Numerical study of soliton stability, resolution and interactions in the 3D Zakharov-Kuznetsov equation. (English) Zbl 1496.65164
Summary: 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 [L.G. Farah et al., “Asymptotic stability of solitary waves of the 3D quadratic Zakharov-Kuznetsov equation”, Preprint, arXiv:2006.00193] 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 are 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.
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 [L.G. Farah et al., “Asymptotic stability of solitary waves of the 3D quadratic Zakharov-Kuznetsov equation”, Preprint, arXiv:2006.00193] 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 are 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.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65H10 | Numerical computation of solutions to systems of equations |
| 35C08 | Soliton solutions |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B20 | Perturbations in context of PDEs |
Keywords:
Zakharov-Kuznetsov equation; soliton stability; soliton resolution; radiation; soliton interactionReferences:
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