The cubic nonlinear fractional Schrödinger equation on the half-line

We study the cubic nonlinear fractional Schr & ouml;dinger equation with L & eacute;vy indices 4/3 < a < 2 posed on the half-line. More precisely, we define the notion of a solution for this model and we obtain a result of local-well-posedness almost sharp in the sense of index of regularity required for the solutions with respect for known results on the full real line R. Also, we prove for the same model that the solution of the nonlinear part is smoother than the initial data and the corresponding linear solution. To get our results we use the Colliander and Kenig approach based on the Riemann-Liouville fractional operator combined with Fourier restriction method of Bourgain (1993) and some ideas of the recent work of Erdo & gbreve;an et al. (2019). The method applies to both focusing and defocusing nonlinearities. As a consequence of our analysis we prove a smoothing effect for the cubic nonlinear fractional Schr & ouml;dinger equation posed in full line R for the case of the low regularity assumption, which was point out at the recent work (Erdo & gbreve;an et al., 2019).