General formulation for electrostatic solitons in multicomponent nonthermal plasmas

A general formulation is proposed for electrostatic solitons in multicomponent plasmas consisting of one cold species described by the fluid model and multiple hot components described by the kinetic equation. For hot species two types of generalized velocity distributions are adopted: the kappa function and the highly nonthermal distribution with two free parameters which incorporate the widely used nonthermal distribution proposed by Cairns et al (1995 Geophys. Res. Lett. 22 2709) as a special limit. The general expressions for the KdV equations and the Sagdeev potential, as well as the conditions for electrostatic solitons with positive or negative polarity, are derived, which may give rise to various solutions for acoustic solitons in multicomponent plasmas with generalized nonthermal velocity distributions.


Introduction
Plasmas consisting of multiple species of charged particles of different mass and electric charge, such as electron-positron-ion plasma or dusty plasma, may exist in many space and astrophysical environments as well as in the laboratory [1][2][3][4][5][6][7][8][9]. Due to the lack of collisions, in multispecies plasmas each component may possess different velocity distributions deviating from the thermal equilibrium. Under some circumstances more than one distribution function may even be necessary to describe the nonthermal behaviors of hot charged particles of the same species arising from different origins [10]. In particular, some observations have shown that the electron distribution in solar system plasma environments often exhibit a high-energy tail and may be described by the superposition of two Maxwellian distributions [11,12] or a non-Maxwellian kappa distribution [13][14][15][16]. Theoretically, it is important to incorporate the nonthermal effects with generalized velocity distributions in the model equations which may suitably describe the properties of collisionless plasmas.
The electrostatic solitary wave (ESW) has been widely observed in space plasma environments [17][18][19][20] and in the laboratory [21,22]. In electron-ion plasmas the formation of the ESW is due to the balance between the nonlinear steepening of the acoustic wave and the dispersive effect arising from the inertia of the cold component. Theoretically, the weakly nonlinear ESW may be described by the Korteweg-de Vries (KdV) equation [23] while the fully nonlinear solutions of the ESW may be solved by the Sagdeev potential analysis [24]. Recently we have extended the theoretical formulation of ESWs to include the nonthermal effects for electron-ion plasmas [25]. In that study the hot component is described by the kappa function, proposed by Cairns et al [26] (referred to as the Cairns et al distribution), where a is a free parameter shaping the distribution and . The latter distribution function may lead to the ESW with depleted density (known as negative solitons) observed by the spacecraft [17,19] while the Maxwellian distribution may give rise to only positive electrostatic solitons. Both types of nonthermal distributions have also been adopted by various authors to study the ESW in various plasma systems, such as the two-electron-temperature plasma [27][28][29], electron-positron-ion plasmas [30][31][32][33][34][35][36], and dusty plasmas [37][38][39].
Hau and Fu [40] found that for the condition κ the kappa function may be expanded as 2 , referred to as the highly nonthermal distribution, which bears a similar form to the Cairns et al distribution and may recover the Maxwellian distribution in the limit κ → ∞. Note that both the Cairns et al distribution and the highly nonthermal distribution may describe the features of nonmonotonic shoulder-like or bump-on-tail velocity distribution [41] while the kappa function is simply monotonic with a high tail feature. All three types of nonthermal distributions may suitably describe the various velocity distributions of nonthermal electrons observed in space plasma environments [10][11][12][13][14][15][16]41]. Chuang and Hau [42] have developed a general formulation for acoustic solitons in three-component plasmas consisting of one cold fluid and two hot components described by the combination of the kappa function and the highly nonthermal distribution function. In their study the charge of each component is a free parameter while the charge neutrality for background plasmas is imposed. In light of the observational evidence for the existence of multiple components in space plasma environments [5][6][7][8][9], a number of theoretical models for electrostatic solitons in four-component plasmas have been developed [43][44][45][46]. In the present study we extend the formulations of Chuang and Hau [25,42] and Chuang [47] to more general cases with multiple (unlimited) hot components described by the combination of the kappa function and the more general nonthermal distribution function of which may give rise to various solutions for electrostatic acoustic solitons in multicomponent plasmas with different velocity distributions, which is a distinct feature of collisionless plasmas. Note that the term 'multicomponent' is more general than the 'multispecies' based on the consideration that the same species, such as electrons, may possess more than one distribution function [6,11,12,48]. The development of such a unified formulation is important not only for nonlinear plasma physics but also for space and astrophysical applications. This paper is organized as follows. In section 2 the model equations, along with the nonthermal distribution functions adopted in the present study, are presented and the corresponding linear dispersion relations for acoustic waves are derived. Section 3 shows the derivation of the KdV equation and section 4 presents the Sagdeev potential analysis. The summary is given in section 5.

Model equations and distribution functions
We consider the multicomponent plasmas comprising one cold component and multiple hot components, described by the fluid and kinetic models, respectively [49][50][51][52]. The model equations include the continuity, momentum and Poisson's equations.
x has been used and the subscript c denotes the cold species. The variables n, m, u, and q represent the number density, mass, flow velocity and electric charge of charged particles, respectively. The subscripts α and β denote the hot species with kappa and general nonthermal distributions, respectively. In particular, for the problem under consideration, the kappa function has the following form [40].  (4) in which α n 0, is the unperturbed number density of α species; Γ is the gamma function; κ is the kappa parameter which for κ → ∞ corresponds to the Maxwellian case. Note that a slightly different kappa function from the above expression has been adopted and applied to study the acoustic solitons in two-component plasmas [53,54]. Comparisons between two different kappa functions and the obtained results have also been made [25,55]. As for the β species we adopt the more general nonthermal distribution with the following form.
the above expression may, respectively, recover the 3D Cairns et al nonthermal distribution and the slightly different nonthermal distribution proposed by Hau and Fu [40], while for = = a a 0 1 2 it describes the Maxwellian distribution. Note that for uniform plasmas, i.e. ϕ( ) = x 0, the distribution functions are only functions of velocity, in particular, The corresponding thermal pressure for kappa distribution may be calculated as , implying that κ > 3/2. While for the more general nonthermal distribution the thermal pressure in uniform plasmas is for various a values, and the highly nonthermal function for various κ′ values. As indicated, for small κ′ values the profile describes the nonmonotonic bump-on-tail distribution while for large κ′ values, say, κ > ′ 10, the kappa function is recovered. Note that both Cairns et al and the highly nonthermal distributions may describe nonmonotonic and bump-on-tail velocity distributions in contrast to the monotonic kappa distribution, and may suitably be termed as the highly nonthermal distribution in the present study. As shown in figure 1, the features for both distributions are not quantitatively the same; in particular, the high tail peak may possibly exceed the central distribution in the Cairns et al case, which is not considered in our previous study for three-component plasmas [42]. The combination of both distributions may give rise to more generalized functions which may flexibly be fitted with the observations or simulations by suitably adjusting the free parameters in equation (6).
Based on the distributions in equations (4) and (5), the number density, and for β species is . Equations(1)-(3), (10) and (11) are the model equations used in this study. In addition, the quasi-neutrality condition for the unperturbed density is imposed, i.e.
. It is important to obtain the characteristic speed of the model equations in uniform plasmas before analyzing the nonlinear model. We assume that all perturbed quantities have the form ω and there is no background flow velocity. Linearization of the model equations then yields: Equations (12) and (13) The plasma frequency for cold species is simply . Note that the effective temperature for multiple components defined in equation (18) differs from the earlier studies and has the advantage of yielding physically meaningful C S,eff and λ D,eff which will be used as the normalization constants in the following calculations. The correct definition for the effective temperature in equation (18) and the corresponding sound speed is important in that the model calculations would not then yield subsonic solitons as obtained in earlier studies [25,30,42,57] which are not physically possible, since steady solitons require the balance between nonlinear steepening and dispersive effects in collisionless plasmas.

KdV equation
We now proceed to adopt the reductive perturbation technique [23] to derive the KdV equation from the model equations for describing the weakly nonlinear ESW. First, we introduce two stretched coordinates, ξ ϵ = ( − ) ′ ′ x u t 1/2 0 and τ ϵ = ′ t 3/2 (where ϵ is a smallness parameter and u 0 is the speed of ESW normalized by C S,eff ) with dimensionless variables,  , and substituting the expanded variables into equations (19)-(23), we obtain the equations of the first order in ϵ and the second order in ϵ, as follows.
for which the parameters b 1 and b 2 are defined as and   (32) in a frame of reference moving with the speed v 0 (normalized by C S,eff ) as as ζ → ∞, where ζ is a transformed coordinate defined as ζ ξ τ = − v 0 . In equation (33) the amplitude is ϕ = v c 3 / m 0 1 and the width is The polarity of ϕ′ ( ) 1 is thus determined by the sign of the coefficient

Sagdeev potential
The fully nonlinear solutions for electrostatic acoustic solitons can be obtained by solving the dimensionless model equations in the stationary frame of reference, ζ = − ′ ′ x M t S , the result being for which the definitions for dimensionless variables are the same as in section 3. The Mach number M S is defined as the moving speed of the ESW with respect to C S,eff , and ϕ ( ) ′ V is the Sagdeev potential. The expressions for ′ α n and ′ β n are given in equations (22) Equation (37) may be integrated as for which the condition of ϕ ϕ ϕ ( ) To form a soliton, the following conditions shall be satisfied: ; and (iii) the number density of cold species in equation (39) has to be real. The The first condition ϕ ( then results in the constraint of M S and its lower limit M LL as  (43) implying that the moving speed of nonlinear solitons has to be larger than the effective sound speed C S,eff . By imposing the third condition, i.e. ′ n h being real, the threshold condition for the normalized electric potential ϕ ϕ crit may be obtained from equation (39) and by replacing ϕ′ with ϕ ′ crit in equation (41) we may obtain the upper limit value of M S .
The polarity condition for nonlinear solitons can be derived from the third derivative of ϕ ( ) ′ V at ϕ = ′ 0 which has the following form In particular, > P 0 SP and < P 0 SP correspond to the solitons with positive (ϕ > ′ 0) and negative (ϕ < ′ 0) potentials, respectively. A comparison between equation (34) and equation (45) shows that = P P 2 SP KdV , i.e. the criterion for the electric polarity of acoustic solitons derived from both the KdV and Sagdeev potential approaches is consistent. If we apply the condition to the electron-ion plasma with electrons being described by the kappa distribution, the condition for anomalous negative ion acoustic solitons is simply κ + ( ) < 2 1/ 4 0 2 , implying no real κ value to satisfy the condition [25]. It can further be shown that the more general nonthermal distribution including the Cairns et al and the highly nonthermal distributions proposed by Hau and Fu [40] may possibly lead to the formation of negative acoustic solitons [42].

Summary
In this paper we have obtained the general formulations for both weakly nonlinear and fully nonlinear electrostatic acoustic solitons in multicomponent nonthermal plasmas. In the model the cold species with finite inertia is described by the fluid model, and the hot components follow the kappa distribution and/or the newly proposed generalized nonthermal distributions which may recover various distribution forms such as the Cairns et al nonthermal distribution adopted in many earlier studies. In the formulation the charges for both cold and hot species are unspecified and the model can suitably be applied to various plasma systems such as the electron-positron-ion and dusty plasmas etc. The present study is an extension of our previous formulation for three-component plasmas [25,42] and may also serve as a unified model for the existing studies of electrostatic solitary waves in two, three and four component plasmas with specified species. The present model may also recover the three models for three-component plasmas discussed in Chuang and Hau [42], respectively, for the parameter values of α α α = , 1 2 , β = 0 (model A), α = 0, β β β = , 1 2 (model B), and α α = 1 , β β = 1 (model C). Note that the correct definition for the effective temperature in multiple components of charged particles with different velocity distributions in equation (18) has yielded a physically meaningful effective sound speed and Debye length used as the normalization quantities in the calculations which are different from the previous definitions [25,42]. As a result, in the KdV and Sagdeev potential analyses the propagation speeds of weakly and fully nonlinear solitons are, respectively, to be the effective sound speed and supersonic (greater than the effective sound speed). One important application of incorporating nonthermal effects in the formulation is due to its capability in producing soliton solutions with anomalous electric polarity. In this study the polarity conditions for both weakly and fully nonlinear solitons are derived, which are essentially the same and can be used to analyze the possible existence of anomalous solitons in general nonthermal multicomponent plasmas. In particular, equation (45) can easily be applied to ordinary electron-ion plasmas which shows that the model with nonthermal electrons being described by the kappa velocity distribution cannot yield anomalous solitons which, however, can be achieved by more general nonthermal distributions [25].