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

TL;DR

This paper demonstrates that solutions to the cubic fractional nonlinear Schrödinger equation become smoother over time, with the degree of smoothing depending on the domain and type of nonlinearity, extending previous results.

Contribution

It introduces a method combining normal form and restricted norm techniques to prove smoothing effects for both focusing and defocusing fractional NLS on the torus and real line.

Findings

Nonlinear part gains a full derivative on the torus with full dispersion.

On the real line, the smoothing is with an epsilon loss.

Extends regularity results to focusing case and real line, improving prior theorems.

Abstract

In this paper we study the cubic fractional nonlinear Schrodinger 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 $\epsilon$ loss. Our result lowers the regularity requirement of a recent theorem of Kappeler et al. 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.