Global bounds for the cubic nonlinear Schr\"odinger equation (NLS) in   one space dimension
  This article is concerned with the small data problem for the cubic nonlinear Schr\"odinger equation (NLS) in one space dimension, and short range modifications of it. We provide a new, simpler approach in order to prove that global solutions exist for data which is small in $H^{0,1}$. In the same setting we also discuss the related problems of obtaining a modified scattering expansion for the solution, as well as asymptotic completeness. 
Mihaela Ifrim and Daniel Tataru
math.AP