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

TL;DR

This work tackles the coupled nonlinear space fractional Ginzburg-Landau equations with a fractional Laplacian, addressing analytical challenges from nonlocal dispersion and strong nonlinearity. It establishes a priori estimates and the well-posedness of weak solutions in appropriate Sobolev spaces, and introduces a fourth-order numerical differential formula for the fractional Laplacian based on refined Riemann-Liouville/Riesz approximations. A two-level implicit time-stepping scheme combined with a Padé-approximation for the exponential operator provides second-order temporal and fourth-order spatial accuracy, with rigorous proofs of boundedness, existence, uniqueness, and convergence, alongside an efficient iterative solver. Numerical experiments validate the theory, demonstrate high accuracy, and reveal the influence of the fractional order α on solution dynamics, underscoring the method’s potential for simulating fractal-media phenomena. The framework offers a robust and efficient approach extendable to other spatial fractional PDEs and to more complex CNLS-type systems.

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.

Figures (4)

• Figure 4.1: The profile of the evolution of $|U|$ and $|V|$ (left), and contour plot of $|U|$ and $|V|$ (right), for different values of $\alpha$ .
• Figure 4.2: Shift of profiles of $|U|$ and $|V|$ at different times $t=0.1,1,5,10$ for different values of $\alpha$ .
• Figure 4.3: Plots of position density of $|U|$ and $|V|$ (left), and contour plot of $|U|$ and $|V|$ (right), for different $\gamma_1$ and $\gamma_2$.
• Figure 4.4: The discrete norm $\left\|U\right\|_h^2+\left\|V\right\|_h^2$ with different $\gamma_1$ and $\gamma_2$ for different values of $\alpha$ .

Theorems & Definitions (47)

• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 2.5
• proof
• Theorem 2.6
• proof
• Lemma 3.1
• Lemma 3.2