Sine-transform-based fast solvers for Riesz fractional nonlinear Schrödinger equations with attractive nonlinearities

This paper presents fast solvers for linear systems arising from the discretization of fractional nonlinear Schrödinger equations with Riesz derivatives and attractive nonlinearities. These systems exhibit complex symmetry, indefiniteness, and a $d$-level diagonal-plus-Toeplitz structure. We propose a Toeplitz-based anti-symmetric and normal splitting iteration method for the equivalent real block linear systems, ensuring unconditional convergence. By integrating this iteration method with sine-transform-based preconditioning, we introduce a novel preconditioner that enhances the convergence rate of Krylov subspace methods. Both theoretical and numerical analyses demonstrate that the new preconditioner exhibits a parameter-free property, and favorable eigenvalue clustering nature of the corresponding preconditioned coefficient matrix, and the associated preconditioned GMRES method converges independently of the mesh size in space and the order of Riesz fractional derivatives.