On the oblique electrostatic waves in a dusty plasma with non-Maxwellian electrons for Saturn’s magnetosphere

Abstract

The characteristics of (non)linear dust-ion acoustic waves (DIAWs) in a collisionless, magnetized dusty plasma are investigated. The current model is composed of (r, q)-distributed electrons along with warm ions and stationary dust grains with negative charge. Both linear and nonlinear waves are considered to progress in x- z plane. The properties of linear waves are studied by deriving the dispersion relation for the plasma parameters of Saturn’s magnetosphere. The fluid equations of the current model are reduced to the universal Korteweg-de Vries (KdV) equation in order to study the characteristics of oblique propagation of DIA solitary waves (DIASWs). The critical point at which the nature/polarity of solitons changes is determined precisely. The influence of various plasma parameters, namely, obliqueness, magnetic field, densities, temperatures, and double spectral indices of the (r, q)-distributed electrons on DIASWs is investigated for Saturn’s magnetosphere. The DIASWs of (r, q)-distributed electrons are also compared with Maxwellian electrons. This work would be helpful to study other astrophysical and laboratory plasma systems where dusty plasmas and (r, q) distribution are predicted.

Keywords

Non-Maxwellian magnetoplasmas dust-ion acoustic waves Saturn’s magnetosphere (r,q)-distributed electrons linear dispersion relation solitary structures

Introduction

The dusty plasma, also known as complex plasma, is like ordinary (electron-ion (e − i)) plasma system with the addition of micro or nano-sized charged dust grains.^1,2 The addition of dust particles alters the usual linear and nonlinear e − i plasma modes, by virtue of the variation of the charge, mass, number, and size of dust particles, to the dust-acoustic waves (DAWs),^3,4 the DIAWs,⁵ the dust-ion acoustic (DIA) rogue waves (DIARWs),⁶ DIA dissipative solitons,⁷ the dust lattice waves,⁸ etc. in the plasma system. The dusty plasma is a widely studied field of research due of its vast existence regimes in laboratory,⁹ as well as in space plasmas, for example, cometary tails, interstellar medium, planetary rings, and terrestrial and planetary magnetospheres.^10,11 The dust grains are initially neutral; however, they attain the negative charge when the electrons are caught by their surface or become positively charged by photoemission or thermionic emission.^12,13

The DIAWs were first predicted theoretically in 1992,⁵ which were later discovered and confirmed experimentally in 1996.¹⁴ Since then, many researchers have extensively investigated the (non)linear DIAWs in space and laboratory dusty plasmas.^15–19 The finite amplitude DIASWs with both positive and negative inertial ions have been studied for both positive and negative solitons.²⁰ Recently, Hassan et al. investigated the DIAWs in a superthermal magnetized collisional plasma in the presence of opposite polarity dust.²¹ They studied the effects of superthermality, adiabaticity, viscosity, magnetic field (MF), obliqueness, the charge on dust and collisions for the solitons and shock waves.

A substantial variation of the Maxwellian distribution has led to the discovery of non-Maxwellian velocity distributions recently^22,23 in both space and astrophysical plasmas including magnetosheath, magnetospheres, ionosphere, sun wind, and interstellar medium.²⁴ The deviations of the velocity distributions include superthermal tails, known as kappa or generalized Lorentzian distribution²⁵ as well as the flat top behavior in bow shocks, magnetosheath and the magnetotail.^26,27 Qureshi et al.²⁸ established a comprehensive and general velocity distribution with flat tops and the high energy tails, named as the generalized (r, q) distribution function. Here, index q determines the particles population in the long tails, whereas the index r ascertains the population of particles at low energies for the flat top of distribution. This distribution function approaches to Maxwellian case for r = 0 and q → ∞ and a kappa distribution²⁹ for r = 0 and q → κ + 1. Recently, a few researchers have investigated the linear as well as the nonlinear propagation properties of various waves in plasma physics by utilizing the (r, q) distribution function to estimate the effect of spectral indices on the waves. Qureshi et al.³⁰ employed the generalized (r, q) distribution to investigate the terrestrial lion roars, primarily studied by using bi-Maxwellian distribution,²⁷ and showed that it agreed with the observed data for (r, q) distribution. Sidra et al. investigated the ion acoustic solitons and periodic structures in upper ionospheric plasmas.³¹ The interaction of DIASWs has been studied for Saturn’s rings with (r, q) distributed electrons with cubic nonlinearity to investigate the influence of spectral indices on interaction regime.³²

Saturn’s magnetosphere (SMS) being the second largest magnetosphere of all the planets in the solar system, discovered for the first time in 1979 by the Pioneer-11 spacecraft, is created due to the interaction of sun wind with the internal MF of the planet.³³ The main subject of research of Cassini mission has been the study of SMS since 2004 which has provided deep insights about Saturn’s inner atmosphere and space around it. The SMS is filled with dusty plasmas due to the planetary rings, as well as its moon, where the major contribution of plasma comes from its moon Enceladus.³⁴ The Saturn rings are specifically important examples of dusty plasma, since they are electron depleted regions due to charging of dust particles by the background electrons.⁵

In this investigation, the DIAWs will be studied in a collisionless magnetoplasma, comprising the warm ions, (r, q) distributed electrons, and stationary dust grains. The motivation for considering the present model is many fold: first, the magnetized plasma exists in SMS, so a realistic model is to consider the MF. Second, the MF affects the charged particles anisotropically for parallel and perpendicular propagation. So, we have considered both directions for propagation of DIAWs. Third, a general double parameter electron distribution is considered which takes into account the flat top and elongated tail effects and can approach kappa and Maxwellian distributions for the limiting cases.

This study is sectioned in following pattern. In the Mathematical Model section, the fluid model equations leading to the oblique propagation of DIAWs in a magnetized collisionless plasma is considered. The Linear Dispersion Relation section discusses the linear mode of DIA waves and Evolutionary Equation and DIASWs section is devoted to derive the evolution equation (KdV equation) by utilizing the reductive perturbation scheme (RPS). The Linear Dispersion Relation section presents the numerical analysis and discussion for the effects of various plasma parameters on the propagation of DIASWs belonging to SMS. Main findings of our results are summarized in the Numerical Results and Discussion section.

Mathematical model

A homogeneous, collisionless complex plasma is considered consisting of warm ions and non-Maxwellian electrons that follow (r, q)-distribution along with stationary dust grains. The dust charge and size of the grains are assumed to be constant. The wave propagates obliquely along x − axis and the z − axis and MF is aligned along the z − axis $(B = B_{0} \hat{z})$ . The equilibrium condition reads n_e0 + z_dn_d0 = n_i0, where z_d represents the average constant charge on dust particles and n_d0, n_i0, and n_e0 depict the unperturbed number densities of the dust, ions, and electrons, respectively. The dynamic equations for the current model in CGS units read

\partial_{t} n_{i} + \tilde{\nabla} . (n_{i} {\tilde{v}}_{i}) = 0,

(1)

\partial_{t} {\tilde{v}}_{i} + ({\tilde{v}}_{i} . \tilde{\nabla}) {\tilde{v}}_{i} = \frac{e}{m_{i}} (\tilde{E} + \frac{1}{c} {\tilde{v}}_{i} \times \tilde{B}) - \tilde{\nabla} P_{i},

(2)

\tilde{\nabla} . \tilde{E} = 4 π e (n_{i} - n_{e} - z_{d} n_{d}) .

(3)

Here,

P_{i} = P_{i 0} {(n_{i} / n_{i 0})}^{γ}

indicates the adiabatic pressure of the warm ions, where

γ = (N + 2) / N (= 5 / 3

for N = 3 for our case) is the adiabatic constant, N are the degrees of freedom, and P_i0 = n_i0T_i is the equilibrium pressure. Also,

\vec{E} \equiv (E_{x}, E_{y}, E_{z}) = - \vec{\nabla} ϕ

represents the electric field, where

\vec{\nabla} = (\partial_{x}, \partial_{y}, \partial_{z})

indicates the Del operator. Normalizing the above equations and writing them in component forms give the following set of equations

\partial_{t} n_{i} + \partial_{x} (n_{i} v_{i x}) + \partial_{z} (n_{i} v_{i z}) = 0,

(4)

\partial_{t} v_{i x} + ▽_{x z} v_{i x} = - \partial_{x} ϕ + Ω v_{i y} - \frac{5}{3} σ n_{i}^{- 1 / 3} \partial_{x} n_{i},

(5)

\partial_{t} v_{i y} + ▽_{x z} v_{i y} = - Ω v_{i x},

(6)

\partial_{t} v_{i z} + ▽_{x z} v_{i z} = - \partial_{z} ϕ - \frac{5}{3} σ n_{i}^{- 1 / 3} \partial_{z} n_{i},

(7)

and the following normalized Poisson’s equation

(\partial_{x}^{2} + \partial_{z}^{2}) ϕ = μ n_{e} - n_{i} + (1 - μ)

(8)

with ▿_xz = (v_ix∂_x + v_iz∂_z), μ = n_e0/n_i0, σ = T_i/T_e, Ω = ω_ci/ω_pi, where ω_ci = eB₀/m_ic and

ω_{p i} = \sqrt{4 π n_{i o} e^{2} / m_{i}}

. Here, n_i and v_i = (v_ix, v_iy, v_iz) denote the ions number density and velocity which are normalized by the corresponding equilibrium number density and ion-acoustic (IA) speed

C_{s} = \sqrt{k_{B} T_{e} / m_{i}}

, respectively. Here, v_ix, v_iy, and v_iz indicate the velocity components in x, y, and z directions, respectively. The electrostatic potential ϕ is scaled by k_BT_e/e, where T_e is the temperature of electrons. The independent variables

(t, x)

are, respectively, normalized by

ω_{p i}^{- 1}

and

λ_{D} (= \sqrt{k_{B} T_{e} / 4 π n_{i 0} e^{2}})

The normalized density of the non-Maxwellian electrons follows the (r, q)-distribution²⁸

n_{e} = 1 + A ϕ + B ϕ^{2} + C ϕ^{3} + \dots,

(9)

with

A = \frac{Γ [R] Γ [q - R] {(q - 1)}^{- 2 R}}{2 β Γ [3 R] Γ [q - 3 R]},

(10)

B = - \frac{Γ [- R] Γ [q + R] {(q - 1)}^{- 4 R}}{8 β^{2} Γ [3 R] Γ [q - 3 R]},

(11)

C = \frac{Γ [- 3 R] Γ [q + 3 R] {(q - 1)}^{- 6 R}}{16 β^{3} Γ [3 R] Γ [q - 3 R]}

(12)

where

\begin{array}{l} β = \frac{3 {(q - 1)}^{- 2 R} Γ [q - 3 R] Γ [3 R]}{2 Γ [q - 5 R] Γ [5 R]}, \\ R = \frac{1}{2 (1 + r)} . \end{array}

Here, Γ[−] is the gamma function and the necessary conditions q > 1 and q(1 + r) > 5/2 appear according to the real values of

(r, q)

Linear dispersion relation

The linear properties of the plasma system under consideration can be studied with the help of the linear dispersion relation (LDR). Accordingly, we utilize the linearized system of the equations (4)–(9) and consider the planar form of perturbations $\sim e^{i (k_{x} x + k_{z} z - ω t)}$ to get the expression between frequency ω and propagation vector k as follows

ω^{4} - [Ω^{2} + k^{2} (\frac{1}{k^{2} + A μ} + \frac{5}{3} σ)] ω^{2} + k_{z}^{2} Ω^{2} (\frac{1}{k^{2} + A μ} + \frac{5}{3} σ) = 0,

(13)

where

k = \sqrt{k_{x}^{2} + k_{z}^{2}}

By solving equation (13), we finally get the LDR

\begin{array}{l} ω_{\pm}^{2} = \frac{1}{2} [Ω^{2} + k^{2} (\frac{1}{k^{2} + A μ} + \frac{5}{3} σ)] \\ \pm \frac{1}{2} \sqrt{{[Ω^{2} + k^{2} (\frac{1}{k^{2} + A μ} + \frac{5}{3} σ)]}^{2} - 4 k_{z}^{2} Ω^{2} (\frac{1}{k^{2} + A μ} + \frac{5}{3} σ)} . \end{array}

(14)

Here, positive and negative signs, respectively, denote the fast and slow modes of DIAW. The fast mode represents the obliquely propagating ion cyclotron wave, since ω₊ → Ω (which is the ion cyclotron frequency) as k → 0. However, the slow mode represents the obliquely propagating ion-acoustic wave (IAW). Additionally, it may be observed from the above expression that the linear properties of the waves depend upon plasma parameters through $(Ω, σ)$ , indices $(r, q)$ through A, and angle of obliqueness by k_z.

Evolutionary equation and DIASWS

For studying the characteristics of DIASWs, the RPS is applied to derive a suitable evolution equation.^35,36 Accordingly, the following stretching is utilized for space and time coordinate

ξ = ϵ^{1 / 2} (l_{x} x + l_{z} z - v_{o} t) and τ = ϵ^{3 / 2} t,

(15)

where v_o indicates the normalized phase velocity and ϵ is the order of nonlinearity

(ϵ ≪ 1)

. Furthermore, the field quantities n_i, ϕ, and v_i are expanded in terms of ϵ as

\begin{array}{c} n_{i} & = 1 + ϵ n_{i}^{(1)} + ϵ^{2} n_{i}^{(2)} + \dots, \\ ϕ & = ϵ ϕ^{(1)} + ϵ^{2} ϕ^{(2)} + \dots, \\ v_{i x} & = ϵ^{3 / 2} v_{x}^{(1)} + ϵ^{2} v_{x}^{(2)} + \dots, \\ v_{i y} & = ϵ^{3 / 2} v_{y}^{(1)} + ϵ^{2} v_{y}^{(2)} + \dots, \\ v_{i z} & = ϵ v_{z}^{(1)} + ϵ^{2} v_{z}^{(2)} + \dots . \end{array}

(16)

Using both the stretching (15) and expansions (16) in the model equations (4)–(9), we get the reduced equations in terms of different powers of ϵ. The equations with the lowest-order of ϵ give us

\begin{array}{l} - v_{o} \partial_{ξ} n_{i}^{(1)} + l_{z} \partial_{ξ} v_{z}^{(1)} = 0, \\ - l_{x} \partial_{ξ} ϕ^{(1)} + Ω v_{y}^{(1)} - \frac{5}{3} σ l_{x} \partial_{ξ} n_{i}^{(1)} = 0, \\ v_{o} \partial_{ξ} v_{y}^{(1)} - Ω v_{x}^{(2)} = 0, \\ - v_{o} \partial_{ξ} v_{z}^{(1)} + \frac{5}{3} σ l_{z} \partial_{ξ} n_{i}^{(1)} + l_{z} \partial_{ξ} ϕ^{(1)} = 0, \end{array}

(17)

and

- n_{i}^{(1)} + A μ ϕ^{(1)} = 0

(18)

By solving the above system, the following normalized phase velocity is obtained

v_{o} = l_{z} \sqrt{\frac{(1 + \frac{5}{3} A σ μ)}{A μ}} .

(19)

The higher-orders of ϵ give us

\begin{array}{l} - v_{o} \partial_{ξ} n_{i}^{(2)} + \partial_{τ} n_{i}^{(1)} + l_{x} \partial_{ξ} v_{x}^{(2)} + l_{z} \partial_{ξ} v_{z}^{(2)} + l_{z} \partial_{ξ} (n_{i}^{(1)} v_{z}^{(1)}) = 0, \\ - v_{o} \partial_{ξ} v_{x}^{(1)} - Ω v_{y}^{(2)} = 0, \\ - v_{o} \partial_{ξ} v_{z}^{(2)} + \partial_{τ} v_{z}^{(1)} + \frac{5}{3} σ l_{z} \partial_{ξ} n_{i}^{(2)} + l_{z} \partial_{ξ} ϕ^{(2)} - \frac{5}{9} σ l_{z} n_{i}^{(1)} \partial_{ξ} n_{i}^{(1)} + l_{z} v_{z}^{(1)} \partial_{ξ} v_{z}^{(1)} = 0, \end{array}

(20)

and

(l_{x}^{2} + l_{z}^{2}) \partial_{ξ}^{2} ϕ^{(1)} = - n_{i}^{(2)} + A μ ϕ^{(2)} + B μ {(ϕ^{(1)})}^{2} .

(21)

After tedious mathematical algebraic steps, we finally obtain the following KdV equation for the obliquely propagating DIAWs

\partial_{τ} ϕ^{(1)} + a ϕ^{(1)} \partial_{ξ} ϕ^{(1)} + b \partial_{ξ}^{3} ϕ^{(1)} = 0,

(22)

with

\begin{array}{l} a = \frac{v_{o}}{2 A} (3 A^{2} μ + 2 B) - \frac{5 σ l_{z}^{2}}{18 A v_{o}} (A^{2} - 6 B), \\ b = \frac{v_{o}}{6 A μ Ω^{2}} (3 (l_{x}^{2} - Ω^{2})) + \frac{5 σ}{v_{o}^{2}} (l_{x}^{2} μ v_{o}^{2} A + l_{z}^{2} Ω^{2}), \end{array}

(23)

A steady-state solution of KdV is obtained³⁷ by using a traveling wave solution $χ = (ξ - u τ)$ for transforming the space and time coordinates to a single coordinate χ. Here, u represents the normalized constant speed of the SW. In order to obtain the localized solutions, we use the appropriate boundary conditions for integration, namely, $ϕ^{(1)} \to 0$ , and derivatives $d ϕ^{(1)} / d χ \to 0$ and $d^{2} ϕ^{(1)} / d χ^{2} \to 0$ as χ → ±∞. Therefore, the stationary localized solitary wave (SW) solution of the KdV equation (22) reads as

ϕ^{(1)} = ϕ_{a m p} \sec h^{2} [Δ (ξ - u τ)],

(24)

where ϕ_amp = 3u/a is the amplitude,

Δ = \sqrt{u / 4 b}

is the inverse width, and u is the velocity of the solitons.

It is worth mentioning here that for the complex plasmas, there is a critical point at which the nonlinearity coefficient a vanishes. At this point, the nonlinearity no more balances the dispersive coefficient for a stable soliton formation. For present dusty plasma, the coefficient a vanishes, that is, a = 0 at the following critical value of the electron concentration

μ_{c} = \frac{5 σ l_{z}^{2}}{27 A^{2} v_{o}^{2}} (A^{2} - 6 B) - \frac{2 B}{3 A^{2}} .

(25)

Thus, the solitary wave changes its polarity (compressive to rarefactive, or vice versa), below and above this critical point. It may be seen that the critical value to the electron concentration μ_c depends upon all the plasma parameters and parameters of electron distribution function, but is independent of the MF B₀ and direction cosine l_z.

Numerical results and discussion

We analyze the structure of both linear and nonlinear electrostatic DIAWs numerically in this section for a dusty magnetized plasma. Both linear and nonlinear properties of DIAWs are greatly influenced by various plasma parameters. To observe the impact of various parameters, we take into account dusty plasma parameters of the SMS,^33,34,38 with MF B₀ = 0.04G, temperature of electrons k_BT_e = (10 − 100)eV, density of electrons n_e0 = (1 − 4) × 10²cm⁻³, dust grains charge and density as z_d = 100 and n_d0 = (0.1 − 10)cm⁻³, respectively. Furthermore, $l_{z} (= l \cos [θ])$ and l_x are cosine coordinates, with $l_{x} = \sqrt{1 - l_{z}^{2}}$ .

First, the influence of dusty plasma parameters, namely, MF B₀, temperature ratio σ(= T_i/T_e), and dust concentration n_d0, on the linear properties of the DIAW can be observed by plotting the LDR (14) as shown in Figure 1. Figure 1 shows the influence of $(B_{0}, σ, n_{d 0})$ on the frequency ω of fast mode versus wave vector k. Here, the standard solid curve is plotted for n_d0 = 0.5cm⁻³, n_e0 = 10²cm⁻³, B₀ = 0.04G, σ = 0.01, q = 2.5, r = 0.3, and θ = 10. One can see from the expression of the dispersion relation $(ω, k)$ that in the absence of B₀, the fast mode remains nonzero and becomes ion-acoustic mode with dispersion relation $ω_{+} \to k \sqrt{5 σ / 3 + 1 / (k^{2} + A μ)}$ . The comparison of standard solid curve with dashed curve (B₀ = 0.042) shows that a slight increase in the B₀ enhances ω of the fast mode significantly. The effect of temperature ratio σ, on the frequency ω of the DIAWs is observed by comparing standard solid curve with dotted curve in Figure 1. It is observed that the frequency of fast mode enhances with σ. This is in contrast to the IAWs, where the existence condition T_i ≪ T_e is fulfilled; thus, increasing T_i would reduce the frequency ω of the wave. Thus, for dusty plasmas the acoustic wave may exist for T_i ∼ T_e in an electron depleted region (n_eo ≪ n_io). Figure 1 further shows the influence of dust concentration n_d0 on frequency ω of fast mode. The comparison of standard solid curve with dotted-dashed curve demonstrates the decrease of frequency ω of the fast wave with the increase of concentration n_d0.

Figure 1.

The frequency ω₊ of fast mode is plotted versus wave vector k to show variation of dust concentration n_d0, MF B₀, and temperature ratio σ. Here, the standard curve is plotted for n_d0 = 0.5cm⁻³, n_e0 = 10²cm⁻³, B₀ = 0.04G, σ = 0.01, q = 2.5, r = 0.3, and θ = 10.

Figure 2 delineates the effects of plasma parameters $(B_{0}, σ, n_{d 0})$ on the linear characteristics of the slow mode of the DIAW versus wave vector k. Here, the standard solid curve represents the curve for the same values as illustrated in Figure 1. The dashed curve plotted for B₀ = 0.06 is compared with the standard solid curve of B₀ = 0.04. The expression of LDR ω depicts that the slow mode does not exist in the absence of B₀. Initially, the effect of increasing B₀ is not prominent; however, for higher values of wave vector k, increasing B₀ would enhance the frequency ω of the slow mode of DIAW. Therefore, we have noted that the B₀ would lead to the increase of the frequency ω of both modes (i.e., fast and slow). However, the effect of change on the slow mode is negligible in comparison of the fast mode. The increase of temperature ratio σ as well as the dust concentration n_d0 enhances ω of the slow mode of the DIAW. In a nutshell, the comparison between both Figures 1 and 2 shows that the fast mode is more like ion cyclotron mode, while the slow mode is more like the thermal/IA mode altering remarkably with the temperature ratio.

Figure 2.

The frequency ω₋ of slow mode is plotted versus wave vector k to show variation of dust concentration n_d0, MF B₀, and temperature ratio σ. Here, the parameters are same as Fig. 1.

Besides the mentioned plasma parameters $(B_{0}, σ, n_{d 0})$ , the indices $(r, q)$ of electron distribution, as well as the angle between the MF and direction of propagation, that is, θ also affect the frequency ω of the DIAW. Figure 3 describes the impact of these parameters on the frequency ω of fast mode versus propagation vector k. Here, the standard solid curve for reference is drawn for q = 2.5, r = 0.3, θ = 10 with n_e0 = 10²cm⁻³, n_d0 = 0.5cm⁻³, B₀ = 0.04G, and σ = 0.1. The comparison of standard and dashed curve shows the enhancement of frequency ω of fast mode with the increase of index q (which is tantamount to the reduction of superthermality) of electron distribution function. The spectral index r determines the number of low energy particles in the phase space or the flatness of the distribution function of electron. The curves demonstrate the increase of fast mode frequency ω by increasing r. Also, the Maxwellian limit (r → 0, q → ∞) of the distribution function is considered in this figure. It is noticed that the normalized frequency ω of fast mode for (r, q) electrons for higher values of (r, q) becomes higher than the Maxwellian ones. However, the frequency ω of the fast mode of Maxwellian case becomes higher than (r, q) case with small values of (r, q), as can be seen by the comparison of standard and Maxwellian curves. Also, Figure 3 demonstrates the influence of angle between MF and propagation vector k, on the frequency ω of the wave. It is observed that any change of the angle θ would lead to great effects on the propagation of both the fast and slow mode. It is thus shown that the normalized frequency for the fast mode increases with increasing θ for the DIAWs leading to a maximum value at θ = 90 with dispersion relation $ω_{+} \to \sqrt{Ω^{2} + k^{2} (5 σ / 3 + 1 / (k^{2} + A μ))}$ .

Figure 3.

The frequency ω₊ of fast mode is plotted versus wave vector k to show variation of q, r and angle θ. Here, the standard curve is plotted for n_d0 = 0.5cm⁻³, n_e0 = 10²cm⁻³, B₀ = 0.04G, σ = 0.1, q = 2.5, r = 0.3, and θ = 10.

Figure 4 illustrates the influence of indices $(r, q)$ , and the angle θ on linear properties of slow mode DIAWs. Here, the reference curve is plotted for the same parameters as the standard curve for the fast mode in Figure 3. The figure shows that both spectral parameters $(r, q)$ enhance the frequency ω of the slow mode. Also, it is noticed that the frequency ω becomes greater for (r, q) case (with higher values of q and r) than the Maxwellian case, in line with the fast mode of the DIAWs, for the given plasma model. However, in contrast to the fast mode, the normalized frequency ω of slow mode decreases with the increase of θ. One can see from the dispersion relation that the normalized frequency of the slow mode completely vanishes for θ = 90° while for θ = 0°, it becomes maximum $ω_{-} = \sqrt{k^{2} (5 σ / 3 + 1 / (k^{2} + A μ))}$ . Therefore, Figure 4 shows the reduction of frequency ω of the slow mode with the enhancement of θ.

Figure 4.

The frequency ω₋ of slow mode is plotted versus wave vector k to show variation of q, r and angle θ. Here, the parameters are same as the Figure 3.

The nonlinear propagation of the DIAWs belonging to SMS also alters for dusty plasma parameters $(B_{0}, σ, n_{d 0})$ as well as spectral indices (r, q), direction cosine (l_z), and velocity u of the DIAW. First, it may be noted from soliton solution (24) that increasing MF does not affect the amplitude of the soliton; however, the width reduces, since the soliton amplitude (width) is independent (dependent) of the MF through the dispersive coefficient b. We may also observe that the positive amplitude of soliton is sensitive to the direction cosine l_z. Therefore, the amplitude as well as the width of DIASW reduces with increasing l_z. This happens since the soliton amplitude is inversely proportional to the nonlinearity coefficient a which increases by increasing l_z. Furthermore, the critical point is independent of B₀ and direction cosine l_z, so we only get the compressive solitons.

Figure 5 presents the variation of the nonlinearity coefficient a with the concentration n_d0. It is shown that the critical point shifts to the lower value of n_d0 with the increase of value of σ. On the other hand, the critical value shifts to the higher values of n_d0 with the increase of both superthermality and flatness parameter, q and r, respectively. The critical value determines the change of polarity of a soliton from compressive to rarefactive. This critical point exists only in multicomponent plasmas, which in our case, depends upon all plasma parameters except MF and direction cosines.

Figure 5.

The nonlinearity(a) is plotted versus dust concentration (n_d0). Here, the standard curve is plotted for n_e0 = 10²cm⁻³, σ = 0.01, q = 2, r = 0.3.

The dusty plasma parameters also significantly alter the nonlinear characteristics of the DIASWs by changing the amplitude (3u/a) and width ( $\sqrt{4 b / u}$ ) of soliton structure. It may be observed from the expressions that the soliton amplitude increases, whereas the soliton width reduces with the speed u. The influence of dust concentration n_d0 on the soliton amplitude is demonstrated in Figure 6. The curves show that the increase of n_d0 leads to enhancing the compressive soliton amplitude. However the polarity of the SW completely changes to rarefactive, if we further increase n_d0. This behavior occurs due to the existence of critical point in our model with dust concentration n_d0. The influence of σ on amplitude of DIASW is depicted in the Figure 7. It is analyzed that the compressive solitons amplitude enhances with increasing σ. However, the nature of soliton changes to rarefactive solitons with increasing σ due to the existence of critical point.

Figure 6.

Electric potential ϕ₁(ξ, τ) of DIASW is plotted across the dust concentration n_d0. Here, n_e0 = 10²cm⁻³, B₀ = 0.04G, σ = 0.1, q = 2, r = 0.3, τ = 1, u = 0.03, and l_z = 0.75.

Figure 7.

Electric potential ϕ₁(ξ, τ) of DIASW is plotted across the temperature ratio σ. Here, n_e0 = 10²cm⁻³, B₀ = 0.04G, n_d0 = 4cm⁻³, q = 2, r = 0.3, τ = 1, u = 0.03, and l_z = 0.75.

Figure 8 shows the effect of double spectral parameters (r, q) on the properties of the DIASWs. The comparison of the curves shows that the increase of the superthermality index q would lead to the reduction of the amplitude of soliton while the flatness index r enhances solitons amplitude. The comparison between the Maxwellian and non-Maxwellian (r, q) cases shows that the soliton amplitude becomes higher for (r, q) case having higher values of q and r than Maxwellian case. However, the soliton amplitude becomes lower for (r, q) electrons having lower r and q values than Maxwellian case.

Figure 8.

Electric potential ϕ₁(ξ, τ) of DIASW is plotted across the spectral indices (r, q). Here, B₀ = 0.04G, n_e0 = 10²cm⁻³, n_d0 = 4cm⁻³, q = 2, r = 0.3, τ = 1, σ = 0.1, u = 0.03, and l_z = 0.75.

Conclusion

The paper presents the study of oblique propagation of both linear and nonlinear DIAWs in a magnetized collisionless dusty plasma having (r, q) distributed electrons. It has been analyzed that the obliqueness in the present model gives two modes of the phase velocity of the DIAWs. The fast mode has been shown to represent the obliquely propagating ion cyclotron wave, since ω₊ → Ω as k → 0, while the slow mode to represent the magnetized obliquely propagating IAW, affected significantly by the temperature ratio. It has been observed that the frequency ω of the fast mode enhances with increasing the propagation angle, while the frequency of slow mode decreases with the increasing of propagation angle. Moreover, the linear properties of the DIAW are greatly influenced by the magnetic field (MF) and double indices of (r, q) distributed electrons. The nonlinear obliquely traveling DIASWs have been studied by deriving the nonlinear partial differential KdV equation. The critical point has been determined analytically and discussed. It has been observed that the critical point depends upon all plasma parameters and double spectral indices; however, it becomes independent of MF and direction cosine. The amplitude of the solitary structure has been shown to depend upon the dust concentration, temperature ratio, magnetic field, direction cosine, superthermality parameter q and the flatness parameter r. The present work may be extended to the laboratory and space plasma systems, where dusty plasmas and (r, q) distributed electrons have been predicted.

Future work: In this plasma model, if the nonplanar geometrical effect, the collisional force between the plasma particles or the higher-order perturbation is/are considered,^39–41 we will get some non-integrable evolution equations which can be solved using some semi-analytical and numerical approaches.^42–48

Footnotes

Acknowledgments

The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R378), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Author’s contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Princess Nourah Bint Abdulrahman University (PNURSP2023R378).

Availability of Data and Material

All data that support the findings of this study are included within the article.

ORCID iD

S. A. El-Tantawy

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