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.

我们在三个空间维度上对Zakharov-Kuznetsov方程的解进行了详细的数值研究。该方程是Korteweg-de Vries方程的三维概括，但不是完全可积分的。这个方程是${\mathrm{大号}}^{\mathrm{2个}}$-subcritical，因此，解决方案在全球范围内都存在，例如， $H^{\mathrm{1个}}$ 能量空间。

我们首先研究大小和对称性各不相同的孤子的稳定性，并显示渐近稳定性和辐射的形成，证实了Farah等人的渐近稳定性结果。（0000）表示较大的初始数据类别。然后，我们研究了不同局部化和衰减速率（包括指数衰减和代数衰减）的解决方案行为，并对该方程中的孤子分辨率猜想给出了肯定的确认。最后，我们研究了各种环境中的孤子相互作用，结果表明，当两个孤子合并成一个孤子时，既有准弹性相互作用又有强相互作用，在所有情况下，总是在负离子的圆锥形区域内发射辐射$X$-方向。