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

• Published in: Journal of scientific computing Vol. 84; no. 3; p. 41
• Main Author: Liu, Ching-Sung
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2020; Springer Nature B.V
• Subjects: Algorithms; Approximation; Boundary conditions; Computational Mathematics and Numerical Analysis; Convergence; Eigenvalues; Eigenvectors; Ground state; Iterative methods; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Parameters; Schrodinger equation; Theoretical; 65F50; Secondary 15B48; Quadratic convergence; Saturable nonlinearity; Ground states; 15B99; Schrödinger equations; Positivity preserving; 65F15; Primary 15A18; matrix
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-020-01297-8

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