Preconditioned Iterative Methods For Discrete Space Fractional Nonlinear Schr(?)dinger Equations

It's well known that the nonlinear Schr(?)dinger equation is one of the most impor-tant equations in quantum mechanics.In recent years,as a generalization of the tra-ditional Schr(?)dinger equation,fractional Schr(?)dinger equation has got more and more attention,especially its numerical solution method.In this paper,we consider the preconditioned iterative methods for uncoupled and coupled space fractional nonlinear Schr(?)dinger equations.Using the Crank-Nicolson scheme,we obtain a series of complex nonlinear equations.In each iteration in Newton method,we need to solve a system of linear equations with Jacobi matrix as coeffi-cient matrix.We focus on the preconditioned Krylov subspace methods for these linear equations.Our contributions are as follows:(1)For the uncoupled space fractional nonlinear Schr(?)dinger equation,we first transform the problem into a real linear system with 2 x 2 block structure.Then,by making use of the structure of the coefficient matrix,we propose a preconditioner which is based on the alternating direction method and circulant matrix.Theoretical analysis is given and numerical results show that the new preconditioner has very good perfor-mance.(2)For the coupled space fractional nonlinear Schr(?)dinger equations,we transform the original problem into a real linear system with 4 x 4 block structure in a similar way.By separating the Toeplitz matrix involved in the coefficient matrix from other matrices,we introduce an alternating direction preconditioner.Numerical experiments are given to show the performance of the new preconditioner.