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

S. Munro, E.J. Parkes

Research output: Contribution to journalArticle

23 Citations (Scopus)

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.
LanguageEnglish
Pages543-552
Number of pages9
JournalJournal of Plasma Physics
Volume70
Issue number5
DOIs
Publication statusPublished - Oct 2004

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solitary waves
perturbation
ion acoustic waves
cold plasmas
traveling waves
magnetic fields
wavelengths
ions
electrons

Keywords

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

Cite this

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

In: Journal of Plasma Physics, Vol. 70, No. 5, 10.2004, p. 543-552.

Research output: Contribution to journalArticle

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AU - Parkes, E.J.

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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.

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