Smoothing for the fractional Schrödinger equation on the torus and the real line

In this paper, we study the cubic fractional nonlinear Schrödinger equation (NLS) on the torus and on the real line. Combining the normal form and the restricted norm methods, we prove that the nonlinear part of the solution is smoother than the initial data. Our method applies to both focusing and defocusing nonlinearities. In the case of full dispersion (NLS) and on the torus, the gain is a full derivative, while on the real line we get a derivative smoothing with an ε loss. Our result lowers the regularity requirement of a recent theorem of Kappeler et al., [19] on the periodic defocusing cubic NLS, and extends it to the focusing case and to the real line. We also obtain estimates on the higher-order Sobolev norms of the global smooth solutions in the defocusing case.