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

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