A positivity preserving iterative method for finding the ground states of saturable nonlinear Schr\"odinger equations

In this paper, we propose an iterative method to compute the positive ground states of saturable nonlinear Schr\"odinger equations. A discretization of the saturable nonlinear Schr\"odinger equation leads to a nonlinear algebraic eigenvalue problem (NAEP). For any initial positive vector, we prove that this method converges globally with a locally quadratic convergence rate to a positive solution of NAEP. During the iteration process, the method requires the selection of a positive parameter $\theta_k$ in the $k$th iteration, and generates a positive vector sequence approximating the eigenvector of NAEP and a scalar sequence approximating the corresponding eigenvalue. We also present a halving procedure to determine the parameters $\theta_k$, starting with $\theta_k=1$ for each iteration, such that the scalar sequence is strictly monotonic increasing. This method can thus be used to illustrate the existence of positive ground states of saturable nonlinear Schr\"odinger equations. Numerical experiments are provided to support the theoretical results.