# The stability of obliquely-propagating solitary-wave solutions to a modified Zakharov-Kuznetsov equation

Munro, S. and Parkes, E.J.
(2004)
*The stability of obliquely-propagating solitary-wave solutions to a modified Zakharov-Kuznetsov equation.*
Journal of Plasma Physics, 70 (5).
pp. 543-552.

## Abstract

In the context of ion-acoustic waves in a magnetized plasma comprising cold ions and non-isothermal electrons, the present authors have previously shown small amplitude, weakly nonlinear waves to be governed by a modified version of the Zakharov-Kuznetsov equation. In this paper, we consider a plane solitary travelling-wave solution to this equation that propagates at an angle $alpha$ to the magnetic field, where $0,{le},alpha,{le},pi$. The multiple-scale perturbation method developed by Allen and Rowlands is used to calculate the growth rate of a small, transverse, long-wavelength perturbation. To first order there is instability for $0,{le},sinalpha,{<},sinalpha_{ m c}$, where the critical angle $alpha_{ m c}$ is identified. At second order, the singularity which apparently occurs in the growth rate at $alpha,{=},alpha_{ m c}$ is removed by using a method devised by Allen and Rowlands; then it is found that there is also instability for $sinalpha,{ge},sinalpha_{ m c}$. A numerical determination for the growth rate is given for the instability range $0,{<},k,{<},3$, where $k$ is the wavenumber of the perturbation. For $k|{ m sec},alpha|,{ll},1$, there is excellent agreement between the analytical and numerical results. The results in this paper agree qualitatively with those of Allen and Rowlands for the Zakharov-Kuznetsov equation.

Item type: | Article |
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ID code: | 2191 |

Keywords: | ion-acoustic waves, magnetized plasma, Zakharov-Kuznetsov equation, Mathematics, Physics, Condensed Matter Physics |

Subjects: | Science > Mathematics Science > Physics |

Department: | Faculty of Science > Mathematics and Statistics Unknown Department |

Depositing user: | Strathprints Administrator |

Date deposited: | 06 Jan 2007 |

Last modified: | 22 Mar 2017 09:19 |

Related URLs: | |

URI: | https://strathprints.strath.ac.uk/id/eprint/2191 |

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