Electrostatic dust-acoustic rogue waves in an electron depleted dusty plasma
J. N. Sikta N. A. Chowdhury A. Mannan S. Sultana A. A. Mamun




^1 Faculty of Science and Information Technology, Daffodil International University, Dhaka-1207, Bangladesh

^2 Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh

^3 Plasma Physics Division, Atomic Energy Centre, Dhaka-1000, Bangladesh

^4 Institut für Mathematik, Martin Luther Universität Halle-Wittenberg, Halle, Germany

^5 Fakultät für Physik und Astronomie, Ruhr-Universittät Bochum, D-44780 Bochum, Germany

E-mail: ^*jebunphy@gmail.com, ^**nurealam1743phy@gmail.com

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 Schrödinger 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.


NLSE modulational instability rogue waves.




§ INTRODUCTION


Opposite polarity (OP) dusty plasma (OPDP) is demonstrated as fully ionized gases consisting of massive  negatively and positively
charged dust grains as well as ions and electrons in presence
of electrostatic and gravitational force fields, and is identified in space, viz., cometary
tails <cit.>, interstellar clouds <cit.>, Planetary rings <cit.>,
solar system <cit.>, Earth polar mesosphere <cit.>, magnetosphere of
Jupiter <cit.>, and laboratory situations, viz., laser-matter interaction
<cit.>. First, Rao et al. <cit.> traced that how the presence
of the  massive dust grains alters the picture of the dynamics of dusty plasma medium (DPM), and also theoretically
predicted new kind of low-frequency dust-acoustic (DA) waves (DAWs). The distinction of low-frequency
DAWs from ion-acoustic waves (IAWs) and experimental verification was, finally, confirmed by
Barkan et al. <cit.> in DPM. The experimental identification
of the DAWs in DPM have mesmerized many plasma physicists to investigate numerous modern electrostatic eigen modes,
viz., dust-drift waves <cit.>, DA solitary waves (DA-SWs) <cit.>,
DAWs <cit.>, dust lattice waves <cit.>,
DA shock waves (DA-SHWs) <cit.>, and dust-ion-acoustic waves (DIAWs) <cit.>
to understand the basic features of the various nonlinear electrostatic structures associated with
the propagation of low frequency electrostatic disturbance.

The mechanisms of the electrons depletion <cit.>,
in which maximum number of electrons are inserted onto the massive negative dust grains from the
background of DPM during the dust charging process, are ubiquitous
in space environments, viz., interstellar clouds <cit.>, F-ring of
Jupiter's magnetosphere <cit.>, Earth polar mesosphere <cit.>,
Saturn <cit.>, cometary tails <cit.>, solar system <cit.>, and also in laboratory DPM.
The attachment of electrons with massive negatively charge dust grains is considered as electron depleted
DP (EDDP) medium (EDDPM). Shukla and Silin <cit.> examined DIAWs in an unmagnetized collisionless EDDPM.
Ferdousi et al. <cit.> considered a two components 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. <cit.> considered
a two components EDDP model for investigation of the propagation of
nonlinear solitary pulses, and found that the height of the negative potential pulses
increases with the number density of ion and dust.
Sahu and Tribeche <cit.> studied electrostatic double-layers (DLs) in an unmagnetized EDDPM having inertial dust grains and inertiales ion,
and reported that both compressive and rarefactive DA-DLs are allowed by their plasma model.
Hossen et al. <cit.> examined DAWs in a multi-component EDDPM
having  inertialess non-thermal ions and inertial massive OP dust grains, and found that the
configuration of DA-DLs and DA-SWs is regourously modified by the existence of the positively charged dust grains.

The super-thermal or κ-distribution <cit.>
can describe the deviation, according to the values of the super-thermal parameter
κ which manifests the presence of the external force fields or wave-particle
interactions, of plasma species from the thermal or Maxwellian distribution. The super-thermal or
κ-distribution exchanges with the Maxwellian distribution when κ tends
to infinity, i.e., κ→∞, and κ-distribution
is normalizable for any kind of values of κ rather than κ>3/2 <cit.>.
Uddin et al. <cit.> numerically analyzed the propagation of nonlinear electrostatic
positron-acoustic waves in a super-thermal plasma, and reported that the amplitude of the
electrostatic positive potential decreases with increasing value of κ.
Shahmansouri and Alinejad <cit.> examined DA-SWs in a DPM having  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 <cit.> examined the speed of the DIA solitons in presence of the super-thermal
particles in a DPM.

The modulational instability (MI), energy localization, and energy redistribution of the carrier
waves are governed by the standard nonlinear Schrödinger equation (NLSE) <cit.>.
Sultana and Kourakis <cit.> examined the electron-acoustic (EA) envelope solitons in a plasma medium having
super-thermal electrons, and found that the stable domain of EA waves decreases with increasing κ.
Ahmed et al. <cit.> reported IAWs in a four-components plasma medium, and
highlighted that the critical wave number (k_c) increases with a decrease in the value of κ.
Saini and Kourakis <cit.> demonstrated the MI of the DAWs a DPM having super-thermal ions,
and obtained that excess super-thermality of the ions leads a narrower envelope solitons.

Recently, Hossen et al. <cit.> considered three components plasma model having
inertial OP dust grains and non-extensive electrons to investigate the propagation of the DA-SWs.
In this paper, we want to develop sufficient extension of previous published work <cit.>
by considering a real and novel four components EDDP model having inertial OP dust grains and inertialess
iso-thermal negatives ions and super-thermal positive ions to examine the MI of DAWs and formation of DA rogue
waves (DARWs).

The layout of the paper is as follows: The governing equations are given in Section <ref>.
The derivation of the standard NLSE is devoted in Section <ref>.
MI and rogue waves are presented in Section <ref>.
Results and discussion are provided in Section <ref>.
A brief conclusion is demonstrated in Section <ref>.


§ BASIC EQUATIONS


We consider a four components plasma medium consisting of inertial positively and negatively
charged massive dust grains, and inertialess positive and  negative ions. At equilibrium, the
quasi-neutrality condition can be expressed as Z_+n_+0+n_20=Z_-n_-0+n_10; where
n_+0, n_10, n_-0, and n_20 are, respectively, the equilibrium number densities of positive dust grains, positive ions,
negative dust grains, and negative ions. Now, after normalization the set of the equations can be written as

    ∂ n_-/∂ t+∂/∂ x(n_-u_-)=0,
    
          ∂ u_-/∂ t+u_-∂ u_-/∂ x=λ_1∂ϕ/∂ x,
    
          ∂ n_+/∂ t+∂/∂ x(n_+u_+)=0,
    
          ∂ u_+/∂ t+u_+∂ u_+/∂ x=-∂ϕ/∂ x,
    
          ∂^2ϕ/∂ x^2=λ_2n_1-λ_3n_2+(1-λ_2+λ_3)n_–n_+,

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 wave speed C_+=(Z_+K_BT_-/m_+)^1/2
(with T_- being the temperature negative ion, m_+ being the positive dust mass,
and K_B being the Boltzmann constant); ϕ is the electrostatic wave potential
normalized by K_BT_-/e (with e being the magnitude of single electron charge); the
time and space variables are normalized by ω_p+^-1=(m_+/4π e^2 Z_+^2n_+0)^1/2
and λ_D+=(k_BT_-/4π e^2 Z_+n_+0)^1/2, respectively; and λ_1=Z_-m_+/Z_+m_-,
λ_2=Z_1n_10/Z_+n_+0, and λ_3=Z_2n_20/Z_+n_+0. We have considered m_-> m_+, Z_->Z_+, and n_-0>n_+0 for our numerical analysis.
Now, the expression for  positive ion number density obeying κ-distribution is given by <cit.>

    n_1=[1 +λ_4ϕ/κ-3/2]^-κ+1/2,

where λ_4 =T_-/T_+ (with T_+ being the temperature positive ion) and T_+>T_-.
The super-thermality of the light positive ion can be represented by the  parameter κ.
The expression for negative ion number density obeying iso-thermal Maxwelliann distribution is given by

    n_2=(ϕ).

Now, by substituting Eqs. (<ref>) and (<ref>) into Eq. (<ref>), and expanding up to third order of ϕ, we get

    ∂^2ϕ/∂ x^2 + n_++λ_3= λ_2 +(1-λ_2+λ_3)n_-
          + T_1ϕ + T_2ϕ^2 + T_3ϕ^3 +···,

where

    T_1=λ_2(2κ-3)+λ_3λ_4(2κ-1)/(2κ-3),
    
          T_2=λ_2(2κ-3)^2-λ_3λ_4^2(2κ-1)(2κ+1)/2(2κ-3)^2,
    
          T_3=λ_2(2κ-3)^3+λ_3λ_4^3(2κ-1)(2κ+1)(2κ+3)/6(2κ-3)^3.



§ 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 components electrons depleted dusty plasma. The stretched co-ordinates, to develop a standard NLSE, can be written as

    ξ=ϵ(x-v_g t),
    
          τ=ϵ^2 t,

where v_g is the group velocity and ϵ (ϵ≪ 1) is a small parameter which measures
the nonlinearity of the plasma medium. Then, the dependent variables can be written as

    n_-=1+∑_m=1^∞ϵ^m∑_l=-∞^∞ n_-l^(m)(ξ,τ) [i l(kx-ω t)],
    
          u_-=∑_m=1^∞ϵ^m∑_l=-∞^∞ u_-l^(m)(ξ,τ) [i l(kx-ω t)],
    
          n_+=1+∑_m=1^∞ϵ^m∑_l=-∞^∞ n_+l^(m)(ξ,τ) [i l(kx-ω t)],
    
          u_+=∑_m=1^∞ϵ^m∑_l=-∞^∞ u_+l^(m)(ξ,τ) [i l(kx-ω t)],
    
          ϕ=∑_m=1^∞ϵ^m∑_l=-∞^∞ϕ_l^(m)(ξ,τ) [i l(kx-ω t)].

For the above consideration, the derivative operators can be recognized as

    ∂/∂ t→∂/∂ t-ϵ v_g ∂/∂ξ
    +ϵ^2∂/∂τ,
    
          ∂/∂ x→∂/∂ x+ϵ∂/∂ξ.

Now, by substituting Eqs. (<ref>)-(<ref>)  into Eqs. (<ref>)-(<ref>), and (<ref>), and
collecting the terms containing ϵ, the first order (m=1 with l=1)  reduced equations can be written as

    n_-1^(1)=-λ_1 k^2/ω^2ϕ_1^(1),
    
          u_-1^(1)=-λ_1 k/ωϕ_1^(1),
    
          n_+1^(1)=k^2/ω^2ϕ_1^(1),
    
          u_+1^(1)=k /ωϕ_1^(1),

these relation provides the dispersion relation for DAWs

    ω^2=k^2+λ_1 k^2-λ_1λ_2 k^2+λ_1λ_3 k^2/k^2+T_1.

The second-order (m=2 with l=1) equations are given by

    n_-1^(2)=-λ_1 k^2/ω^2ϕ_1^(2)-2iλ_1k(v_gk-ω)/ω^3∂ϕ_1^(1)/∂ξ,
    
          u_-1^(2)=-λ_1 k/ωϕ_1^(2)-iλ_1(v_gk-ω)/ω^2∂ϕ_1^(1)/∂ξ,
    
          n_+1^(2)=k^2/ω^2ϕ_1^(2)+2ik(v_gk-ω)/ω^3∂ϕ_1^(1)/∂ξ,
    
          u_+1^(2)=k/ωϕ_1^(2)+i(v_gk-ω)/ω^2∂ϕ_1^(1)/∂ξ,

with the compatibility condition

    v_g=ω+ωλ_1-ωλ_1λ_2+ωλ_1λ_3-ω^3/k+kλ_1-kλ_1λ_2+kλ_1λ_3                            .

The coefficients of ϵ for m=2 with l=2 provide the second
order harmonic amplitudes which are found to be proportional to |ϕ_1^(1)|^2

    n__2^(2)=T_4|ϕ_1^(1)|^2,
    
          u_-2^(2)=T_5 |ϕ_1^(1)|^2,
    
          n_+2^(2)=T_6|ϕ_1^(1)|^2,
    
          u_+2^(2)=T_7 |ϕ_1^(1)|^2,
    
          ϕ_2^(2)=T_8 |ϕ_1^(1)|^2,

where

    T_4=λ_1k^2(3λ_1k^2-2ω^2T_8)/2ω^4,
    
          T_5=λ_1k(λ_1k^2-2ω^2T_8)/2ω^3,
    
          T_6=k^2(2ω^2T_8+3k^2)/2ω^4,
    
          T_7=k(2ω^2T_8+k^2)/2ω^3,
    
          T_8=2T_2ω^4-3k^4(1+λ_2λ_1^2-λ_1^2-λ_3λ_1^2)/F1-2ω^4(4k^2+T_1),

where F1=2ω^2k^2(1+λ_1-λ_1λ_2+λ_1λ_3). 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

    n_-0^(2)=T_9|ϕ_1^(1)|^2,
    
          u_-0^(2)=T_10|ϕ_1^(1)|^2,
    
          n_+0^(2)=T_11ϕ_1^(1)|^2,
    
          u_+0^(2)=T_12|ϕ_1^(1)|^2,
    
          ϕ_0^(2)=T_13|ϕ_1^(1)|^2,

where

    T_9=λ_1^2k^2(ω+2kv_g)-T_13λ_1ω^2/v_g^2ω^3,
    
          T_10=λ_1(λ_1k^2- ω^2T_13)/v_gω^2,
    
          T_11=ω(k^2+ω^2 T_13)+2v_gk^3/v_g^2ω^3,
    
          T_12=k^2+ω^2 T_13/v_gω^2,
    
          T_13=2v_g(v_gω^3T_2-k^3)-ω k^2+(1-λ_2+λ_3)× F2/ω^3+λ_1ω^3-λ_1λ_2ω^3+λ_1λ_3ω^3-T_1ω^3v_g^2,

where F2=ω k^2λ_1^2+2v_g k^3 λ_1^2. Finally, the third harmonic modes (m=3) and (l=1) with the help of Eqs. (<ref>)-(<ref>), give a set of equations, which can be reduced to the following NLSE:

    i∂Φ/∂τ+P∂^2Φ/∂ξ^2+Q|Φ|^2Φ=0,

where Φ=ϕ_1^(1) for simplicity. In Eq. (<ref>), P is the dispersion coefficient which can be written as

    P=3v_g(v_g k-ω)/2ω k,

and also Q is the nonlinear coefficient which can be written as

    Q=3ω^3T_3+2ω^3T_2(T_8+T_13)-F3/2k^2+2λ_1 k^2-2λ_1λ_2 k^2+2λ_1λ_3 k^2

where

    F3=ω k^2(T_6+T_11)+2k^3(T_7+T_12)
    
          +(ωλ_1 k^2-ωλ_1λ_2k^2+ωλ_1λ_3 k^2)(T_4+T_9)
    
          +(2λ_1 k^3-2λ_1λ_2k^3+2λ_1λ_3 k^3)(T_5+T_10).

It may be noted here that both P and Q are function 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 DPM.






§ MODULATIONAL INSTABILITY AND ROGUE WAVES


The nonlinear property of the plasma medium as well as the formation and stability conditions of the DAWs
in an EDDPM can be organized according to the sign of dispersive (P) and nonlinear
(Q) coefficients of the standard NLSE (<ref>). The sign of the dispersive coefficient is always negative for
any kinds of wave number k in 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 (figure is not included). Both P and Q have same
sign (i.e., positive or negative) then they reflect a modulationally unstable parametric regime (i.e., P/Q>0) whereas both P and Q have
opposite sign (i.e., positive and negative) then they reflect a modulationally stable parametric regime (i.e., P/Q<0) for the DAWs in presence
of the external perturbation. 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 critical waves number (k_c) <cit.>.

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 <cit.>

    Φ(ξ,τ)= √(2P/Q)[4(1+4iPτ)/1+16P^2τ^2+4ξ^2-1 ](2iPτ).

We have numerically analyzed Eq. (<ref>) in Figs. <ref> and <ref> to understand
how the nonlinear properties of a four components EDDPM as well as the configuration of the
DARWs associated DAWs have been changed by the various 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 Fig. <ref>, 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 the κ. Figure <ref>
indicates the effects of the number density as well as the charge state of the positive dust grains and
negative ions (via λ_2=Z_1n_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 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 unstable (stable) parametric regime of the DAWs increases with Z_+ (Z_1) for a fixed negative ion and positive dust
number density.

We have numerically analyzed first-order rogue waves [by using Eq. (<ref>)] in Figs. <ref>
and <ref>. Figure <ref> highlights the effects of mass of the positive and negative ions
as well as their charge states (via λ_2=Z_-m_+/Z_+m_-) in recognizing the shape of the DARWs in
an electron depleted four components EDDPM, and it can be manifested from this figure that (a) the
increase in the value of λ 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_2; (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_-. Figure <ref> reflects how the number density and charge state of the
light positive ion and heavy negative dust grains (via λ_3=Z_2n_20/Z_+n_+0) contribute to generate
highly energetic rogue waves in a four components 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 charge state and number density. The physics of this result is that
the nonlinearity of a four components electron depleted plasma medium increases (decreases) with positive dust grain (positive ion) number density
as well as with positive dust grain (positive ion) charge state.


§ CONCLUSION


In this present article, we have examined the modulationally stable and unstable parametric regimes of DAWs, and the DARWs in the unstable parametric regime
of DAWs by employing standard NLSE in a four components EDDPM. The numerical analysis can explain the
dependency of the stability conditions as well as the configuration of DARWs associated with DAWs in the modulationally unstable parametric regime.
To conclude, the results obtained from this investigation should be useful in understanding the basic properties of these
rouge waves predicted to be formed in electron depleted unmagnetized opposite polarity dusty plasma systems like mesospherere, F-rings of saturn, and
cometary atmosphere, etc.


§ ACKNOWLEDGEMENTS

A. Mannan and S. Sultana gratefully acknowledge the support from the Alexander von Humboldt Foundation for via their Postdoctoral Fellowship.
99
Hossen2016a M. M. Hossen, et al., Eur. Phys. J. D 70, 252 (2016).

Hossen2016b M. M. Hossen, et al., Phys. Plasmas 23, 023703 (2016).

Hossen2017 M. M. Hossen, et al., High Energ. Dens. Phys. 24, 9 (2017).

Shukla1992 P. K. Shukla and V. P. Silin, Phys. Scr. 45, 508 (1992).

Shahmansouri2013 M. Shahmansouri and H. Alinejad, Phys. Plasmas 20, 033704 (2013).

Rao1990 N. N. Rao, et al., Planet. Space Sci. 38, 543 (1990).

Barkan1995 A. Barkan, et al., Phys. Plasmas 2, 3563 (1995).

Shukla1991 P. K. Shukla, et al., J. Geophys. Res. 96, 21343 (1991).

Melandso1996 F. Melandso, Phys. Plasmas 3, 3890 (1996).

Ferdousi2015 M. Ferdousi, et al., Astrophys. Space Sci. 43, 360 (2015).

Mamun1996 A. A. Mamun, et al., Phys. Plasma 3, 702 (1996).

Dialynas2009 K. Dialynas, et al., J. Geophys. Res. 114, A01212 (2009).

Mayout2012 S. Mayout and M. Tribeche, J. Plasma Phys. 78, 657 (2012).

Sahu2012 B. Sahu and M. Tribeche, Astrophys. Space Sci. 341, 573 (2012).

Ferdousi2017 M. Ferdousi, et al., Eur. Phys. J. D. 71, 102 (2017).

Vasyliunas1968 V. M. Vasyliunas, J. Geophys. Res. 73, 2839 (1968).

Uddin2015 M. J. Uddin, et al., Phys. Plasmas 22, 062111 (2015).

Shahmansouri2012 M. Shahmansouri and H. Alinejad, Phys. Plasmas 19, 123701 (2012).

Kourakis2011 I. Kourakis and S. Sultana,  AIP Conf. Proc. 1397, 86 (2011).

Sultana2011 S. Sultana, et al., Plasma Phys. Control. Fusion 53, 045003 (2011).

Ahmed2018 N. Ahmed, et al., Chaos 28, 123107 (2018).

Gill2010 T. S. Gill, et al., Phys. Plasmas 17, 013701 (2010).

Saini2018 N. S. Saini and I. Kourakis, Phys. Plasmas 15, 123701 (2018).

C1 N. A. Chowdhury, et al., Chaos 27, 093105 (2017).

C2 M. H. Rahman, et al., Chinese J. Phys. 56, 2061 (2018).

C3 N. A. Chowdhury, et al., Phys. plasmas 24, 113701 (2017).

C4 M. H. Rahman, et al., Phys. Plasmas 25, 102118 (2018).

C5 N. A. Chowdhury, et al., Vacuum 147, 31 (2018).

C6 R. K. Shikha, et al., Eur. Phys. J. D 73, 177 (2019).

C7 N. A. Chowdhury, et al., Plasma Phys. Rep. 45, 459 (2019).

C8 S. Jahan, et al., Commun. Theor. Phys. 71, 327 (2019).

C9 N. A. Chowdhury, et al., Contrib. Plasma Phys. 58, 870 (2018).

C10 M. Hassan, et al., Commun. Theor. Phys. 71, 1017 (2019).

Akhmediev2009a N. Akhmediev, et al., Phys. Rev. E 80, 026601 (2009).

Ankiewiez2009b A. Ankiewicz, et al., Phys. Lett. A 373, 3997 (2009).




