Nonlinear singular perturbations of the fractional Schr\"odinger equation in dimension one

The paper discusses nonlinear singular perturbations of delta type of the fractional Schr"odinger equation $\mathrm{ı}{\mathrm{\partial }}_t\psi ={(-\mathrm{△})}^s\psi$, with $s\in (\frac{1}{2},1]$, in dimension one. Precisely, we investigate local and global well posedness (in a strong sense), conservations laws and existence of blow-up solutions and standing waves.