Conservative numerical schemes for the nonlinear fractional Schrödinger equation

This paper deals with the Crank-Nicolson Fourier collocation method for the nonlinear fractional Schrödinger equation containing a fractional derivative. We prove that at each discrete time the method preserves the discrete mass and energy conservation laws. The existence, uniqueness and convergence of the numerical solution are also investigated. In particular, we show that the method has the second-order accuracy in time and the spectral accuracy in space. Since the proposed schemes are implicit, they are solved by an iteration algorithm with FFT. Two examples illustrate the efficiency and accuracy of the numerical schemes.