Global well-posedness of cubic fractional Schr\"odinger equations in one dimension

TL;DR

This paper establishes global existence and scattering for small solutions of one-dimensional cubic fractional Schrödinger equations with fractional power between 1/3 and 1, extending previous results to a broader range of fractional powers.

Contribution

It introduces a modified dispersive estimate in weighted Sobolev spaces for fractional Schrödinger equations, enabling analysis for a wider range of fractional powers.

Findings

Proved global existence of solutions for fractional powers in (1/3,1)

Established modified scattering behavior of solutions

Extended previous results to a broader fractional range

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.