Regularized splitting spectral method for space-fractional logarithmic Schrödinger equation

Abstract

In this paper, we study a regularized Lie-Trotter splitting spectral method for a regularized space-fractional logarithmic Schrödinger equation by introducing a small regularized parameter 0<ε≪1. The regularized method can be used to avoid numerical blow-up in the space-fractional logarithmic Schrödinger equation. The regularized space-fractional logarithmic Schrödinger equation is proved to approximate the space-fractional logarithmic Schrödinger equation with linear convergence rate O(ε). The proposed numerical scheme can preserve the discrete mass and step-energy for regularized space-fractional logarithmic Schrödinger equation. The first order convergence in time for the numerical method is rigorous proved for the regularized space-fractional logarithmic Schrödinger equation. Due to the appearance of fractional Laplace operator with order α, we can adjust the parameters of order to make the equation show much more dynamic characteristics than the classical logarithmic Schrödinger equation. Numerical simulations for 1D case based on the Fourier spectral approximation in space are presented to validate the theoretical analysis.