A Positivity Preserving Iterative Method for Finding the Ground States of Saturable Nonlinear Schrödinger Equations

Abstract

In this paper, we propose an iterative method to compute the positive ground states of saturable nonlinear Schrodinger equations. A discretization of the saturable nonlinear Schrodinger 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 kth 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 Schrodinger equations. Numerical experiments are provided to support the theoretical results.