A numerical method for the fractional Zakharov-Kuznetsov equation

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 by replacing the Laplacian with a fractional Laplacian of order $\alpha\in(0,2]$, thereby covering the classical ZK equation $\alpha=2$, the higher-dimensional Benjamin--Ono--ZK equation $\alpha=1$, and weaker fractional-dispersion regimes $0<\alpha<1$. We first propose a semi-discrete FSG scheme in space that preserves the discrete analogues of mass, momentum, and Hamiltonian energy. Using periodic Kato--Ponce product and commutator estimates, we prove local-in-time uniform Sobolev bounds and strong convergence of the semi-discrete approximations to the unique strong solution in $C([0,\bar T];L^2_{\mathrm{per}}(\Omega))$, for the initial condition in $H^s_{\mathrm{per}}(\Omega)$, $s\geq 2+\alpha$, and, as by product, we show that the existence and uniqueness of fZK equation in $L^\infty(0,\bar T;H^s_{\mathrm{per}}(\Omega))\cap W^{1,\infty}(0,\bar T;L^2_{\mathrm{per}}(\Omega))$. We then introduce a modified projection adapted to the fractional transport dispersive operator and prove optimal spatial error estimates of order $\mathcal O(N^{-r})$ for $r>2+\alpha$, together with exponential convergence for analytic solutions. An integrating-factor fourth-order four-stage Runge--Kutta time discretization is used to integrate the stiff fractional dispersive part exactly, and a fourth-order temporal error estimate is obtained under a high-regularity nonlinear stability assumption. Numerical experiments illustrate the accuracy, fractional-order dependence, and fully discrete conservation drift of the method.