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A linearly implicit conservative difference scheme is applied to discretizethe attractive coupled nonlinear Schr\"odinger equations with fractionalLaplacian. Complex symmetric linear systems can be obtained, and the systemmatrices are indefinite and Toeplitz-plus-diagonal. Neither efficientpreconditioned iteration method nor fast direct method is available to dealwith these systems. In this paper, we propose a novel matrix splittingiteration method based on a normal splitting of an equivalent real block formof the complex linear systems. This new iteration method convergesunconditionally, and the quasi-optimal iteration parameter is deducted. Thecorresponding new preconditioner is obtained naturally, which can beconstructed easily and implemented efficiently by fast Fourier transform.Theoretical analysis indicates that the eigenvalues of the preconditionedsystem matrix are tightly clustered. Numerical experiments show that the newpreconditioner can significantly accelerate the convergence rate of the Krylovsubspace iteration methods. Specifically, the convergence behavior of therelated preconditioned GMRES iteration method is spacial mesh-size-independent,and almost fractional order insensitive. Moreover, the linearly implicitconservative difference scheme in conjunction with the preconditioned GMRESiteration method conserves the discrete mass and energy in terms of a givenprecision.

A fast normal splitting preconditioner for attractive coupled nonlinear Schr\"odinger equations with fractional Laplacian