A Modified Center-of-Mass Conservation Law in Finite-Domain Simulations of the Zakharov--Kuznetsov Equation

TL;DR

This paper analyzes conservation laws in finite-domain simulations of the Zakharov--Kuznetsov equation, identifies boundary-induced drift in a key quantity, and proposes a modified conservation law that accurately captures center-of-mass motion.

Contribution

It introduces a modified center-of-mass quantity that remains conserved in finite-domain numerical simulations of the ZK equation, improving the understanding of boundary effects.

Findings

Most conserved quantities are well preserved in simulations.

Boundary flux causes drift in the original center-of-mass quantity.

The modified quantity remains conserved and accurately describes center-of-mass motion.

Abstract

We investigate conservation laws of the two-dimensional Zakharov--Kuznetsov (ZK) equation, a natural higher-dimensional and non-integrable extension of the Korteweg--de Vries equation. The ZK equation admits three scalar conserved quantities -- mass, momentum, and energy -- represented as $I_1$, $I_2$, and $I_3$, as well as a vector-valued quantity $\bm{I}_4$. In high-accuracy numerical simulations on a finite double-periodic domain, most of these quantities are well preserved, while a systematic temporal drift is observed only in the $x$-component $I_{4x}$. We show that the nontrivial evolution of $I_{4x}$ originates from an explicit boundary-flux contribution, which is induced by fluctuations of the solution and its spatial derivatives at the domain boundaries. We successfully identify the source of the inaccuracy in the numerical solutions. Motivated by this analysis, we define a modified center-of-mass quantity $I_{4x}^{\mathrm{mod}}$ and demonstrate its conservation numerically for single-pulse configurations. The modified quantity thus provides a consistent conservation law for the ZK equation and yields an appropriate description of center-of-mass motion in finite-domain numerical simulations.