Local well-posedness for cubic fractional Schr\"odinger equations with derivatives on the right-hand side

Abstract

For $s \in (\frac{1}{2},1]$ we investigate well-posedness of the equation \[ \left ( i \partial_t + (-\Delta)^{s} \right ) u = \left (|D|^{1-2s} |u|^2 \right)\ |D|^{2s-1} u \] under small initial data $\|u(0)\|_{H^{\frac{n-2s}{2}}(\mathbb{R}^n)} \ll 1$. This equation is a model equation for for $s$-Schr\"odinger map equation \[ \partial_t \psi = \psi \wedge (-\Delta)^s \psi: \quad \psi: \mathbb{R}^n \times \mathbb{R} \to \mathbb{S}^{2}, \]