Modulational instability of dust-ion-acoustic waves in pair-ion plasma having non-thermal non-extensive electrons
M.K. Islam, A.A. Noman, J. Akter, N.A. Chowdhury, A. Mannan, T.S. Roy, M. Salahuddin, A.A. Mamun
aa r X i v : . [ phy s i c s . p l a s m - ph ] F e b Modulational instability of dust-ion-acoustic waves in pair-ionplasma having non-thermal non-extensive electrons
M.K. Islam ∗ , , A.A. Noman ∗∗ , , J. Akter ∗∗∗ , , N.A. Chowdhury † , , A. Mannan †† , , ,T.S. Roy ‡ , , M. Salahuddin § , , and A.A. Mamun §§ , Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh Plasma Physics Division, Atomic Energy Centre, Dhaka-1000, Bangladesh Institut f¨ur Mathematik, Martin Luther Universit¨at Halle-Wittenberg, Halle, Germany Department of Physics, Bangladesh University of Textiles, Tejgaon Industrial Area, Dhaka, Bangladeshe-mail: ∗ [email protected], ∗∗ [email protected], ∗∗∗ [email protected], † [email protected], †† [email protected], ‡ [email protected], § su [email protected], §§ mamun [email protected] Abstract
The modulational instability (MI) criteria of dust-ion-acoustic (DIA) waves (DIAWs) havebeen investigated in a four-component pair-ion plasma having inertial pair-ions, inertialess non-thermal non-extensive electrons, and immobile negatively charged massive dust grains. A non-linear Schr¨odinger equation (NLSE) is derived by using reductive perturbation method. Thenonlinear and dispersive coe ffi cients of the NLSE can predict the modulationally stable and un-stable parametric regimes of DIAWs and associated first and second order DIA rogue waves(DIARWs). The MI growth rate and the configuration of the DIARWs are examined, and it isfound that the MI growth rate increases (decreases) with increasing the number density of thenegatively charged dust grains in the presence (absence) of the negative ions. It is also observedthat the amplitude and width of the DIARWs increase (decrease) with the negative (positive) ionmass. The implications of the results to laboratory and space plasmas are briefly discussed. Keywords:
NLSE, Modulational instability, Rogue waves, Pair-ion plasma.
1. Introduction
The ubiquitous existence of pair-ion (PI) in astrophysical environments such as upper regionsof Titan’s atmosphere [1, 2, 3, 4, 5], cometary comae [6], the ( H + , O − ) and ( H + , H − ) plasmas inthe D and F-regions of Earth’s ionosphere [1, 2, 3, 4, 5], and also in the laboratory experimentsnamely, neutral beam sources [7], plasma processing reactors [8], and Fullerene ( C + , C − ) [9]have enormously attracted the plasma physicists to examine nonlinear electrostatic modes inPI plasma. The properties of the laboratory PI plasma have also been observed by a number ofauthors [9, 10, 11]. The dynamics of the PI plasma system and the configuration of the associatednonlinear electrostatic structures have been rigorously changed by the presence of negativelycharged massive dust grains. Misra [4] examined dust-ion-acoustic (DIA) shock waves in PIplasma in the presence of immobile negatively charged dust grains, and reported that the heightof the shock front seems to decrease with the number density of dust grains. Mushtaq et al. [5] considered a plasma model having inertialess non-thermal electrons, inertial positive and Preprint submitted to “Contribution to Plasma Physics” February 4, 2021 egative ions, and immobile negatively charged massive dust grains to investigate DIA solitarywaves, and found that the amplitude of the solitary waves increases with increasing the numberdensity and charge state of the negative dust grains.The highly energetic particles associated with long tail in space plasmas have been identifiedby Viking spacecraft [12] and Freja satellite [13]. Cairns et al. [14] proposed non-thermal veloc-ity distribution for these energetic particles in space plasmas, and observed electrostatic solitarystructures. On the other hand, Tsallis [15] introduced non-extensive q -distribution for explain-ing the high energy tail in space plasma. The parameter q in the non-extensive q -distributiondescribes the deviation of the plasma particles from the thermally equilibrium state. A numberof authors [16, 17, 18] have considered non-thermal non-extensive distribution for investigat-ing nonlinear electrostatic waves. Dutta and Sahu [18] examined DIA waves (DIAWs) by con-sidering inertialess non-thermal non-extensive electrons, inertial PI, and stationary negativelycharged dust grains, and highlighted that the angular frequency of the DIAWs decreases withnon-extensivity of electrons but increases with non-thermality of electrons.Rogue waves (RWs) are the results of modulational instability (MI) of the carrier waves, andare governed by the rational solution of standard nonlinear Schr¨odinger equation (NLSE) [19,20, 21], and have been observed in fiber optics [22], water waves [23], and plasmas [24], etc.Bailung et al. [24] experimentally observed the RWs in a multi-component plasma, and foundthat the amplitude of the RWs is three times greater in comparison with surrounding normalwaves. Abdelwahed et al. [3] observed the e ff ects of the negative ion on the RWs in PI plasmaand determined that the negative ion number density increases the nonlinearity as well as theamplitude and width of the RWs in PI plasma. Bains et al. [25] analyzed the MI of ion-acousticwaves (IAWs) and identified that the critical wave number ( k c ) increases with q . El-Labany et al. [26] investigated the MI of the IAWs and associated RWs in a three-component plasma systemhaving inertialess iso-thermal electrons and inertial PI, and found that the amplitude and widthof the RWs increase with the number density and mass of the negative ion. Javidan and Pakzad[27] considered a three-component dusty plasma containing inertial ions, inertialess electrons,and immobile dust grains, and examined the instability of the IAWs, and demonstrated that theinstability growth rate decreases with the increase in the value of negative dust number density.To the best knowledge of the authors, no attempt has been made to study the MI of DIAWs andassociated DIA RWs (DIARWs) in a four-component plasma having inertialess non-thermal non-extensive electrons, inertial positive and negative ions, and stationary negatively charged massivedust grains. The aim of the present investigation is, therefore, to derive NLSE and investigateMI criteria of DIAWs in a four-component plasma, and to observe the e ff ects of various plasmaparameters to the formation of first and second order DIARWs.This manuscript is organized in the following way: The governing equations are described inSec. 2. The standard NLSE is derived in Sec. 3. The MI of the DIAWs and associated DIARWsare examined in Sec. 4. The conclusion of our present work is provided in Sec. 5.
2. Governing Equations
We consider a four-component PI plasma medium having inertial positive ion (charge q + = eZ + and mass m + ), inertial negative ion (charge q − = − eZ − and mass m − ), inertialess non-thermalnon-extensive electron (charge q e = − e and mass m e ), and immobile negatively charged massivedust grains (charge q d = − eZ d and mass m d ). Overall, the charge neutrality condition for ourplasma model can be written as Z + n + = n e + Z − n − + Z d n d , where Z + ( Z − ) is the charge state2f the positive (negative) ion, and Z d is the negatively charged massive dust grains’ charge state.The propagation of the DIAW is governed by the following equations: ∂ n + ∂ t + ∂∂ x ( n + u + ) = , (1) ∂ u + ∂ t + u + ∂ u + ∂ x = − ∂φ∂ x , (2) ∂ n − ∂ t + ∂∂ x ( n − u − ) = , (3) ∂ u − ∂ t + u − ∂ u − ∂ x = µ ∂φ∂ x , (4) ∂ φ∂ x = (1 − µ − µ ) n e − n + + µ n − + µ , (5)where n + ( n − ) is the positive (negative) ion number density normalized by it’s equilibrium value n + ( n − ); u + ( u − ) is the positive (negative) ion fluid speed normalized by the ion-acousticwave speed C + = ( Z + k B T e / m + ) / with T e being the non-thermal non-extensive electron tem-perature and k B being the Boltzmann constant; φ is the electrostatic wave potential normal-ized by k B T e / e ; the time and space variables are normalized by ω − p + = ( m + / π Z + e n + ) / and λ D + = ( k B T e / π Z + e n + ) / , respectively. Other plasma parameters can be written as µ = Z − m + / Z + m − , µ = Z − n − / Z + n + , and µ = Z d n d / Z + n + . Now, the expression for thenumber density of electrons following non-thermal non-extensive distribution [16] can be writ-ten as n e = (cid:2) + A φ + B φ (cid:3) × (cid:2) + ( q − φ (cid:3) ( q + q − , (6)where the parameter q stands for the strength of non-extensive system and the coe ffi cients A and B are defined by A = − q α/ (3 − q + q + α ) and B = − A (2 q − α is a parameterdetermining the number of non-thermal electrons in the model. Williams et al. [17] discussedthe range and the validity of ( q , α ) for solitons. In the limiting case ( q → α = q → α , φ , we get ∂ φ∂ x + n + = + µ + µ n − + γ φ + γ φ + γ φ + · · · , (7)where γ = [(1 − µ − µ )(2 A + q + / , γ = [(1 − µ − µ ) { B + ( q + A − q + } ] / ,γ = [(1 − µ − µ ) { B ( q + − ( q + q − A − q + } ] / . The terms containing γ , γ , and γ in Eq. (7) are due to the contribution of the non-thermalnon-extensive electrons.
3. Derivation of the NLSE
To study the MI of the DIAWs, we want to derive the NLSE by employing the reductiveperturbation method (RPM) and for that case, first we can write the stretched co-ordinates in the3orm [28, 29, 30] ξ = ǫ ( x − v g t ) , (8) τ = ǫ t , (9)where v g is the group speed and ǫ (0 < ǫ <
1) is a small parameter measuring the weakness ofthe dispersion. Then, we can write the dependent variables as [28, 29, 30] n + = + ∞ X m = ǫ m ∞ X l = −∞ n ( m ) + l ( ξ, τ ) exp[ il ( kx − ω t )] , (10) n − = + ∞ X m = ǫ m ∞ X l = −∞ n ( m ) − l ( ξ, τ ) exp[ il ( kx − ω t )] , (11) u + = ∞ X m = ǫ m ∞ X l = −∞ u ( m ) + l ( ξ, τ ) exp[ il ( kx − ω t )] , (12) u − = ∞ X m = ǫ m ∞ X l = −∞ u ( m ) − l ( ξ, τ ) exp[ il ( kx − ω t )] , (13) φ = ∞ X m = ǫ m ∞ X l = −∞ φ ( m ) l ( ξ, τ ) exp[ il ( kx − ω t )] , (14)where k and ω are the real variables representing the carrier wave number and frequency, respec-tively. The derivative operators can be written as ∂∂ t → ∂∂ t − ǫ v g ∂∂ξ + ǫ ∂∂τ , (15) ∂∂ x → ∂∂ x + ǫ ∂∂ξ . (16)Now, by substituting Eqs. (8)-(16) into Eqs. (1)-(4) and (7), and collecting the terms containing ǫ , the first order (when m = l =
1) reduced equations can be written as u (1) + = k ω φ (1)1 , n (1) + = k ω φ (1)1 , u (1) − = − µ k ω φ (1)1 , n (1) − = − µ k ω φ (1)1 , these relations provide the dispersion relation of DIAWs ω = k (cid:0) + µ µ (cid:1) k + γ . (17)We have numerically analyzed Eq. (17) to examine the dispersion properties of DIAWs fordi ff erent values of q and α . The results are displayed in Fig. 1, which shows that (a) for smallwave number, the angular frequency of the DIAWs exponentially increases and for large wavenumber, the dispersion curves become saturated (both left and right panel); (b) the ω decreaseswith q (left panel), and this result is a good agreement with the result of Dutta and Sahu [18];and (c) as we increase the non-thermality of the electron then the ω also increases (right panel),and this result also coincides with the work of Dutta and Sahu [18]. The second order equations4 igure 1: The variation of ω with k for di ff erent values of q when α = . ω with k fordi ff erent values of α when q = . µ = . µ = .
7, and µ = . (when m = l =
1) are given by u (2) + = k ω φ (2)1 + i ( v g k − ω ) ω ∂φ (1)1 ∂ξ , n (2) + = k ω φ (2)1 + ik ( v g k − ω ) ω ∂φ (1)1 ∂ξ , u (2) − = − k µ ω φ (2)1 − i µ ( v g k − ω ) ω ∂φ (1)1 ∂ξ , n (2) − = − µ k ω φ (2)1 − ik µ ( v g k − ω ) ω ∂φ (1)1 ∂ξ , and with the compatibility condition, the group velocity of the DIAWs can be written as v g = ω (1 − ω + µ µ ) k (1 + µ µ ) . (18)The variation of v g with k can be seen from the left panel of Fig. 2, and it is clear from this figurethat the group velocity increases with increasing the value of α . Therefore, one can concludethat the group velocity of DIAWs increases as the non-thermality of electrons increases. Thecoe ffi cients of ǫ when m = l = | φ (1)1 | n (2) + = µ | φ (1)1 | , u (2) + = µ | φ (1)1 | , n (2) − = µ | φ (1)1 | , u (2) − = µ | φ (1)1 | , φ (2)2 = µ | φ (1)1 | , (19)where µ = k + k ω µ ω , µ = k + k ω µ ω , µ = k µ − k ω µ µ ω ,µ = k µ − k ω µ µ ω , µ = k − γ ω − µ µ k ω (4 k ω + γ ω − k − k µ µ ) . Now, m = l = m = l = n (2) + = µ | φ (1)1 | , u (2) + = µ | φ (1)1 | , n (2) − = µ | φ (1)1 | , u (2) − = µ | φ (1)1 | , φ (2)0 = µ | φ (1)1 | , (20)5 igure 2: The variation of v g with k for di ff erent values of α when q = . P / Q with k for di ff erent values of q when α = . µ = . µ = .
7, and µ = . where µ = k v g + k ω + µ ω v g ω , µ = k + µ ω v g ω , µ = k v g µ + k µ ω − µ µ ω v g ω ,µ = k µ − µ µ ω v g ω , µ = γ v g ω + k µ µ (2 kv g + ω ) − k (2 kv g + ω ) ω (1 + µ µ − γ v g ) . Finally, the third harmonic modes, when m = l =
1, with the help of Eqs. (17)-(20) give aset of equations which can be reduced to the standard NLSE: i ∂ Φ ∂τ + P ∂ Φ ∂ξ + Q | Φ | Φ = , (21)where Φ = φ (1)1 , for simplicity. P and Q are the dispersion and nonlinear coe ffi cients of theNLSE, respectively, P = v g ( v g k − ω )2 ω k , (22) Q = γ ω ( µ + µ ) − k ω ( µ + µ + µ µ + µ µ ) + γ ω − k ( µ + µ + µ µ + µ µ )2 k (1 + µ µ ) . (23)The space and time evolution of the DIAWs in PI plasma are directly governed by the coe ffi cients P and Q , and indirectly governed by the di ff erent plasma parameters such as q , α , µ , µ , and µ .
4. Modulational Instability and Rogue Waves
The stable and unstable parametric regimes of the DIAWs are determined by the sign of thedispersion ( P ) and nonlinear ( Q ) coe ffi cients of the standard NLSE [31, 32, 33, 34]. When P igure 3: The variation of Γ g with k M for di ff erent values of µ when q = α = Γ g with k M for di ff erent values of µ when q = . α = . k = . φ = . µ = .
5, and µ = . Γ g with k M for di ff erent values of µ when µ = Γ g with k M for di ff erent values of µ when µ = . k = . φ = . α = . q = . µ = . and Q have same sign (i.e., P / Q > P and Q have opposite sign (i.e., P / Q < P / Q against k yieldsstable and unstable parametric regimes of DIAWs. The point, at which transition of P / Q curveintersects with k -axis, is known as threshold or critical wave number k ( = k c ) [31, 32, 33, 34]. Wehave numerically analyzed the variation of P / Q with k for di ff erent values of q in the right panelof Fig. 2, and it can be seen from this figure that (a) both modulationally stable and unstableparametric regimes of DIAWs can be obtained; (b) the modulationally stable parametric regime7 igure 5: The variation of | Φ | with ξ for di ff erent values of µ when q = α = | Φ | with ξ for di ff erent values of µ when q = . α = . k = . τ = µ = .
5, and µ = . | Φ | with ξ for di ff erent values of µ (both left and right panel). Other plasma parameters are k = . τ = α = . q = . µ = .
7, and µ = . of the DIAWs increases with an increase in the value of q , and this result is a good agreementwith the result of Bains et al. [25]; and (c) the modulationally unstable parametric regime allowsto generate first and second order DIARWs. When P / Q > k M < k c , the MI growth rate( Γ g ) is given by [34, 35] Γ g = | P | k M s k c k M − , (24)8 igure 7: The variation of | Φ | with ξ for di ff erent values of µ (both left and right panel). Other plasma parameters are k = . τ = α = . q = . µ = .
7, and µ = . k = . τ = α = . q = . µ = . µ = .
7, and µ = . where k M is the modulated wave number. Now, we have graphically shown how Γ g varies with k M for di ff erent values of µ in Fig. 3. It is obvious from Fig. 3 that (a) the growth rate initiallyincreases with k M and becomes maximum, then reduces to zero; (b) under the consideration ofiso-thermal electrons (i.e., q = α = Γ g as well as nonlinearity of the plasma system increases (decreases)with increasing the number density of negative (positive) dust grains (ion) for their constantcharge state under the consideration of non-thermal non-extensive electrons (right panel); (d)the maximum value of the growth rate increases rigorously in the presence of non-thermal non-extensive electrons than the presence of iso-thermal electrons.The existence of the negative ions in the PI plasma rigourously changes the dynamics of thesystem. Fig. 4 describes the variation of Γ g with k M for di ff erent values of µ . It is obvious from9his figure that (a) when we neglect the contribution of negative ions from our plasma system thatmeans only inertial positive ions, inertialess non-thermal non-extensive electrons, and immobiledust grains are present in the plasma system, then the maximum value of Γ g decreases withthe increase of negative dust grain number density, and similar e ff ect has also been observedby Javidan and Pakzad [27] in a three-component plasma system (left panel); (b) but when weinclude negative ions in the plasma system, that means a four-component PI plasma system,then Γ g increases with increasing the negatively charged dust grains (right panel). So, the quasi-neutrality condition and associated dynamics of the plasma system are fully changed by negativeions.The first-order rational solution of Eq. (21), which can predict the concentration of largeamount of energy in a small region of the modulationally unstable parametric regime ( P / Q > Φ ( ξ, τ ) = s PQ (cid:20) + iP τ )1 + P τ + ξ − (cid:21) exp(2 iP τ ) . (25)We have plotted Eq. (25) in Figs. 5 and 6 to understand the nonlinear properties of the PI plasmasystem as well as the mechanism to the formation of DIARWs associated with DIAWs in themodulationally unstable parametric regime. Figure 5 indicates that (a) the amplitude and widthof the DIARWs increase with an increase in the value of µ (both left and right panel); (b) underthe consideration of iso-thermal electrons, as we increase (decrease) the value of negative (posi-tive) ion number density, the amplitude and width of the DIARWs increase (decrease) when theircharge state remain constant (left panel), and this result is similar with the result of El-Labany et al. [26]; (c) similarly, the amplitude and width of the DIARWs increase with increasing(decreasing) negative (positive) ion number density when the plasma system has non-extensivenon-thermal electrons, and this result coincides with the result of Abdelwahed et al. [3]; (d) thedirection of the variation of amplitude and width has not been changed with µ under the consid-eration of iso-thermal or non-thermal non-extensive electrons, but the variation of amplitude andwidth is severely happened for non-thermal non-extensive electrons than iso-thermal electrons.It can be seen from the literature that the PI plasma system can support these conditions: m − > m + (i.e., H + , O − ) [2, 3], m − = m + (i.e., H + , H − ) [2, 3], and m − < m + (i.e., Ar + , F − ) [2, 3].So, in our present investigation, we have graphically observed the variation of the electrostaticwave potential with µ in Fig. 6 under the consideration of m − > m + (i.e., µ < m − < m + (i.e., µ > et al. [26].The second-order rational solution of NLSE can be written as [36, 37] Φ ( ξ, τ ) = r PQ (cid:20) + G + iM T (cid:21) exp( iP τ ) , (26)10here G ( ξ, τ ) = − ξ − P ξτ ) − P τ ) − ξ − P τ ) + , M ( ξ, τ ) = − P τ h ξ + P ξτ ) + P τ ) − ξ + P τ ) − i , D ( ξ, τ ) = ξ + ξ ( P τ ) + ξ ( P τ ) + ξ + P τ ) − P ξτ ) + ξ + P τ ) + . We have graphically shown Eq. (26) in Fig. 7. The nature of the second order DIARWs is verysensitive to the change of mass and charge state of the PI under the consideration of m − > m + (i.e., µ < m − < m + (i.e., µ > ξ -axis;(e) the second (first) order DIARWs has three (one) local maxima.
5. Conclusion
We have studied an unmagnetized realistic PI plasma system having inertialess non-thermalnon-extensive electrons, inertial negative and positive ions, and immobile negatively chargeddust grains. The RPM is used to derive the NLSE, and the nonlinear and dispersive coe ffi cientsof the NLSE determine the stable and unstable parametric regimes of DIAWs. The results thathave been found from our investigation can be summarized as follows: • The angular frequency of the DIAWs decreases (increases) with the increase in the value ofnon-extensivity (non-thermality) of the electrons. • Both modulationally stable and unstable parametric regimes of DIAWs have been observed. • The modulational instability growth rate increases (decreases) with increasing the numberdensity of the negatively charged dust grains in the presence (absence) of the negative ions. • The amplitude and width of the DIARWs associated with DIAWs increase (decrease) withthe negative (positive) ion mass.It may be noted here that the gravitational and magnetic e ff ects are very important but beyondthe scope of our present work. In future and for better understanding, someone can investigatethe nonlinear propagation in a four-component PI plasma by considering the gravitational andmagnetic e ff ects. The results of our present investigation will be useful in understanding thenonlinear phenomena in both astrophysical environments such as upper regions of Titan’s atmo-sphere [1, 2, 3, 4, 5], cometary comae [6], the ( H + , O − ) and ( H + , H − ) plasmas in the D andF-regions of Earth’s ionosphere [1, 2, 3, 4, 5], and also in laboratory plasmas namely, neutralbeam sources [7], plasma processing reactors [8], and Fullerene ( C + , C − ) [9], etc.11 cknowledgements Authors would like to acknowledge “UGC research project 2018-2019” for their financialsupports to complete this work.
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