Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $\mathbb{R}$

Abstract

We consider the Cauchy problem for a family of semilinear defocusing Schrödinger equations with monomial nonlinearities in one space dimension. We establish global well-posedness and scattering. Our analysis is based on a four-particle interaction Morawetz estimate giving \emph{a priori} $L_{t,x}^8$ spacetime control on solutions.