Growth rate of instability of obliquely propagating ion-acoustic solitons in a magnetized non-thermal plasma
01 Feb 2001-Journal of Plasma Physics (Cambridge University Press (CUP))-Vol. 65, Iss: 2, pp 131-150
TL;DR: In this paper, the growth rate of instability is obtained correct to order k 2, where k is the wavenumber of a long-wavelength plane-wave perturbation, and the case where the coefficient of the nonlinear term in the KdV--ZK equation vanishes is also considered.
Abstract: The Korteweg-de Vries-Zakharov-Kuznetsov (KdV-ZK) equation, governing the behaviour of long-wavelength weakly nonlinear ion-acoustic waves propagating obliquely to an external uniform magnetic field in a non-thermal plasma, admits soliton solutions having a sech 2 profile. The higher-order growth rates of instability are obtained using the method developed by Allen and Rowlands [J. Plasma Phys. 50, 413 (1993); 53, 63 (1995)]. The growth rate of instability is obtained correct to order k 2 , where k is the wavenumber of a long-wavelength plane-wave perturbation. The case where the coefficient of the nonlinear term in the KdV--ZK equation vanishes is also considered.
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TL;DR: In this article, the Sagdeev potential approach has been used to study the arbitrary amplitude dust acoustic solitary waves and double layers in nonthermal plasma consisting of negatively charged dust grains, non-thermal ions, and isothermal electrons including the effect of dust temperature.
Abstract: A computational scheme has been developed to study the arbitrary amplitude dust acoustic solitary waves and double layers in nonthermal plasma consisting of negatively charged dust grains, nonthermal ions, and isothermal electrons including the effect of dust temperature. The Sagdeev potential approach, which is valid to study the arbitrary amplitude solitary waves and double layers, has been employed. The computation has been carried out over the entire interval of β1:0≤β1<βM. This β1 is a parameter associated with the nonthermal distribution of ions and βM is the upper bound of β1. Depending on the nature of existence of solitary waves and double layers, the interval for β1 can be broken up into four disjoint subintervals holding the other parameters fixed. By nature of existence of solitary waves and double layers, it is meant that in some subinterval only negative potential solitary waves can exist, whereas in another both negative and positive potential solitary waves can coexist along with a double ...
29 citations
TL;DR: In this article, it was shown that the compressive or rarefactive nature of the ion-acoustic solitary wave solution of the KdV-ZK equation does not depend on the ion temperature.
Abstract: The Korteweg-de Vries-Zakharov-Kuznetsov (KdV-ZK) equation describes the nonlinear behaviour of long-wavelength weakly nonlinear ion-acoustic waves propagating obliquely to an external uniform (space independent) static (time independent) magnetic field in a plasma consisting of warm adiabatic ions and a superposition oftwo distinct population of electrons, one due to Cairns et al. (1995 Geophys. Res. Lett. 22, 2709), which generates the fast energetic electrons, and the other the well-known Maxwell-Boltzman distributed electrons. It is found that the compressive or rarefactive nature of the ion-acoustic solitary wave solution of the KdV-ZK equation does not depend on the ion temperature if σ c 1, where σ c is a function of β 1, n sc and σ sc . This β 1 is the non-thermal parameter associated with the non-thermal velocity distribution function of electrons (Cairns et al.), n sc is the ratio of the unperturbed number density of the isothermal electrons to that of the non-thermal electrons and σ sc is the ratio of the average temperature of the non-thermal electrons to that of the isothermal electrons. The KdV-ZK equation describes compressive or rarefactive ion-acoustic solitary wave according to whether σ c 1. When 0 ≤ σ c ≤ 1, the KdV-ZK equation describes compressive or rarefactive ion-acoustic solitary wave according to whether a > σ c or a < σ c , where a is the ratio of the average temperature of ions to the effective temperature of electrons. If a takes the value at with 0 ≤ σ c ≤ 1, the coefficient of the nonlinear term of the KdV-ZK equation vanishes and for this case the nonlinear evolution equation of the ion-acoustic wave is a modified KdV-ZK (MKdV-ZK) equation. It is found that the four-dimensional parameter space, originated from the physically admissible values of the four-parameters β 1, a, σ sc and n sc of the present extended plasma system, can be decomposed into five mutually disjoint subsets with respect to the critical values of the different parameters, and the nonlinear behaviour of the same ion acoustic wave in those subsets can be described by different modified KdV-ZK equations. A general method of perturbation of the dependent variables has been developed to obtain the different evolution equations. The applicability of the different evolution equations and their solitary wave solutions (along with the conditions for their existence) have been investigated analytically and graphically.
21 citations
TL;DR: In this article, the authors used the multiple-scale perturbation method developed by Allen and Rowlands to calculate the initial growth rate of a small, transverse, long-wavelength perturbations to these solitary-wave solutions.
Abstract: In certain circumstances, small amplitude, weakly nonlinear ion-acoustic waves in a magnetized plasma are governed by a Zakharov-Kuznetsov equation or by a reduced form of the equation. Both equations have a plane solitary travelling-wave solution that propagates at an angle αto the magnetic field. The multiple-scale perturbation method developed by Allen and Rowlands is used to calculate the initial growth rate of a small, transverse, long-wavelength perturbation to these solitary-wave solutions. Previous results in the literature are corrected. A numerical determination of the growth rate is given. For k[mid R:] secα[mid R:][double less-than sign]1, where k is the wavenumber of the perturbation, there is excellent agreement between our analytical and numerical results.
18 citations
Cites background or methods or result from "Growth rate of instability of obliq..."
...The second aim of the present paper is to attempt to apply the method correctly and to point out the errors in Bandyopadhyay and Das (2001)....
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...Ghosh and Das (1994) derived (1.4) for a closely related scenario. (Equation (1.3) can be obtained from (1.4) by suitably rescaling the variables, or by artificially setting A′ = −1, B′ = 1 and C ′ = 1.) Incidentally, the ZK and rZK equations occur in other physical contexts. For example, Melkonian and Maslowe (1989) have shown that, under different circumstances, long waves on thin films may be governed by either the ZK equation or the rZK equation in their two-dimensional form (i.e. with no y dependence); Nozaki (1981) has shown that vortices in plasma drift waves are governed by the twodimensional ZK equation....
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...In terms of the notation in the present paper, Bandyopadhyay and Das (2001) obtained γ1 as given by (2.28), and hence γ1r as in (3.1), by two methods: firstly they applied the consistency condition (2.50) with j = 2 and secondly they solved (2.18) for δu2 and then removed the exponentially secular…...
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...Ghosh and Das (1994) derived (1.4) for a closely related scenario. (Equation (1.3) can be obtained from (1.4) by suitably rescaling the variables, or by artificially setting A′ = −1, B′ = 1 and C ′ = 1.) Incidentally, the ZK and rZK equations occur in other physical contexts. For example, Melkonian and Maslowe (1989) have shown that, under different circumstances, long waves on thin films may be governed by either the ZK equation or the rZK equation in their two-dimensional form (i....
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...The calculation of γ up to second order was considered by Bandyopadhyay and Das (2001)....
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TL;DR: In this article, the authors derived the alternative solitary wave solution of variable width instead of fixed width of the combined MKdV-KdVV-ZK equation along with the condition for its existence and found that this solution assumes the sech profile of the modified Korteweg-de Vries-Korteteg-De Vries,Zakharov-Kuznetsov) equation, when the coefficient of the nonlinear term of the KdV -ZK (KortewEG-deVries-Zakarov-
Abstract: The purpose of this paper is to present the recent work of Das et al. [J. Plasma Phys. 72, 587 (2006)] on the existence and stability of the alternative solitary wave solution of fixed width of the combined MKdV-KdV-ZK (Modified Korteweg-de Vries-Korteweg-de Vries-Zakharov-Kuznetsov) equation for the ion-acoustic wave in a magnetized nonthermal plasma consisting of warm adiabatic ions in a more generalized form. Here we derive the alternative solitary wave solution of variable width instead of fixed width of the combined MKdV-KdV-ZK equation along with the condition for its existence and find that this solution assumes the sech profile of the MKdV-ZK (Modified Korteweg-de Vries-Zakharov-Kuznetsov) equation, when the coefficient of the nonlinear term of the KdV-ZK (Korteweg-de Vries-Zakharov-Kuznetsov) equation tends to zero. The three-dimensional stability analysis of the alternative solitary wave solution of variable width of the combined MKdV-KdV-ZK equation shows that the instability condition and the first order growth rate of instability are exactly the same as those of the solitary wave solution (the sech profile) of the MKdV-ZK equation, when the coefficient of the nonlinear term of the KdV-ZK equation tends to zero.
15 citations
TL;DR: In this paper, the authors considered the nonlinear dynamics of ion-acoustic waves in a plasma consisting of warm adiabatic ions and non-thermal electrons having a vortex-like velocity distribution function (due to the presence of trapped electrons), immersed in a uniform (space independent) and static (time-independent) magnetic field.
Abstract: . The solitary structures of the ion-acoustic waves have been considered in a plasma consisting of warm adiabatic ions and non-thermal electrons (due to the presence of fast energetic electrons) having a vortex-like velocity distribution function (due to the presence of trapped electrons), immersed in a uniform (space-independent) and static (time-independent) magnetic field. The nonlinear dynamics of ion-acoustic waves in such a plasma is governed by the Schamel's modified Korteweg-de Vries-Zakharov-Kuznetsov (S-ZK) equation. This equation admits solitary wave solutions having a profile sech 4 . When the coefficient of the nonlinear term of this equation vanishes, the vortex-like velocity distribution function of electrons simply becomes the non-thermal velocity distribution function of electrons and the nonlinear behaviour of the same ion-acoustic wave is described by a Korteweg-de Vries-Zakharov-Kuznetsov (KdV-ZK) equation. This equation admits solitary wave solutions having a profile seeh 2 . A combined S-KdV-ZK equation more efficiently describes the nonlinear behaviour of an ion-acoustic wave when the vortex-like velocity distribution function of electrons approaches the non-thermal velocity distribution function of electrons, i.e. when the contribution of trapped electrons tends to zero. This combined S-KdV-ZK equation admits an alternative solitary wave solution having a profile different from either sech 4 or sech 2 . The condition for the existence of this alternative solitary wave solution has been derived. It is found that this alternative solitary wave solution approaches the solitary wave solution (the sech 2 profile) of the KdV-ZK equation when the contribution of trapped electrons tends to zero. The three-dimensional stability of these solitary waves propagating obliquely to the external uniform and static magnetic field has been investigated by the multiple-scale perturbation expansion method of Allen and Rowlands. The instability condition and the growth rate of the instability have been derived at the lowest order. It is also found that the instability condition and growth rate of instability of the alternative solitary waves are exactly the same as those of the solitary waves as determined from the KdV-ZK equation (the sech 2 profile) when the contribution of trapped electrons tends to zero.
12 citations