A numerical method for the fractional Zakharov-Kuznetsov equation

TL;DR

The paper develops a fully discrete Fourier spectral Galerkin method for the two-dimensional fractional Zakharov-Kuznetsov equation with a fractional Laplacian of order $α∈[1,2]$ on a periodic domain. A structure-preserving semi-discrete scheme conserves discrete mass, momentum, and energy, and a modified projection yields optimal spatial convergence, including exponential rates for analytic data. Time discretization uses an integrating-factor RK4 approach to handle stiffness from the fractional dispersion, achieving $O(Δt^4)$ temporal accuracy with stability analyzed in the linearized setting. The authors prove well-posedness, uniform convergence of the semi-discrete solution, and provide comprehensive numerical experiments demonstrating spectral convergence and robust performance across fractional orders. This work offers the first complete numerical analysis for the fZK model and establishes a reliable framework for long-time simulations in nonlinear, nonlocal dispersive systems.

Abstract

This paper develops a fully discrete Fourier spectral Galerkin (FSG) method for the fractional Zakharov-Kuznetsov (fZK) equation posed on a two-dimensional periodic domain. The equation generalizes the classical ZK model to incorporate nonlocal dispersion through a fractional Laplacian of order $α\in [1,2]$. We first propose a semi-discrete FSG scheme in space that preserves the discrete analogs of mass, momentum, and energy. The existence and uniqueness of semi-discrete solutions are established. Using compactness arguments, we prove the uniform convergence of the semi-discrete approximations to the unique solution of the fZK equation for the periodic initial data in $H^{1+α}_{\mathrm{per}}(Ω)$. The method achieves spectral convergence of order $\mathcal{O}(N^{-r})$ for initial data in $H^r_{\mathrm{per}}$ with $r \geq α+1$, and exponential convergence for analytic solutions utilizing a modified projection. An efficient integrating-factor Runge-Kutta time discretization is designed to handle the stiff fractional term, and an error analysis is presented. Numerical experiments validate the theoretical results and demonstrate the method's effectiveness across various fractional orders.

Figures (2)

• Figure 6.1: Comprehensive validation of the FSG method \ref{['fsg:fZK']} for the fZK equation ($\alpha=2$) \ref{['eqn:fZK']}. Panel (a) shows the cross-sectional wave profile at $y=0$. Panel (b) displays the 2D contour plot. Panel (c) provides a 3D surface visualization of the solitary wave profile. Panel (d) presents the spectral convergence analysis. Parameters: $c=1$, $\theta=0$, $T=10$, domain $\Omega = [-20\pi,20\pi]^2$ and time step $\Delta t = 1/(N\|u\|_{L^{\infty}(\Omega)})$.
• Figure 6.2: Soliton interaction dynamics for the fZK equation ($\alpha=2$). Panel (a) shows the wave profile at $y=0$ for different fractional orders $\alpha\in\{1.2,1.5,1.9,2.0\}$ at final time $T=60$. Panel (b) displays the 3D surface visualization of the initial two-soliton configuration ($T=0$). Panel (c) provides a 3D surface visualization during collision ($T=30$). Panel (d) presents the 3D surface visualization after collision ($T=60$). Parameters: $c_1=0.5$, $c_2=0.2$, $\theta_1=0$, $\theta_2=0$, domain $\Omega = [-20\pi,20\pi]^2$, spatial resolution $N=512$, and time step $\Delta t = 1/(N\|u_0\|_{L^{\infty}(\Omega)})$.

Theorems & Definitions (23)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3: Inverse and Sobolev Inequalities
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• Lemma 3.3