Numerical Study of Zakharov–Kuznetsov Equations in Two Dimensions

Abstract

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.