A conservative difference scheme for solving the strongly coupled nonlinear fractional Schrödinger equations

• The strongly coupled nonlinear space fractional Schrödinger equation is studied numerically.
• An implicit difference scheme with the discrete conservative properties is constructed.
• The solvability, boundedness and convergence in the maximum norm are proved.
• A three-level linear difference scheme with two identities is also presented.
• The performance of both schemes are verified numerically.

This paper focuses on numerically solving the strongly coupled nonlinear space fractional Schrödinger equations. First, the laws of conservation of mass and energy are given. Then, an implicit difference scheme is proposed, under the assumption that the analytical solution decays to zero when the space variable x tends to infinity. We show that the scheme conserves the mass and energy and is unconditionally stable with respect to the initial values. Moreover, the solvability, boundedness and convergence in the maximum norm are established. To avoid solving nonlinear systems, a linear difference scheme with two identities is proposed. Several numerical experiments are provided to confirm the theoretical results.