• DOI:10.1007/s00332-021-09680-x
• Corpus ID: 211171893

Numerical Study of Zakharov–Kuznetsov Equations in Two Dimensions

@article{Klein2020NumericalSO,
  title={Numerical Study of Zakharov–Kuznetsov Equations in Two Dimensions},
  author={Christian Klein and Svetlana Roudenko and Nikola M. Stoilov},
  journal={Journal of Nonlinear Science},
  year={2020},
  volume={31}
}

• C. Klein, S. Roudenko, N. Stoilov
• Published 18 February 2020
• Materials Science
• Journal of Nonlinear Science

We present a detailed numerical study of solutions to the (generalized) Zakharov–Kuznetsov equation in two spatial dimensions with various power nonlinearities. In the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}$$\end{document}-subcritical case, numerical evidence is presented for the stability of solitons and the soliton resolution for generic initial data. In the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-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 L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}$$\end{document}-critical case, the blow-up appears to be similar to the one in the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}$$\end{document}-critical generalized Korteweg–de Vries equation with the profile being a dynamically rescaled soliton.