Global well-posedness of cubic fractional Schrödinger equations in one dimension

Abstract: In this paper, we consider the Cauchy's problem of global existence and scattering behavior of small, smooth, and localized solutions of cubic fractional Schr\"odinger equations in one dimension, \begin{equation*} \mathrm{i} \partial_t u- (-\Delta)^{\frac{\alpha}{2}} u=c_|u|^2u, \end{equation} where $\alpha \in (\frac{1}{3},1), c_* \in \mathbb{R}$. Our work is a generalization of the result due to Ionescu and Pusateri \cite{IP}, where the case $\alpha=\frac{1}{2}$ was considered. The highlight in this paper is to give a modified dispersive estimate in weighted Sobolev spaces for cubic fractional Schr\"odinger equations, which could be used for $ \alpha \in (\frac{1}{3},1)$. Based on this modified dispersive estimate, we prove the global existence and modified scattering behavior of solutions combining space-time resonance and bootstrap arguments.