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 $M_{p,p^{\mathrm{\prime }}}$ 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.