Now-a-days, the study of dusty plasma (DP) is one of the most rapidly growing branches in plasma physics due to their existence in space, viz., planetary rings,^[1] cometary tails,^[2] Jupiterʼs magnetosphere,^[2] lower part of the Earthʼs atmosphere^[1] and also in laboratory plasmas.^[3–7] The DP have generally considered to be an ensemble of negatively charged dust grains, free electrons, and ions. However, the co-existence of opposite polarity (OP) dust grains in plasmas introduces a new DP model called OP DP (OPDP) whose main constituent species are positively and negatively charged warm massive dust grains.^[2,6] The exclusive property of this OPDP, which makes it completely unique from other plasmas (viz., electron ion and electron-positron plasmas), is that the ratio of the size of positively charged dust grains to that of negatively charged dust grains can be smaller^[8] or larger^[9] or equal to unity.^[9] There are three main processes by which dust grains become positively charged: (a) Secondary emission of electrons from the surface of the dust grains; (b) Thermionic emission induced by the radiative heating; (c) Photoemission in the presence of a flux of ultraviolet photons.^[1,10]

The researchers have focused on wave dynamics, specifically, dust-acoustic (DA) waves (DAWs), dust-acoustic rogue waves (DARWs), and dust ion-acoustic waves (DIAWs) in understanding electrostatic density perturbations and potential structures (viz., shock, soliton, envelope solitons,^[11–12] and rogue waves^[13–15]) in DP. Rao et al.^[14] first theoretically predicted the existence of very low frequency DAWs (where the inertia is provided by the dust mass and restoring force is provided by the thermal pressure of electrons and ions) in comparison with the electron and ion thermal velocities and this theoretical prediction has been conclusively verified by Barken et al.^[4] There is also direct evidence for the co-existence of both positively and negatively charged dust grains in different regions of space plasmas (viz., cometary tails,^[2] upper mesosphere,^[2] and Jupiterʼs magnetosphere,^[10] etc.) and laboratory devices (viz., direct current and radio-frequency discharges,^[1] plasma processing reactors,^[16] fusion plasma devices, and solid-fuel combustion products,^[1] etc). The novelty of this OPDP has attracted numerous authors^[17–21] to investigate the linear and nonlinear propagation of electrostatic waves. Sayed and Mamun^[2] studied the finite solitary potential structures that exist in OPDP. El-Taibany^[17] examined the DAWs in inhomogeneous four component OPDP, and observed that only compressive soliton is created corresponding to fast DA mode.

In space and astro-physical situations, if the plasma species move very fast compared to their thermal velocities^[22] then the Maxwellian distribution is no longer valid to explain the dynamics of these plasma species. For that reason, Tsallis proposed the non-extensive statistics,^[23] which is the generalisation of Boltzmann-Gibbs-Shannon entropy. The importance of Tsallis statistics is that it can easily describe the long range interactions of the electron-ion in DP system.^[13–15] The research regarding modulational instability (MI) of DAWs in nonlinear and dispersive mediums has been increasing significantly due to their existence in astrophysics, space physics^[11–15] as well as in application in many laboratory situations.^[11] A large number of researchers have used the nonlinear Schrödinger equation (NLSE), which governs the dynamics of the DAWs, to study the formation of the envelope solitons or rogue waves^[13–15] in DP. Bains et al.^[13] investigated the MI of the DAWs in non-extensive DP. Moslem et al.^[14] have studied the MI of the DAWs in three component DP in presence of the non-extensive electrons and ions, and have found that the threshold wave number (k_c) increases with q. Duan et al.^[19] have investigated the criteria for MI of the DAWs and the formation of envelope solitons in OPDP. Zaghbeer et al.^[20] have reported DARWs in a four component OPDP. Gill et al.^[21] have studied MI of DAWs in a four component OPDP, and have found that the presence of positive dust grains significantly modify the domain of the MI and localized envelope solitons. To the best knowledge of the authors, no attempt has been made to study the MI and corresponding dark and bright envelope solitons associated with the DAWs in a four component OPDP in presence of OP warm adiabatic dust grains. The aim of the present investigation is therefore to extend the work of Gill et al.^[21] by examining the conditions for the MI of the DAWs (in which inertia is provided by the OP warm dust masses and restoring force is provided by the thermal pressure of q-distributed electrons and iso-thermal ions) in four component OPDP.

The manuscript is organized as the following fashion: The governing equations of our plasma model are provided in Sec. 2. The NLSE is derived in Sec. 3. The stability of DAWs is examined in Sec. 4. Envelope solitons is presented in Sec. 5. Finally, a brief discussion is provided in Sec. 6.

In this paper, we consider a collisionless, fully ionized, unmagnetized four component dusty plasma system composed of q-distributed electrons (charge −e, mass m_e), iso-thermal ions (charge + e, mass m_i) and inertial warm negatively charged dust grains (charge , mass m₁) as well as positively charged warm dust grains (charge , mass m₂), where z₁ (z₂) is the charge state of the negatively (positively) charged warm dust particles. The negatively and positively charged warm dust grains can be displayed by continuity and momentum equations, respectively, as:



where n₁ (n₂) is the number densities of the negatively (positively) charged warm dust grains; t (x) is the time (space) variable; u₁ (u₂) is the fluid speed of the negatively (positively) charged warm dust species; e is the magnitude of the charge of the electron; φ is the electrostatic wave potential; p₁ (p₂) is the adiabatic pressure of the negatively (positively) charged warm dust grains. The system is enclosed through Poissonʼs equation as
where n_i and n_e are, respectively, the ion and electron number densities. The quasi-neutrality condition at equilibrium can be written as
where n_i0, n₂₀, , and n₁₀ are the equilibrium number densities of the iso-thermal ions, positively charged warm dust grains, q-distributed electrons, and negatively charged warm dust grains, respectively. Now, in terms of normalized variables, namely, , ; (with being the sound speed of the negatively charged warm dust grains); , (with T_i being the temperature of the iso-thermal ion); (with being the plasma frequency of the negatively charged warm dust grains); (with being the Debye length of the negatively charged warm dust grains); , , ; (with p₁₀ being the equilibrium adiabatic pressure of the negatively charged warm dust grains and , where N is the degree of freedom, for one-dimensional case, N = 1 so that γ = 3); (with T₁ being the temperature of the negatively charged warm dust grains and is the Boltzmann constant); (with p₂₀ being the equilibrium adiabatic pressure of the positively charged warm dust grains) and (with T₂ being the temperature of the positively charged warm dust particles). After normalization, the governing equations (1)–(5) can be written as




where , , , , and . It may be noted here that we have considered for our numerical analysis , , and T_e, , T₂. The number densities of the non-extensive q-distributed^[15] electron can be given by the following normalized equation
where (with T_e being the temperature of the non-extensive q-distributed electron and ) and q is the non-extensive parameter describing the degree of non-extensivity, i.e., q = 1 indicates the Maxwellian distribution, whereas refers to the super-extensivity, and the opposite condition corresponds to the sub-extensivity.^[15] The number densities of the iso-thermally distributed^[15] ion can be represented as
Now, by substituting Eqs. (12) and (13) into Eq. (11), and extending up to the third order in ϕ, we can obtain
where
The left hand side of Eq. (14) is the contribution of electron and ion species.

We will use the reductive perturbation method (RPM) to derive the NLSE for studying the MI of the DAWs in OPDP. Now, the stretched co-ordinate^[15] can be defined as

where V_g is the envelope group velocity and ϵ is a small but real parameter. The dependent variables^[15] can be written as




where and k (ω) is the carrier wave number (frequency). The derivative operators in the above equations are considered as follows:

Now, by substituting Eqs. (15)–(23) into Eqs. (7)–(10), and,and Eq. (14) and collecting power term of ϵ, the first order approximation (m = 1) with the first harmonic (l = 1) provides the following relation




where and . Now, these equations can be reduced to the following pattern



where and . Therefore, the dispersion relation for the DAWs can be written as
where , , and . The condition must be satisfied in order to obtain real and positive values of ω. Normally, two types of DA modes exist, namely, fast ( ) and slow ( ) DA modes according to the positive and negative sign of Eq. (33). Now, we have studied the dispersion properties by depicting ω with k in Figs. 1 and 2 which clearly indicates that (a) The fast DA mode exponentially increases with k for its lower range, but a saturation starts after a certain value of k; (b) The value of increases exponentially with the increasing values of z₂ for fixed value of z₁, n₂₀, and n₁₀ (see in Fig. 1); (c) On the other hand, the slow DA mode linearly increases with k; (d) The decreases with the increase of z₂ for the fixed value of z₁, n₂₀, and n₁₀ (see in Fig. 2). This result agrees with the result of previous published work.^[6,18] It is important to mention that in fast DA mode, both positive and negative warm dust species oscillate in phase with electrons and ions. Whereas in slow DA mode, only one of the inertial massive dust components oscillate in phase with electrons and ions, but the other species are in anti-phase with them.^[15] Next, with the help of second-order (m = 2 with l = 1) equations, we obtain the expression of V_g as

Fig. 1

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	Fig. 1 The variation of ωf with k for different values of β, along with α = 1.2, δ = 0.3, μi = 1.4, σ1 = 0.0001, σ2 = 0.001, and q = 1.8.

Fig. 2

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	Fig. 2 The variation of with k for different values of β, along with α = 1.2, δ = 0.3, μi = 1.4, σ1 = 0.0001, σ2 = 0.001, and q = 1.8.

From the next order of ϵ, we can get the second-harmonic mode of the carrier wave with m = 2 and l = 2 as




where
Now, we consider the expressions for (m = 3, l = 0) and (m = 2, l = 0), which lead to the zeroth harmonic modes. Finally, we get




where
Now, we can obtain the standard NLSE from the third harmonic (m = 3, l = 1) modes with the help of Eqs. (29)–(44) which can be written as
where for simplicity and the dispersion (P) and nonlinear (Q) coefficient are, respectively, written as
where

The DAWs are modulationally stable against external perturbation when . On the other hand, when , the DAWs are modulationally unstable against external perturbation. When , the corresponding value of k ( = k_c) is called the critical or threshold wave number (k_c) for the onset of MI. The variation of P/Q with k for μ_i and α are shown in Figs. 3 and 4, respectively, which clearly indicate that (a) The value of k_c increases with the increase of n_i0 for fixed value of z₁ and n₁₀; (b) On the other hand, k_c value decreases with the increase of m₂ for fixed value of m₁, z₂, and z₁. The growth rate (Γ) of the modulationally unstable region for the DAWs (when and ) can be written as^{[6,11–12,15,18]}

Fig. 3

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	Fig. 3 The variation of P/Q with k for different values of μi, along with α = 1.2, β = 0.07, δ = 0.3, σ1 = 0.0001, σ2 = 0.001, q = 1.8, and ωf.

Fig. 4

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	Fig. 4 The variation of P/Q with k for different values of α, along with β = 0.07, δ = 0.3, μi = 1.4, σ1 = 0.0001, σ2 = 0.001, q = 1.8, and ωs.

Now, we have graphically shown how the Γ varies with for different values of α and q in Figs. 5–9. It is obvious from Figs. 5–7 that (a) Within three limits of q (q = 1, q=+ve, and q=−ve), the maximum value of Γ increases with the increases in the value of z₂ for fixed values z₁, m₁, and m₂ (via α); (b) So, the effects of the α on the maximum value of the growth rate is independent from the various limits of q. The physics of this result is that the nonlinearity, which leads to increase the maximum value of the growth rate of DAWs, increases with the increase in the values of α.

Fig. 5

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	Fig. 5 The variation of Γ with for different values of α (when q = 1.0), along with β = 0.07, δ = 0.3, μi = 1.4, σ1 = 0.0001, σ2 = 0.001, , k = 0.6, and ωf.

Fig. 6

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	Fig. 6 The variation of Γ with for different values of α (when q = 1.5), along with β = 0.07, δ = 0.3, μi = 1.4, σ1 = 0.0001, σ2 = 0.001, , k = 0.6, and ωf.

Fig. 7

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	Fig. 7 The variation of Γ with for different values of α (when q = −0.6), along with β = 0.07, δ = 0.3, μi = 1.4, σ1 = 0.0001, σ2 = 0.001, , k = 0.6, and ωf.

Fig. 8

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	Fig. 8 The variation of Γ with for q ( ), along with α = 1.2, β = 0.07, δ = 0.3, μi = 1.4, σ1 = 0.0001, σ2 = 0.001, , k = 0.6, and ωf.

Fig. 9

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	Fig. 9 The variation of Γ with for q ( ), along with α = 1.2, β = 0.07, δ = 0.3, μi = 1.4, σ1 = 0.0001, σ2 = 0.001, , k = 0.6, and ωf.

The effects of non-extensivity of the electrons on the MI growth rate can be observed from Figs. 8 and 9, and it is obvious from these figures that (a) The maximum value of Γ decreases (decreases) with the increase in the values of q for the limits of ( ); (b) So, the variation of Γ with respect to is independent on the sign of the q. This result agrees with the result of previous published work.^[18]

The bright (when ) and dark (when ) envelope solitonic solutions, respectively, can be written as^[24–27]

where ψ₀ indicates the envelope amplitude, U is the traveling speed of the localized pulse, W is the pulse width which can be written as , and Ω₀ is the oscillating frequency for U = 0. The bright (by using Eq. (47)) and dark (by using Eq. (48)) envelope solitons are depicted in Figs. 10 and 11, respectively.

Fig. 10

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	Fig. 10 The variation of Re(Φ) with ξ for k = 0.6 (bright envelope solitons), along with α = 1.2, β = 0.07, δ = 0.3, μi = 1.4, σ1 = 0.0001, σ2 = 0.001, τ = 0, ψ0 = 0.008, q = 1.5, Ω0 = 0.4, U = 0.4, and ωf.

Fig. 11

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	Fig. 11 The variation of Re(Φ) with ξ for k = 0.2 (dark envelope solitons), along with α = 1.2, β = 0.07, δ = 0.3, μi = 1.4, σ1 = 0.0001, σ2 = 0.001, τ = 0, ψ0 = 0.008, q = 1.5, Ω0 = 0.4, U = 0.4, and ωf.

We have studied an unmagnetized realistic space dusty plasma system consists of q-distributed electrons, iso-thermal ions, positively charged warm dust grains as well as negatively charged warm dust grains. The RPM is used to derive the NLSE. The results that have been found from our investigation can be summarized as follows:

(i) The fast DA mode increases exponentially with z₂ for fixed value of z₁, n₂₀, and n₁₀ (via β). On the other hand, the slow DA mode linearly decreases with the increase of z₂ for the fixed value of z₁, n₂₀, and n₁₀ (via β).

(ii) The DAWs is modulationally stable (unstable) in the range of values of k in which the ratio P/Q is ( ).

(iii) The value of k_c increases with the increase of n_i0 for fixed value of z₁ and n₁₀ (via μ_i and for fast mode). On the other hand, k_c value decreases with the increase of m₂ for fixed value of m₁, z₂, and z₁ (via α and for slow mode).

(iv) The value of Γ increases with α for fixed value of q (within three ranges of q, namely, , q = 1, and ). So, the variation of Γ with α is independent of possible values of q.

(v) The maximum value of Γ decreases (decreases) with the increase in the values of q for the limits ( ). So, the growth rate is independent on the sign of the q.

The results of our present investigation will be useful in understanding the nonlinear phenomena both in space (viz., Jupiters magnetosphere,^[2] upper mesosphere, and comets tails,^[2] etc.) and laboratory (viz., direct current and radio-frequency discharges, plasma processing reactors, fusion plasma devices,^[1] and solid-fuel combustion products,^[1] etc.) plasma system containing q-distributed electrons, iso-thermal ions, negatively and positively charged massive warm dust grains in OPDP medium.