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

JF - Journal of Plasma Physics

SN - 0022-3778

IS - 5

ER -