On existence of global solutions of the one-dimensional cubic NLS for initial data in the modulation space Mp,q(R)

We prove global existence for the one-dimensional cubic non-linear Schrödinger equation in modulation spaces Mp,p' for p sufficiently close to 2. In contrast to known results, [9] and [14], our result requires no smallness condition on initial data. The proof adapts a splitting method inspired by work of Vargas-Vega, Hyakuna-Tsutsumi and Grünrock to the modulation space setting and exploits polynomial growth of the free Schödinger group on modulation spaces.