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

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.

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Cited by in corpus (6)

• Dynamics of solutions in the generalized Benjamin-Ono equation: a numerical study
• Asymptotic stability of a finite sum of solitary waves for the Zakharov-Kuznetsov equation
• Dynamics of the collision of two nearly equal solitary waves for the Zakharov-Kuznetsov equation
• Whitham modulation theory for the Zakharov-Kuznetsov equation and transverse instability of its periodic traveling wave solutions
• Numerical study of the transverse stability of line solitons of the Zakharov-Kuznetsov equations
• Spectral property for the 2D Zakharov-Kuznetsov equation