Local well-posedness for cubic fractional Schrödinger equations with derivatives on the right-hand side

Abstract

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