A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation

In this paper, we propose a conservative linearized difference scheme for the nonlinear fractional Schrödinger equation. The scheme efficiently avoids the time consuming iteration procedure necessary for the nonlinear scheme and thus is time saving relatively. It is rigorously proved that the scheme is mass conservative and uniquely solvable. Then employing mathematical induction, we further show that the proposed scheme is convergent at the order of O(τ2 + h2) in the l2 norm with time step τ and mesh size h. Moreover, an extension to coupled nonlinear fractional Schrödinger systems is presented. Finally, numerical tests are carried out to corroborate the theoretical results and investigate the impact of the fractional order α on the collision of two solitons.