The Cairns-distributed electrons and positive ions in a plasma made up of a negative dust fluid are taken into account to examine the presence of arbitrary amplitude dust-acoustic solitons. It has previously been noted that nonthermal ions and thermal electrons generate both compressive and rarefactive solitons. In this paper, we noticed that nonthermal electrons together with nonthermal ions are responsible for producing the rarefactive solitons. It is found that the Sagdeev potential strongly depends on plasma parameters, such as nonthermal index α and Mach number, which, in turn, influence the Sagdeev potential and solitons significantly. We also found that the critical match number and height of soliton increase with the nonthermal parameter α. We further note that the Sagdeev potential as a function of potential φ becomes more negative and the amplitude of the soliton also enhances as the value of Mach number rises. It is concluded that the model presented here based on nonthermal ions and electrons in a negative dust fluid provides a worthy interpretation for electrostatic solitons observed in space plasmas.

Electrostatic solitary structures (ESS) have long been believed to be observed in the Earth’s magnetosphere, such as the plasma sheet boundary layer (PBSL),1,2 the magnetosheath,3 significant currents within the accelerated auroral region,4 the bow shock,5 and the solar wind.6–8 Solitons are mathematically defined as solutions to nonlinear partial differential equations with the following characteristics:9 (i) the solution must show a wave with a stable form, (ii) the solution must be localized, which means that it either converges to a steady value at infinity or decays exponentially to zero, and (iii) the soliton maintains its identity when interacting with other solitons. Typically, two types of ESS are seen in space plasma environments, compressive and rarefactive structures.10 In general, compressive solitary structures match up with positive electrostatic potentials. Such compressive solitary structures are found to propagate at rates of a few thousand km/s in the plasma sheet boundary (PSB) layer, parallel to the background magnetic field. Two stream and bump-on-tail instabilities have been identified as the underlying causes of the formation of compressive solitary structures.11 Furthermore, rarefactive solitary structures correspond with the negative electrostatic potential. These negative potential structure scales resemble the dispersive measure of electron-acoustic waves in many ways.12 It was later studied that the rarefactive isolated structures are expressions of steepened electron acoustic waves.13 

The analysis of plasmas containing enormous negatively and positively charged dust has become the priority of researchers in recent decades owing to the frequent existence of this plasma in labs,14 spaces,15 and astrophysical atmospheres.16 Dusty (complex) plasma is an electron–ion plasma that also incorporates a charged component of micrometer-sized dust particles. The presence of positive/negative dust components in plasmas contributes to the system’s complexity. Complex systems are effective in capturing the physical conditions of a variety of cosmic environments,13,14,17 including planetary rings, interstellar molecular clouds,14 and circumsolar and interplanetary mediums.13 In addition, the enormous charged dust particles play a role in the formation of novel eigenmodes, such as the dust-acoustic (DA) mode, which has been later proven experimentally by Barkan et al.17 Plentiful research on dust-acoustic-waves (DAWs) in a complex plasma through three components (dust, electrons, and ions) has also been reported.18–20 In addition, the existence of positively charged dust particles was discovered in several space environments, including Jupiter’s magnetosphere,21 the upper mesosphere, cometary tails, and plasma experiments.22 The system related to the charged dust size has previously been described by Chow et al.23 The mechanisms include photoemission when ultraviolet light is present, secondary emission of electrons from the surface of dust particles, and thermionic emission brought on by radiative heating.

However, several investigations of fast-moving ions and electrons in the interplanetary environment show that their velocity distribution is not in thermal equilibrium. The standard Maxwellian distribution thus becomes insufficient to examine the behavior of superthermal particles properly. For instance, in the Earth’s bow shock, nonthermal ions were noted.24 Satellite observations have shown that Mars’ upper ionosphere is losing energetic ions.25 Fast protons have also been spotted recently near the moon,26 and a novel kappa distribution27 can be employed to predict plasma behavior. The kappa distribution is taken into account when wave–particle interaction occurs in space and astrophysical environments,28 for example, the solar wind, planetary ionosphere, thermosphere, magnetosphere, and interstellar medium. Viking spacecraft29 and the Freja satellite30 performed observations in 1995 and observed the rarefaction using the ion number density. Cairns et al.31 introduced the Cairns distribution, a type of nonthermal distribution, to explain the observations of the Viking and Freja satellites. The satellite observation29,31 based theoretical studies examined the existence of cavitons in plasma structures when the particles are not thermally balanced. Many researchers in the field of nonlinear plasma have used velocity distributions that are not thermal.31 

In a two-component system involving Boltzmannian ions, it was predicted that the electron’s nonthermal character would be essential to the propagation of solitary waves. The observations made by the Freja spacecraft may be explained by the concurrent occurrence of solitary waves, both with positive and with negative potentials. Mamun et al.32 discovered that only one single wave was supported when considering a thermal plasma and both positive and negative potential structures can exist simultaneously. In a two-component plasma that contained negatively charged dust and nonthermal ions, Gill and Kaur33 looked into nonlinear dust acoustic waves. Incorporating dust thermal properties into the system, they observed that the temperature of dust particles affects the rarefactive and compressive solitons that exist simultaneously in the nonthermal plasma. The drift speed of the electron beam determines how quickly the solitary structure’s polarity changes. More recently, Abid et al.10 have observed through particle-in-cell (PIC) simulation that rarefactive ESS (convergent electric field formation) form for small values of beam velocity and agree with theoretically predicted negative potential solitary structures.11 It has been further confirmed through fluid simulations.

Nonthermal distribution significantly changed the characteristics of solitary waves based on the exact pseudopotential equation studied from the analytical model and its numerical solution.34 However, there is no soliton formation for some nonthermal parameter values in their investigations. Ion temperature affects both the Mach number and the lowest nonthermal parameter for the coexistence of compressive and rarefactive ion-acoustic solitary waves.35 In a complex plasma containing positive dust and nonthermal electrons, existing conditions for double-layer structures have been found in the lower-altitude ionosphere of the Earth.36 Later, using the pseudopotential approach to study compressive and rarefactive solitons, it was discovered that the necessary cutoff values of the nonthermal parameter and Mach number depended on the dust temperature.37 In many astrophysical and space systems, there is also relative streaming among charged particles and their surroundings. Mahmood and Saleem employed a dust–ion and dust–ion–electron model, with electrons and ions being Boltzmannian.38 The study found that when the velocity of the dust stream increases, the amplitude of the dust acoustic solitary structures decreases significantly. Moreover, it was discovered that the size of the single structure is smaller when the dust plasma contains free electrons. This demonstrates the necessity of researching electrons in dust/ion plasmas. Maharaj et al.39 investigated the two-component dust–ion framework using Boltzmannian electrons and discovered that the coexistence of solitons is affected by both the temperature and streaming velocity of the dust grains.

The formation of rarefactive soliton in the presence of Cairns-distributed electrons and ions has not been studied yet. Therefore, the primary goal of the current work is to study the solitary electrostatic structure in the presence of non-Maxwellian Cairns-distributed electrons and ions with negatively charged drifting dust particles. A detailed numerical analysis has been performed to obtain the formation of rarefactive soliton. It has been found that only rarefactive solitons can occur in the presence of positive nonthermal ions. This paper is organized as follows: In Sec. II, we present a model distribution and governing equations for the dust-acoustic (DA) waves. The numerical results and discussion are explained in Sec. III, and in Sec. IV, we summarize our results.

On the basis of the observations made by Freja2 and Viking spacecraft,31 Cairns et al.31 made a novel nonthermal distribution identified as Cairns distribution, having the following form:
fsV=Nsα1VTsπ3/21+αVs4VTs4expVs2VTs2,
(1)
where VTs[=2Ts/ms1/2] is the thermal speed (where Ts and ms are the temperature and mass of plasma sth species, respectively). The nonthermal parameter α quantifies the number of nonthermal constituent parts in the plasma system, and α1 = 1 + 3α. It has been noticed here that the Cairns distribution function agrees with the Maxwellian distribution function under the limit α = 0. By integrating Eq. (1) over the velocity space fsVdV, we obtained the number densities of electrons and ions, respectively, as
ni=ni01+βφ+βφ2eφ
(2)
and
ne=ne01+βφ+βφ2eδφ,
(3)
where β=α/1+4α and δ = Ti/Te. We take into consideration a complex plasma with various components: negatively charged drifting dust grains in the occurrence of electrons and ions that follow the Cairn distribution. However, the charge neutrality at equilibrium can be expressed as
ni0=ne0+Zdnd0,
(4)
where the unperturbed number densities for ions, electrons, and dust grains are ni0, ne0, and nd0, respectively, while the number of electrons that exist on a particular dust grain is Zd. The continuity, momentum, and pressure balance equations are employed to determine the dynamics of the dust as
ndt+ndud=0,
(5)
udt+udud=φτndP,
(6)
and
Pt+udP+γPud=0,
(7)
and the Poisson equation closes the system
2φ=neni+nd,
(8)
where τ=Td/ZdTi is the normalized temperature of the dust particle, the dust density nd is normalized by nd0, neni is the number density of electron (ion) normalized by Zdnd0, and γ=2+N/N, with N, denotes the degree of freedom. The dust velocity ud is normalized by dust-acoustic speed Cd=ZdkBTi/md1/2, the dust pressure P is normalized by nd0kBTdφ, the space variable is normalized by λd(=kBTi/4πZdnd0e21/2, and the time is normalized by ωpd1(=md/4πZd2nd0e21/2, where ωpd1 is the reciprocal of dust frequency. For one-dimensional propagation (1D) and γ = 3, the above-mentioned set of equations can be expressed as
ndt+udndx=0,
(9)
udt+ududx=φxτndPt,
(10)
Pt+udPx=3Pudx=0,
(11)
2φx2=nd+ne0Zdnd01+βφ+βφ2eδφni0Zdnd01+βφ+βφ2eφ=0.
(12)
In order to obtain a solitary wave solution, we employ the stationary frame ξ = (xMt) on the system of equations, where M denotes the Mach number. Moreover, the velocity is normalized by dust acoustic speed Cd. Equations (9)(12) can now be expressed as
Mndξ+udndξ=0,
(13)
Mudξ+ududξ=φξτndPξ,
(14)
MPξ+udPξ=3Pudξ=0,
(15)
2φξ2=nd+ne0Zdnd01+βφ+βφ2eδφni0Zdnd01+βφ+βφ2eφ=0.
(16)
Equations (13) and (15) can be integrated by using boundary conditions udud0, φ → 0 (ud0 is the dust drift velocity at equilibrium), nd → 0, and P → 0 at ξ±; then, the value of dust number density is obtained as nd=ud0M/udM and P=nd3. By inserting the value of nd into Eq. (14) and after some algebraic calculations, we get the following bi-quadratic equation:
3τnd4Mud02+3τ+2φnd2+Mud02=0.
(17)
The solution of nd can be expressed as
nd=τ12τ01+2φMud0τ121+2φMud02τ124τ02τ14,
(18)
where
τ0=3τMud02 and τ1=1+τ02.
(19)
The normalized equilibrium dust drift speed ud0=0 setting in Eqs. (18) and (19) agrees with that in equations of Ref. 37. By substituting Eq. (18) into Eq. (16), we obtain
d2φdξ2=τ12τ01+2φMud02τ121+2φMud02τ1224τ02τ1412+ne0Zdnd01+βφ+βφ2eδφni0Zdnd01+βφ+βφ2eφ.
(20)
Thus, Eq. (20) can be indicated in the form of Sagdeev potential Vφ as
12dφdξ2+Vφ=0,
(21)
where the Vφ is known by
Vφ=ni0Zdnd01+3β+3βφ+βφ2eφne0Zdnd01δ1+1δβ2δ1β2δ1φ+βφ2eδφMud02τ0expθ2+13exp3θ2+C1.
(22)
Here, C1 is the constant of integration. Following Mendoza-Briceno et al.,37 to enable the limit τ → 0, i.e., to consider the cold dust in the Sagdeev potential, we define θ as follows:
θ=lnτ12τ021+2φMud02τ12+4τ04τ141+2φMud02τ1221.
(23)
By using Eq. (23), the Sagdeev potential can be expressed as
Vφ=ni0Zdnd01+3β+3βφ+βφ2eφne0Zdnd01δ1+1δβ2δ1β2δ1φ+βφ2eδφMud02σ121+2φMud02τ12+1+2φMud0224τ02τ141222ττ131+2φMud02τ12+1+2φMud0224τ02τ1432+C1.
(24)
The constant C1 can now be calculated for the limit, i.e., Vφ=0 at φ = 0, as
C1=ni0Zdnd01+3β+ne0Zdnd01δ1+1δβ2δ1Mud02σ121+14τ02τ141222ττ131+14τ02τ1432.
(25)
Equation (24) is the Sagdeev-potential Vφ that represents a plasma consisting of electrons and ions following nonthermal Cairns distribution with a negatively charged dust fluid. The results obtained by Shah et al.35 are exactly congruent with Eq. (24) if the electrons follow the Maxwellian distribution function. Moreover, in the absence of unperturbed electron density and dust drift speed, i.e., ne0 = 0 and ud0 = 0, respectively, Eq. (24) agrees with Eq. (29) of Ref. 37. The solitary wave solution can be obtained from Eq. (24) under the following conditions: (i) Vφ=0 at φ = 0, (ii) dVdξ=0 at φ = 0, (iii) d2Vφdξ2<0 at φ = 0, which means that the fixed point at the origin is unstable, and (iv) dVdξ<0 at φ = φmin so that φmin < φ < 0. This generates the rarefactive solitons, i.e., solitons having negative potential, where φmin is the minimum value of the potential.40 Condition (iii) given above gives the critical value of the Mach number Mc above which the solitary solution can be obtained and is given as follows:
Mc=1+3τni0Zdnd01β+3τne0Zdnd0β+δni0Zdnd01β+ne0Zdnd0β+δ+ud0.
(26)
The above-mentioned Eq. (26) represents the critical Mach number above which the soliton structures can be obtained. Therefore, when the Sagdeev potential Eq. (24) satisfies conditions (i)–(iv), rarefactive solitary structures can be obtained when the Mach number M is larger than the critical Mach number Mc given in Eq. (26). We note that if we consider Maxwellian electrons, Eq. (26) reduces to Eq. (31) of Ref. 39. In addition, by applying the conditions ne0 = 0 and ud0 = 0, Eq. (26) reduces to the condition obtained in Ref. 37. It should also be noted that, in contrast, Mamun et al.18 found compressive solitons in a plasma comprising a negative dust fluid and Boltzmannian ions.

This work is motivated by theoretical investigations37,39,41,42 and experimental observations17 of low-phase velocity dust acoustic (DA) waves. It has been investigated that non-Maxwellian ion distribution mainly affects the solitary structures and supports the existence of large amplitude positive and negative potential solitary structures.39,43 The current study is focused on the influence of nonthermal ions and electrons on the formation of solitary structures. We numerically examine the effect of Cairns distribution [Eq. (1)] for numerous values of the nonthermal parameter α that play a great part in the formation of solitary structures. Figure 1 shows the three-dimensional (3D) picture of the distribution function given in Eq. (1). It is shown in Fig. 1 that the high energy particles in the tail of the distribution increase with the nonthermal parameter α(= 0.0–0.4).

FIG. 1.

3D plot of Cairns-distribution.

FIG. 1.

3D plot of Cairns-distribution.

Close modal

Figure 2 is plotted for an equilibrium plasma with the nonthermal spectral index set to zero in Eq. (24). The upper panel of Fig. 2 shows the Sagdeev potential structures, and the lower panel shows the corresponding solitons for various Mach numbers M(= 1.051, 1.056, 1.061, and 1.066).

FIG. 2.

The Sagdeev potential Vφ vs φ (upper panel) and the corresponding rarefactive solitons (lower panel) for different values of Mach number and α = 0.0. Other numerical values used are δ = 1, τ = 0.02, ne0 = 0.1Zdnd, and ud0 = 0.1Cd.

FIG. 2.

The Sagdeev potential Vφ vs φ (upper panel) and the corresponding rarefactive solitons (lower panel) for different values of Mach number and α = 0.0. Other numerical values used are δ = 1, τ = 0.02, ne0 = 0.1Zdnd, and ud0 = 0.1Cd.

Close modal

We note that the multi-species (electrons and positive ions) nonthermal plasma supports only negative potential solitary structures, which is in contrast to the results of Maharaj et al.,39 where they revealed that only positive potential solitary structures could exist in a thermal plasma. From Eq. (26), we calculate the value of Mc=1.045 for fixed parameters, such as τ = 0.02, α = 0, ne0 = 0.1Zdnd, and ud0 = 0.1Cd and found that the solitons of negative potential start to appear for M > Mc. It agrees well with the past investigations37,39,41 that thermal dusty plasma only supports solitons of negative potential structures. Using a model that includes a negative dust fluid and Boltzmannian ions, Mamun et al.18 found that the thermal plasma could only support solitary waves of negative potential. In Fig. 3, we study a nonthermal plasma (nonthermal electrons and ions) with a minimum value of the nonthermal parameter α = 0.1. The only negative potential solitons are observed. It has been found that nonthermal ions can produce compressive and rarefactive solitons only when electrons follow the thermal distribution.39 However, in Fig. 3, it is also observed that if nonthermal electrons are present along with the nonthermal ions, only rarefactive solitons can exist. It is emphasized that the findings of the present work help in elaborating the nonlinear characteristics of localized electrostatic structures in a variety of astrophysical dusty plasma environments, including the planetary rings, such as Saturn’s rings;44 cometary dusty plasma, such as Halley’s comet;45 and the interstellar medium,46 where negatively charged dust particles and thermal/nonthermal ions and electrons exist. In Fig. 4, we plot Vφ vs φ for a fixed value of the nonthermal parameter α = 0.2 for both the electrons and ions using different values of M (= 1.41,1.42,1.43, and 1.44) while the other parameters are kept the same as in Fig. 2. However, the value of the critical Mc derived from Eq. (26) is 1.3207 for this case. We found that the rarefactive solitons start to appear at M > Mc. Figure 5 is plotted for the value of the nonthermal parameter α = 0.38. In this figure, it is found that with the increase in the nonthermal parameter, the amplitude of solitons also increases as compared to Figs. 3 and 4 for the same values of other parameters.

FIG. 3.

The Sagdeev potential Vφ vs φ (upper panel) and the corresponding rarefactive solitons (lower panel) for different values of Mach number and α = 0.1. The other numerical values are the same as in Fig. 2.

FIG. 3.

The Sagdeev potential Vφ vs φ (upper panel) and the corresponding rarefactive solitons (lower panel) for different values of Mach number and α = 0.1. The other numerical values are the same as in Fig. 2.

Close modal
FIG. 4.

The Sagdeev potential Vφ vs φ (upper panel) and the corresponding rarefactive solitons (lower panel) for different values of Mach number and α = 0.2. The other numerical values are the same as in Fig. 2.

FIG. 4.

The Sagdeev potential Vφ vs φ (upper panel) and the corresponding rarefactive solitons (lower panel) for different values of Mach number and α = 0.2. The other numerical values are the same as in Fig. 2.

Close modal
FIG. 5.

The Sagdeev potential Vφ vs φ (upper panel) and the corresponding rarefactive solitons (lower panel) for different values of Mach number and α = 0.38. The other numerical values are the same as in Fig. 2.

FIG. 5.

The Sagdeev potential Vφ vs φ (upper panel) and the corresponding rarefactive solitons (lower panel) for different values of Mach number and α = 0.38. The other numerical values are the same as in Fig. 2.

Close modal

It is also seen that by increasing α, the critical Mach number also increases, which is Mc = 1.549 82 in this case. By increasing the values of M(= 1.552, 1.554, and 1.556, 1.558), the soliton increases in amplitude but decreases in width. It is noted from Figs. 25 that with an increase in nonthermal particles, i.e., α(= 0.0, 0.1, 0.2, and 0.38), the negative potential soliton is becoming taller and decreases in width, but no positive potential counterpart exists.

Figure 6 shows the implications of dust temperature τ=Td/ZdTi on the critical Mach number Mc as a function of the nonthermal parameter α [see Eq. (26)]. We found that the dependence of Mc on the nonthermal index, α, is linear as found in past studies.36,37 In Fig. 6, for fixed α, we observed that the impact of enhancing the dust temperature τ=Td/ZdTi is to enhance the critical match number that agrees with the past observation.36,37

FIG. 6.

The critical Mach number vs nonthermal parameter (α) for different values of dust temperature τ(= 0.00, 0.02, 0.04, and 0.06). The other numerical values are the same as in Fig. 2.

FIG. 6.

The critical Mach number vs nonthermal parameter (α) for different values of dust temperature τ(= 0.00, 0.02, 0.04, and 0.06). The other numerical values are the same as in Fig. 2.

Close modal

The same effect has been observed on the critical Mach number for different values of dust drift velocity ud0 as shown in Fig. 7, in which, with the increase in dust drift velocity, the critical match number increases linearly. In Fig. 8, the critical Mach number against the nonthermal parameter α is shown for various values of the nonthermal electron number density ne0=0.0,0.1,0.2,0.3. It is observed that the critical match number Mc varies inversely with the electron number density. For larger values of α, the effect of electron number density is more prominent than for smaller values and shows a non-monotonic effect with an increase in electron number density.

FIG. 7.

The critical Mach number vs nonthermal parameter (α) for different values of dust drift speed ud0/Cd(= 0.00, 0.05, 0.10, and 0.15). The other numerical values are the same as in Fig. 2.

FIG. 7.

The critical Mach number vs nonthermal parameter (α) for different values of dust drift speed ud0/Cd(= 0.00, 0.05, 0.10, and 0.15). The other numerical values are the same as in Fig. 2.

Close modal
FIG. 8.

The critical Mach number vs nonthermal parameter (α) for different values of electron number density ne0(= 0.0, 0.1, 0.2, and 0.3). The other numerical values are the same as in Fig. 2.

FIG. 8.

The critical Mach number vs nonthermal parameter (α) for different values of electron number density ne0(= 0.0, 0.1, 0.2, and 0.3). The other numerical values are the same as in Fig. 2.

Close modal

We have investigated the existence of rarefactive electrostatic solitons in a plasma model containing nonthermal Cairns-distributed electrons and ions in a negatively charged dust fluid. For nonthermal electrons and ions with a nonthermal parameter α > 0.0, we consider a region in a parametric space, where solitons of negative potential are usually found. Numerical results show that the variation of plasma parameters, such as α and M, significantly affect the Sagdeev potential and the corresponding solitons. We found that (i) with the increase in the nonthermal parameter α, the height of the solitons increases and the width increases, (ii) by increasing α, the critical match number also increases, and (iii) by increasing the values of the Mach number, the Sagdeev potential becomes more negative and, as a consequence, the amplitude of the soliton increases. It is suggested that the findings of this study will be useful in understanding the nonlinear behavior of electrostatic waves in space plasmas, where the structures of negative potential associated with the nonthermal distribution of constituent species are observed.44–46 

This work was supported by Collaborative Innovation Program of Hefei Science Center (CAS, Grant No. CX2140000018), the Funding for Joint Lab of Applied Plasma Technology (Grant No. JL06120001H), and Higher Education Commission (HEC), Pakistan, Project No. 7558/Punjab/NRPU/R&D/HEC/2017.

The authors have no conflicts to disclose.

A. A. Abid: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Software (equal). Wu Zhengwei: Formal analysis (equal); Supervision (equal); Writing – review & editing (equal). Abdullah Khan: Methodology (equal); Software (equal). M. N. S. Qureshi: Conceptualization (equal); Methodology (equal); Software (equal); Writing – review & editing (equal). Amin Esmaeili: Software (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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