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

TL;DR

This paper proves the global existence of solutions for the one-dimensional cubic NLS in certain modulation spaces without requiring small initial data, using a novel splitting method and properties of the Schr"odinger group.

Contribution

It extends global existence results to larger initial data in modulation spaces by adapting a splitting method and analyzing the Schr"odinger group behavior.

Findings

Global solutions exist for initial data in $M_{p,p'}$ spaces near $p=2$.

No smallness condition on initial data is needed for these solutions.

The method relies on polynomial growth properties of the Schr"odinger group.

Abstract

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.