Global Solutions of Modified One-Dimensional Schrödinger Equation

Abstract

In this paper, we consider the modified one-dimensional Schrödinger equation: ( Dt−F(D) ) u=λ|u|u, where F(ξ) is a second order constant coefficients classical elliptic symbol, and with smooth initial datum of size ε≪1. We prove that the solution is global-intime, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we get a one term asymptotic expansion for u when t→+∞. AMS subject classifications: Primary: 35Q55; Secondary: 35A01, 35B40, 35S50