Analysis of the damped nonlinear space-fractional Schrödinger equation

In this paper, we verify the unique existence of the global smooth solution of the damped nonlinear space-fractional Schrödinger (DNFS) equation and show it follows a conformal mass conservation law. We propose a conformal mass-preserving linearized scheme. It is rigorously proved that this scheme preserves the discrete conformal mass. Furthermore, we prove that the proposed scheme admits a unique solution and is of second order convergence in space and first order convergence in time. Some numerical experiments are carried out to validate the theoretical analysis.