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 -subcritical case, numerical evidence is presented for the stability of solitons and the soliton resolution for generic initial data. In the -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 -critical case, the blow-up appears to be similar to the one in the -critical generalized Korteweg-de Vries equation with the profile being a dynamically rescaled soliton.
Cite
@article{arxiv.2002.07886,
title = {Numerical study of Zakharov-Kuznetsov equations in two dimensions},
author = {C. Klein and S. Roudenko and N. Stoilov},
journal= {arXiv preprint arXiv:2002.07886},
year = {2021}
}