Crank-Nicolson Fourier spectral methods for the space fractional nonlinear Schrodinger equation and its parameter estimation

In this paper, the Crank-Nicolson Fourier spectral approximations for solving the space fractional nonlinear Schrodinger equation are proposed. Firstly, the numerical formats of the Crank-Nicolson Fourier Galerkin and Fourier collocation methods are established. The fast Fourier transform technique is applied to practical computation. Secondly, Convergence with spectral accuracy in space and second-order accuracy in time is verified for both Galerkin and collocation approximations. Moreover, a rigorous analysis of the conservation for the Crank-Nicolson Fourier Galerkin fully discrete system is derived. Thirdly, the Bayesian method is presented to estimate the fractional derivative order and the coefficient of nonlinear term based on the spectral format of the direct problem. Finally, some numerical examples are given to confirm the theoretical analysis.