On 1D Mass-subcritical Nonlinear Schrödinger equations in modulation spaces $M^{p, p'} (p<2)$

Abstract: We establish global well-posedness for the 1D mass-subcritical nonlinear Schr\"odinger equation $$iu_t +u_{xx} \pm |u|^{\alpha-1}u=0 \quad (1< \alpha<5)$$ for large Cauchy data in modulation spaces $M^{p,\frac{p}{p-1}}(\mathbb R)$ with $4/3<p<2$ and $p$ sufficiently close to $2$. This complements the work of Vargas-Vega (2001) and Chaichenets et al. (2017), where they established the result for $p$ larger than $2$. The proof adopts Bourgain's high-low decomposition method inspired by the work of Vargas-Vega (2001) and Hyakuna-Tsutsumi (2012) to the modulation space setting and exploits generalized Strichartz estimates in Fourier-Lebesgue spaces.