On existence of global solutions of the one-dimensional cubic NLS for initial data in the modulation space $M_{p,q}(\mathbb{R})$

Abstract

We prove global existence for the one-dimensional cubic non-linear Schrödinger equation in modulation spaces $M_{p,p^{\prime }}$ 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 Schrödinger group on modulation spaces.

13 pages

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