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

TL;DR

This paper addresses how finite-domain simulations of the two-dimensional Zakharov--Kuznetsov equation affect its conservation laws. It shows that while scalar invariants $I_1$, $I_2$, and $I_3$ and the $y$-component of the center-of-mass vector $m{I}_4$ are well preserved, the $x$-component $I_{4x}$ experiences a boundary-induced drift. The authors derive the boundary flux term and introduce a modified invariant $I_{4x}^{ ext{mod}}(t)=I_{4x}(t)- ext{(flux accumulation)}$, which restores conservation for isolated pulses and yields consistent centroid dynamics. They demonstrate that centroid motion inferred from the modified invariant reflects uniform translation, and show that a KdV example exhibits a similar correction, indicating broad applicability of this finite-domain conservation-law modification to dispersive PDEs.

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.

Figures (7)

• Figure 1: Snapshots of a traveling solitary pulse of the ZK equation: the initial condition at $t=0$ (left) and the numerical solution at $t=10$ (right). The pulse propagates stably without noticeable deformation. Weak radiative tails, though barely visible at this scale, extend across the computational domain and generate nonzero boundary values.
• Figure 2: Time evolution of the center-of-mass quantities: $x$-component $I_{4x}$ (left) and $y$-component $I_{4y}$ (right). The $I_{4y}$ remains conserved to high accuracy, whereas the $I_{4x}$ exhibits a systematic drift induced by boundary flux contributions associated with weak radiative tails.
• Figure 3: Comparison between the numerically evaluated time derivative $\Delta I_{4x}/\Delta t$ (left) and the analytically derived boundary flux contribution $\mathcal{A}(t)$ defined in Eq. \ref{['eq:I4A']} (right). The two quantities exhibit the same temporal behavior over the entire simulation interval, confirming that the observed drift of $I_{4x}$ originates from the boundary flux contribution.
• Figure 4: Time evolution of the deviation from the initial value for the center-of-mass-type quantity. The solid line labeled "Original" represents $\lvert I_{4x}(t) - I_{4x}(0) \rvert$, while the line labeled "Modified" corresponds to $\lvert I_{4x}^{\mathrm{mod}}(t) - I_{4x}(0) \rvert$.
• Figure 5: Comparison of centroid velocities computed from the original definition ($v_\textrm{c}^{\mathrm{con}}$, based on $I_{4x}$) and from the modified definition ($v_\textrm{c}^{\mathrm{mod}}$, based on $I_{4x}^{\mathrm{mod}}$). Only the modified definition yields a nearly constant centroid velocity, consistent with uniform translational motion of an isolated pulse.