Locally Optimal Preconditioned Conjugate Gradient Method for Computing Ground State of Space-Fractional Nonlinear Schrödinger Equation

Abstract

In this paper, we investigate numerical methods for computing the ground state of space-fractional nonlinear Schrödinger equation. We focus on extending the locally optimal preconditioned conjugate gradient method, originally designed for linear eigenvalue problems, to address this nonlinear fractional framework. Through comparative numerical experiments with the discrete gradient flow method, we demonstrate that the locally optimal preconditioned conjugate gradient method achieves significantly superior efficiency in many scenarios. This advantage is particularly pronounced in multidimensional problems and cases involving lower fractional derivative orders. The results highlight the adaptability of the locally optimal preconditioned conjugate gradient approach to eigenproblem of fractional nonlinear systems, offering a robust and high efficient computational alternative for high-dimensional or low-order fractional derivative settings.