On existence of global solutions of the one-dimensional cubic NLS for   initial data in the modulation space $M_{p,q}(\mathbb R)$
  We prove global existence for the one-dimensional cubic non-linear Schr\"odinger equation in modulation spaces $M_{p,p'}$ for $p$ sufficiently close to $2$. In contrast to known results, our result requires no smallness condition on initial data. The proof adapts a splitting method inspired by work of Vargas-Vega and Hyakuna-Tsutsumi to the modulation space setting and exploits polynomial growth of the free Schr\"odinger group on modulation spaces. 
Leonid Chaichenets, Dirk Hundertmark, Peer Kunstmann and Nikolaos   Pattakos
math.AP