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arXiv · 2002.07886

Numerical study of Zakharov-Kuznetsov equations in two dimensions

C. Klein·S. Roudenko·N. Stoilov
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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.

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C. Klein, S. Roudenko, N. Stoilov. 2020-02-18. Numerical study of Zakharov-Kuznetsov equations in two dimensions. https://doi.org/10.1007/s00332-021-09680-x

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