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

Christian Klein (1) , Svetlana Roudenko (2) , Nikola Stoilov (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 UNF - University of North Florida [Jacksonville]

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 Farah et al. (0000) 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. (C) 2021 Elsevier B.V. All rights reserved.

Mots clés: Zakharov-Kuznetsov equation; Soliton stability; Soliton resolution; Radiation; Soliton interaction

Domaines: Mathématiques [math]; Physique [physics]

Christian Klein, Svetlana Roudenko, Nikola Stoilov. Numerical study of soliton stability, resolution and interactions in the 3D Zakharov–Kuznetsov equation. Physica D: Nonlinear Phenomena, 2021, 423, pp.132913. ⟨10.1016/j.physd.2021.132913⟩. ⟨hal-03303233⟩