Global existence and scattering for rough solutions to generalized   nonlinear Schr\"odinger equations on $\R$
  We consider the Cauchy problem for a family of semilinear defocusing Schr\"odinger 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} $L^8_{t,x}$ spacetime control on solutions. 
J. Colliander, J. Holmer, M. Visan, X. Zhang