Numerical study of soliton stability, resolution and interactions in the 3D Zakharov-Kuznetsov equation

TL;DR

This paper provides a comprehensive numerical analysis of the 3D Zakharov-Kuznetsov equation, confirming soliton stability, exploring decay behaviors, and examining soliton interactions, thus advancing understanding of this non-integrable, subcritical PDE.

Contribution

It offers new numerical evidence for soliton stability, decay patterns, and interaction dynamics in the 3D Zakharov-Kuznetsov equation, extending prior theoretical results.

Findings

Confirmed asymptotic stability of solitons with various perturbations.

Supported the soliton resolution conjecture for different decay rates.

Demonstrated both quasi-elastic and merging soliton interactions with radiation emission.

Abstract

We present a detailed numerical study of solutions to the Zakharov-Kuznetsov equation in three spatial dimensions. The equation is a three-dimensional generalization of the Korteweg-de Vries equation, though, not completely integrable. This equation is $L^2$-subcritical, and thus, solutions exist globally, for example, in the $H^1$ energy space. We first study stability of solitons with various perturbations in sizes and symmetry, and show asymptotic stability and formation of radiation, confirming the asymptotic stability result in \cite{FHRY2020} for a larger class of initial data. We then investigate the solution behavior for different localizations and rates of decay including exponential and algebraic decays, and give positive confirmation toward the soliton resolution conjecture in this equation. Finally, we investigate soliton interactions in various settings and show that there is both a quasi-elastic interaction and a strong interaction when two solitons merge into one, in all cases always emitting radiation in the conic-type region of the negative $x$-direction.