### Abstract

Language | English |
---|---|

Pages | 543-552 |

Number of pages | 9 |

Journal | Journal of Plasma Physics |

Volume | 70 |

Issue number | 5 |

DOIs | |

Publication status | Published - Oct 2004 |

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### Keywords

- ion-acoustic waves
- magnetized plasma
- Zakharov-Kuznetsov equation

### Cite this

*Journal of Plasma Physics*,

*70*(5), 543-552. https://doi.org/10.1017/S0022377803002769

}

*Journal of Plasma Physics*, vol. 70, no. 5, pp. 543-552. https://doi.org/10.1017/S0022377803002769

**The stability of obliquely-propagating solitary-wave solutions to a modified Zakharov-Kuznetsov equation.** / Munro, S.; Parkes, E.J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The stability of obliquely-propagating solitary-wave solutions to a modified Zakharov-Kuznetsov equation

AU - Munro, S.

AU - Parkes, E.J.

PY - 2004/10

Y1 - 2004/10

N2 - 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.

AB - 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.

KW - ion-acoustic waves

KW - magnetized plasma

KW - Zakharov-Kuznetsov equation

UR - http://www.maths.strath.ac.uk/~caas35/m&pJPP04.pdf

UR - http://dx.doi.org/10.1017/S0022377803002769

U2 - 10.1017/S0022377803002769

DO - 10.1017/S0022377803002769

M3 - Article

VL - 70

SP - 543

EP - 552

JO - Journal of Plasma Physics

T2 - Journal of Plasma Physics

JF - Journal of Plasma Physics

SN - 0022-3778

IS - 5

ER -