Crank–Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative

In this paper, the Crank–Nicolson (CN) difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative is studied. The existence of this difference solution is proved by the Brouwer fixed point theorem. The stability and convergence of the CN scheme are discussed in the L2L2 norm. When the fractional order is two, all those results are in accord with the difference scheme developed for the classical non-fractional coupled nonlinear Schrödinger equations. Some numerical examples are also presented.