We present a detailed numerical study of solutions to the (generalized) Zakharov–Kuznetsov equation in two spatial dimensions with various power nonlinearities. In the \(L^{2}\)-subcritical case, numerical evidence is presented for the stability of solitons and the soliton resolution for generic initial data. In the \(L^2\)-critical and supercritical cases, solitons appear to be unstable against both dispersion and blow-up. It is conjectured that blow-up happens in finite time and that blow-up solutions have some resemblance of being self-similar, i.e., the blow-up core forms a rightward moving self-similar type rescaled profile with the blow-up happening at infinity in the critical case and at a finite location in the supercritical case. In the \(L^{2}\)-critical case, the blow-up appears to be similar to the one in the \(L^{2}\)-critical generalized Korteweg–de Vries equation with the profile being a dynamically rescaled soliton.

我们提出了在（二维）Zakharov–Kuznetsov方程在具有不同功率非线性的两个空间维度上求解的详细数值研究。在\（L ^ {2} \）次临界情况下，提供了数值证据，证明了孤子的稳定性和一般初始数据的孤子分辨率。在\（L ^ 2 \）临界和超临界情况下，孤子对分散和爆炸似乎都不稳定。推测爆炸发生在有限的时间，并且爆炸解与自相似有些相似，即爆炸芯形成向右移动的自相似类型的重新缩放轮廓，爆炸发生在临界情况下的无穷大和超临界情况下的有限位置。在\（L ^ {2} \）中临界情况下，爆炸类似于\（L ^ {2} \）临界广义Korteweg-de Vries方程中的爆炸，轮廓是动态缩放的孤子。