Mathematical analysis and numerical simulation of coupled nonlinear space-fractional Ginzburg-Landau equations

TL;DR

This paper provides a rigorous mathematical analysis and develops a high-order numerical scheme for coupled nonlinear space-fractional Ginzburg-Landau equations, addressing challenges posed by fractional derivatives and nonlinearity.

Contribution

It introduces a novel fourth-order numerical method and proves its stability, convergence, and well-posedness for the complex coupled fractional system.

Findings

Proved a priori estimates and well-posedness of the system

Developed a fourth-order implicit difference scheme

Demonstrated the efficiency and accuracy through numerical examples

Abstract

The coupled nonlinear space fractional Ginzburg-Landau (CNLSFGL) equations with the fractional Laplacian have been widely used to model the dynamical processes in a fractal media with fractional dispersion. Due to the existence of fractional power derivatives and strong nonlinearity, it is extremely difficult to mathematically analyze the CNLSFGL equations and construct efficient numerical algorithms. For this reason, this paper aims to investigate the theoretical results about the considered system and construct a novel high-order numerical scheme for this coupled system. We prove rigorously an a priori estimate of the solution to the coupled system and the well-posedness of its weak solution. Then, to develop the efficient numerical algorithm, we construct a fourth-order numerical differential formula to approximate the fractional Laplacian. Based on this formula, we construct a high-order implicit difference scheme for the coupled system. Furthermore, the unique solvability and convergence of the established algorithm are proved in detail. To implement the implicit algorithm efficiently, an iterative algorithm is designed in the numerical simulation. Extensive numerical examples are reported to further demonstrate the correctness of the theoretical analysis and the efficiency of the proposed numerical algorithm.