Electrostatic dust-acoustic rogue waves in an electron depleted dusty plasma

The formation of the gigantic dust-acoustic rouge waves (DARWs) in an electron depleted unmagnetized opposite polarity dusty plasma system is theoretically observed for the first time. The nonlinear Schr\"{o}dinger equation (derived by utilizing the reductive perturbation method) has been analytically as well as numerically analyzed to identify the basic features (viz., height, thickness, and modulational instability, etc.) of DARWs. The results obtained form this investigation should be useful in understanding the basic properties of these rouge waves which can predict to be formed in electron depleted unmagnetized opposite polarity dusty plasma systems like mesosphere, F-rings of Saturn, and cometary atmosphere, etc.

The mechanism of electron depletion is considered as the maximum number of electrons that are inserted onto the massive negative dust grains from the background of DPM during the dust charging process, and the medium in which the electron depletion process occurs is known as electron depleted DPM (EDDPM) [21][22][23]. The laboratory observation of low phase velocity DAWs is associated with significant depletion of the electron number density, suggesting that the wave dynamics is governed by the inertia of the dust fluid and the pressure of inertialess ions only [12,[24][25][26]. The electron depletion process can also be observed in astrophysical environments, viz., Earth's mesosphere (where the D-region plasma could suffer from electron density depletion) [27], cometary tails [4], Jupiter's magnetosphere [5,6], and F-ring of Saturn [28], etc. Shukla and Silin [20] examined DIAWs in an unmagnetized collisionless EDDPM. Ferdousi et al. [19] considered a two-component plasma system containing inertialess positively charged ions and inertial negatively charged dust grains to investigate DA-SHWs, and found that the model supports both positive and negative electrostatic potentials. Mamun et al. [21] considered a two-component EDDPM to investigate the propagation of a nonlinear solitary pulse, and found that the height of the negative potential pulse increases with the number density of ions. Sahu and Tribeche [29] studied electrostatic double-layers (DLs) in an unmagnetized EDDPM with inertial dust grains and inertialess ions, and reported that both compressive and rarefactive DA-DLs are allowed by the plasma model. Hossen et al. [16] examined DAWs in a multi-component EDDPM with inertialess non-thermal ions and inertial massive OP dust grains, and found that the configuration of DA-DLs and DA-SWs is rigourously modified by the existence of positively charged dust grains.
The super-thermal κ-distribution [30][31][32][33][34][35][36][37] can describe the deviation, according to the values of the super-thermal parameter κ which manifests the presence of external force fields or wave-particle interactions, of plasma species from the thermally equilibrium state or Maxwellian distribution. The super-thermal κ-distribution exchanges with the Maxwellian distribution when κ tends to infinity, i.e., κ → ∞, and κ-distribution is normalizable for any values of κ rather than κ > 3/2 [32][33][34][35][36][37]. Uddin et al. [32] numerically analyzed the propagation of nonlinear electrostatic positron-acoustic waves in a superthermal plasma, and reported that the amplitude of the electrostatic positive potential decreases with an increasing value of κ. Shahmansouri and Alinejad [33] examined DA-SWs in a DPM with super-thermal plasma species, and found that the depth of the potential well decreases with the increase in the value of κ. Kourakis and Sultana [34] examined the speed of the DIA solitons in the presence of super-thermal particles in a DPM.
The modulational instability (MI) of the carrier waves, energy re-distribution and localization are governed by the standard nonlinear Schrödinger equation (NLSE) [35][36][37][38][39][40][41][42]. Sultana and Kourakis [35] examined the electron-acoustic (EA) envelope solitons in a plasma medium with super-thermal electrons, and found that the stable domain of EA waves decreases with increasing κ. Ahmed et al. [36] reported IAWs in a four-component plasma medium, and highlighted that the critical wave number (k c ) increases with a decrease in the value of κ. Saini and Kourakis [38] demonstrated the MI of the DAWs as a DPM with super-thermal ions, and obtained that excess super-thermality of the ions leads to a narrower envelope of solitons.
Recently, Hossen et al. [15] considered a three-component plasma model with inertial OP dust grains and non-extensive electrons to investigate the propagation of the DA-SWs. In this paper, we aim to extend on previous published work [15] by considering a real and novel four-component EDDPM with inertial OP dust grains and inertialess iso-thermal negative ions and super-thermal positive ions to examine the MI of DAWs and the formation of DA rogue waves (DARWs).
The layout of the paper is as follows: The governing equations are given in Section 2. The derivation of the standard NLSE is demonstrated in Section 3. The MI and rogue waves are presented in Section 4. The results and discussion are provided in Section 5. A brief conclusion is presented in Section 6.

Basic Equations
We consider a four-component EDDPM consisting of inertial negatively charged dust grains (mass m − ; charge q − = −eZ − ), positively charged dust grains (mass m + ; charge q + = eZ + ), inertialess negatively charged ions (mass m 1 ; charge q 1 = −eZ 1 ), and positively charged ions (mass m 2 ; charge q 2 = eZ 2 ); where Z − , Z + , Z 1 , and Z 2 are the charge state of the negative dust, positive dust, negative ion, and positive ion, respectively, and e is the magnitude of a single electron charge. At equilibrium, the quasi-neutrality condition can be expressed as Z − n −0 + Z 1 n 10 = Z + n +0 + Z 2 n 20 ; where n −0 , n 10 , n +0 , and n 20 are, respectively, the equilibrium number densities of negative dust grains, negative ions, positive dust grains, and positive ions. Now, after normalization, the set of basic equations can be written as ∂n + ∂t where n − and n + are the negative and positive dust grains number density normalized by their equilibrium value n −0 and n +0 , respectively; u − and u + are the negative and positive dust fluid speed normalized by DA wave speed C + = (Z + k B σ 1 /m + ) 1/2 (with σ 1 being the temperature negative ion and k B being the Boltzmann constant); φ is the electrostatic wave potential normalized by k B σ 1 /e; the time and space variables are normalized by ω −1 and n −0 > n +0 for our numerical analysis. The expression for heavy negative ion number density obeying iso-thermal Maxwelliann distribution is given by The expression for light positive ion number density obeying super-thermal κ-distribution is given by [36] where λ 4 = σ 1 /σ 2 (with σ 2 being the temperature positive ion) and σ 2 > σ 1 . The superthermality of the light positive ion can be represented by the parameter κ. Now, by substituting Equations (6) and (7) into Equation (5), and expanding up to the third order of φ, we get where 3 .
We note that the terms containing T 1 , T 2 , and T 3 are due to the contribution of positive and negative ions.

Derivation of the NLSE
The reductive perturbation method is applicable to derive the standard NLSE as well as to study the MI of the DAWs in a four-component EDDPM. The stretched co-ordinates, to develop a standard NLSE, can be written as [39][40][41][42] where v g is the group velocity and is a small parameter which measures the nonlinearity of the plasma medium. Then, the dependent variables can be written as [39][40][41][42] For the above consideration, the derivative operators can be represented as [39][40][41][42] ∂ ∂t Now, by substituting Equations (9)-(17) into Equations (1)-(4), and (8), and collecting the terms containing , the first order (m = 1 with l = 1) reduced equations can be written as these relations provide the dispersion relation for DAWs The second-order (m = 2 with l = 1) equations are given by with the compatibility condition The coefficients of for m = 2 with l = 2 provide the second-order harmonic amplitudes which are found to be proportional to |φ where . Now, we consider the expression for (m = 3 with l = 0) and (m = 2 with l = 0), which leads the zeroth harmonic modes. Thus, we obtain where Finally, the third harmonic modes (m = 3) and (l = 1) of Equations (1)-(4), and (8), and with the help of Equations (18)- (26) give a set of equations which can be reduced to the following NLSE: where Φ = φ (1) 1 for simplicity. In Equation (27), P is the dispersion coefficient which can be written as and also Q is the nonlinear coefficient which can be written as . It may be noted here that both P and Q are functions of various plasma parameters such as k, λ 1 , λ 2 , λ 3 , λ 4 , and κ. So, all the plasma parameters are used to maintain the nonlinearity and the dispersion properties of the EDDPM.

Modulational Instability and Rogue Waves
The nonlinear property of DPM as well as the stability conditions of the DAWs in an EDDPM can be determined according to the sign of dispersive (P) and nonlinear (Q) coefficients of the standard NLSE (27) [39][40][41][42]. The sign of the dispersive coefficient is always negative for any kind of wave number k in the "P vs. k curve", while the sign of the nonlinear coefficient is positive for small values of k and negative for large values of k in "Q vs. k curve" (figures are not included). Both P and Q have the same sign (i.e., positive or negative), then they determine a modulationally unstable parametric regime (i.e., P/Q > 0) of DAWs in the "P/Q vs. k curve", and when both P and Q have the opposite sign (i.e., positive and negative), then they determine a modulationally stable parametric regime (i.e., P/Q < 0) of DAWs in the presence of external perturbation [39][40][41][42]. The stable and unstable parametric regimes are differentiated by a vertical line at which P/Q → ∞ (i.e., Q = 0 and because P is always negative), and the wave number for which P/Q → ∞ is known as the critical waves number (k ≡ k c ) [39][40][41][42]. We have numerically analyzed the modulationally stable and unstable parametric regimes of DAWs in Figure 1. The modulationally unstable parametric regime of the DAWs allows to generate highly energetic and mysterious DARWs associated with DAWs, and the governing equation of the puzzling DARWs can be written as [43,44] We have numerically analyzed Equation (28) in Figure 2 to understand how the nonlinear properties of a four-component EDDPM and the configuration of the DARWs associated DAWs have been changed by the variation of different plasma parameters.

Results and Discussion
The variation of P/Q with k for different values of κ as well as the stable and unstable parametric regimes of the DAWs can be observed in the left panel of Figure 1, and it is obvious from this figure that (a) DAWs become modulationally stable for small values of k, while unstable for large values of k; (b) the k c decreases with the increase in the value of κ. The right panel of Figure 1 indicates the effects of the number density and charge state of negative ions and positive dust grains (via λ 2 = Z 1 n 10 /Z + n +0 ) to recognize the stable and unstable parametric regimes of the DAWs, and it can be seen from this figure that (a) the modulationally stable parametric regime increases with λ 2 ; (b) the modulationally stable parametric regime increases with an increase in the value of negative ion number density (n 10 ) while it decreases with an increase in the value of the positive dust grains number density (n +0 ) for a constant value of Z 1 and Z + ; (c) the modulationally stable parametric regime of the DAWs increases (decreases) with Z 1 (Z + ) for a fixed negative ion and positive dust number density.
We have numerically analyzed first-order rogue waves [using Equation (28)] in Figure 2. The left panel of Figure 2 highlights the effects of the mass and charge state of the positive and negative dust grains (via λ 1 = Z − m + /Z + m − ) in recognizing the shape of the DARWs in an EDDPM, and it can be manifested from this figure that (a) the increase in the value of λ 1 does not only cause to change the amplitude of the DARWs but also causes to change the width of the DARWs; (b) the amplitude and width of the DARWs decrease with the increase in the value of λ 1 ; (c) actually, the nonlinearity of the plasma medium as well as the amplitude of the DARWs increases with increasing m − , while the nonlinearity as well as the amplitude of the DARWs decreases with increasing m + for a fixed value of Z + and Z − . The right panel of Figure 2 reflects how the number density and charge state of the positive ion and dust grains (via λ 3 = Z 2 n 20 /Z + n +0 ) can contribute to generate highly energetic rogue waves in a four-component EDDPM. The amplitude and width of the electrostatic DARWs associated with DAWs decreases with an increase in the value of positive ion charge state and number density, while increases with an increase in the value of positive dust grain charge state and number density. The physics of this result is that the nonlinearity of a four-component EDDPM increases (decreases) with positive dust grain (negative ion) number density as well as with positive dust grain (negative ion) charge state.

Conclusions
In this present article, we have considered a four-component EDDPM with inertial opposite polarity dust grains and inertialess positive and negative ions. A nonlinear Schrödinger equation (NLSE) is derived by employing the reductive perturbation method. We have theoretically predicted the modulationally stable and unstable parametric regimes of DAWs, and associated DARWs in the modulationally unstable parametric regime of DAWs in an EDDPM. The results that have been found from our investigation can be summarized as follows: • Both modulationally stable and unstable parametric regimes of DAWs are allowed by the plasma model. • The critical wave number (k c ) decreases with the increase in the value of κ.

•
The nonlinearity of the plasma medium as well as the amplitude of the DARWs increases (decreases) with increasing negative (positive) dust mass.
It may be noted here that the gravitational effect is very important to consider in the governing equations but is beyond the scope of our present work. However, electrons' depletion process can be observed in astrophysical OPDP mediums, viz., Earth's mesosphere (where the D-region plasma could suffer from electron density depletion) [27], cometary tails [4], Jupiter's magnetosphere [5,6], F-ring of Saturn [28], and also laboratory EDDPM [12,[24][25][26]. To conclude, the results obtained from the present investigation will be useful in the future to understand the MI of DAWs and associated DARWs in Earth's mesosphere (where the D-region plasma could suffer from electron density depletion) [27], cometary tails [4], Jupiter's magnetosphere [5,6], F-ring of Saturn [28], and laboratory EDDPM [12,[24][25][26].