linear Schrödinger Equations

Abstract

We study the local in time solvability of the initial value problem (IVP) of the one dimensional fully nonlinear Schrödinger equation. Under appropriate assumptions on the nonlinearity (regularity and ellipticity) and on the initial data (regularity and decay at infinity), we establish the existence and uniqueness of solutions of the IVPin weighted Sobolev spaces. The equation can be reduced to its quasi-linear version by taking space derivative. The desired results are obtained by combining a change of variables, energy estimates, and the artificial viscosity method.