  A linearly implicit conservative difference scheme is applied to discretize
the attractive coupled nonlinear Schr\"odinger equations with fractional
Laplacian. Complex symmetric linear systems can be obtained, and the system
matrices are indefinite and Toeplitz-plus-diagonal. Neither efficient
preconditioned iteration method nor fast direct method is available to deal
with these systems. In this paper, we propose a novel matrix splitting
iteration method based on a normal splitting of an equivalent real block form
of the complex linear systems. This new iteration method converges
unconditionally, and the quasi-optimal iteration parameter is deducted. The
corresponding new preconditioner is obtained naturally, which can be
constructed easily and implemented efficiently by fast Fourier transform.
Theoretical analysis indicates that the eigenvalues of the preconditioned
system matrix are tightly clustered. Numerical experiments show that the new
preconditioner can significantly accelerate the convergence rate of the Krylov
subspace iteration methods. Specifically, the convergence behavior of the
related preconditioned GMRES iteration method is spacial mesh-size-independent,
and almost fractional order insensitive. Moreover, the linearly implicit
conservative difference scheme in conjunction with the preconditioned GMRES
iteration method conserves the discrete mass and energy in terms of a given
precision.
