Dust-ion-acoustic rogue waves in dusty plasma having super-thermal electrons
A.A. Noman, M.K. Islam, M. Hassan, S. Banik, N.A. Chowdhury, A. Mannan, A.A. Mamun
aa r X i v : . [ phy s i c s . p l a s m - ph ] J a n Dust-ion-acoustic rogue waves in dusty plasma having super-thermal electrons
A.A. Noman , ∗ , M.K. Islam , ∗∗ , M. Hassan , ∗∗∗ , S. Banik , , † , N.A. Chowdhury , ‡ , A. Mannan , , § , and A.A. Mamun , §§ Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh Health Physics Division, Atomic Energy Centre, Dhaka-1000, Bangladesh Plasma Physics Division, Atomic Energy Centre, Dhaka-1000, Bangladesh Institut f¨ur Mathematik, Martin Luther Universit¨at Halle-Wittenberg, 06009 Halle, Germanye-mail: ∗ [email protected], ∗∗ [email protected], ∗∗∗ [email protected], † [email protected] ‡ [email protected], § [email protected], §§ mamun [email protected] Abstract
The standard nonlinear Schr¨odinger equation (NLSE) is one of the elegant equations to find the information about the modulationalinstability criteria of dust-ion-acoustic (DIA) waves (DIAWs) and associated DIA rogue waves (DIARWs) in a three-componentdusty plasma medium having inertialess super-thermal kappa distributed electrons, and inertial warm positive ions and negativedust grains. It can be seen that under the consideration of inertial warm ions along with inertial negatively charged dust grains,the plasma system supports both fast and slow DIA modes. The charge state and number density of the ion and dust grain areresponsible to change the instability conditions of the DIAWs and the configuration of DIARWs. These results are to be consideredthe cornerstone for explaining the real puzzles in space and laboratory dusty plasmas.
Keywords:
Dust-ion-acoustic waves; modulational instability; rogue waves
1. Introduction
The size, mass, charge and ubiquitous existence of mas-sive dust grains in both space (viz., cometary tails [1, 2, 3, 4],magnetosphere [3], ionosphere [3], aerosols in the astrosphere[2, 4], planetary rings [1], Earth’s ionospheres [1], nebula,and interstellar medium [4], etc.) and laboratory (viz., ac-discharge, plasma crystal [4], Q-machine, nano-materials [5],and rf-discharges [4], etc.) plasmas do not only change the dy-namics of the dusty plasma medium (DPM) but also change themechanism of the formation of various nonlinear electrostaticexcitations, viz., dust-acoustic (DA) solitary waves (DASWs)[6], DA shock waves (DA-SHWs) [7], dust-ion-acoustic (DIA)solitary waves (DIASWs) [1, 2, 3], DIA shock waves (DI-ASHWs), and DIA rogue waves (DIARWs), etc.The activation of the long range gravitational and Coulombforce fields is the main cause to generate non-equilibriumspecies [8] as well as the high energy tail in space environ-ments, viz., terrestrial plasma-sheet [9, 10], magneto-sheet, au-roral zones [9, 10], mesosphere, radiation belts [9, 10], mag-netosphere, and ionosphere [9, 10], etc. The Maxwellian ve-locity distribution function, in which the dynamics of the non-equilibrium species is not considered, is not enough to describethe intrinsic mechanism of the high energy tail in space envi-ronments [8]. While the super-thermal / κ -distribution function,in which the dynamics of the non-equilibrium species is alsoconsidered, is suitable for explaining the high energy tail inspace environments [8]. The non-equilibrium properties of thespecies are recognized by the magnitude of κ in super-thermal κ -distribution [8]. The κ -distribution is normalizable for a rangeof values of κ from κ > / κ → ∞ , and the non-equilibrium properties of the species is considerable when the value of κ tends to 3 / et al. [1] observed that the ve-locity of the DIASWs increases with decreasing the value of κ in DPM. Shahmansouri and Tribeche [3] considered an electrondepleted DPM having super-thermal plasma species to investi-gate DASWs, and reported that the amplitude of the DASWsincreases while the width of the DASWs decreases with a de-crease in the value of κ that means the super-thermality of theplasma species leads a narrower and spiky solitons. Ferdousi et al. [7] numerically observed that the height of the positiveDASHWs increases while negative DASHWs decreases withincreasing the value of κ in a multi-component DPM in the pres-ence of super-thermal electrons.The nonlinear and dispersive properties of plasma mediumare the prime reasons to organize the modulational instability(MI) criteria of various kinds of waves in the presence of ex-ternal perturbation, and are governed by the standard nonlin-ear Schr¨odinger equation (NLSE) which can be derived by em-ploying reductive perturbation method (RPM) [11, 12, 13, 14,15, 16, 17, 18, 19]. The rational solution of the NLSE is alsoknown as freak waves, giant waves, or rogue waves (RWs) inwhich a large amount of energy can concentrate into a smallarea, and the height of the RWs is almost three times greaterthen the height of associated normal carrier waves. Initially,RWs are identified only in the ocean and are considered as adestructive sign of nature which can sink the ship or destroy thehouse in the bank of the ocean. Now-a-days, RWs can also beobserved in optics, stock market, biology, and plasma physics,etc. Gill et al. [6] investigated the MI of the DAWs in a multi-component DPM and found that the critical wave number ( k c ) Preprint submitted to “Contributions to Plasma Physics” January 8, 2021 hich defines the modulationally stable and unstable paramet-ric regimes decreases with the increase in the value of κ . Amin et al. [11] studied the propagation of nonlinear electrostaticDAWs and DIAWs, and their MI in a three-component DPM.Jukui and He [12] demonstrated the amplitude modulation ofspherical and cylindrical DIAWs. Saini and Kourakis [13] con-sidered a three-component DPM having inertial highly chargedmassive dust grains and inertialess electrons and ions to studythe MI of DAWs, and highlighted that the angular frequencyof the DAWs increases with the super-thermality of the plasmaspecies and the stable parametric regime decreases with an in-crease in the value of negative dust number density.The outline of the paper is as follows: The governing equa-tions describing our plasma model are presented in section 2.The derivation of NLSE is demonstrated in section 3. The Mod-ulational instability and rogue waves is described in section 4.Results and discussion are devoted in section 5. Conclusion isprovided in section 6.
2. Governing Equations
We consider an unmagnetized, fully ionized and collisionlessthree-component DPM comprising super-thermal electrons,positively charged inertial warm ions and negatively chargeddust grains. At equilibrium, the overall charge neutrality condi-tions of our plasma system can be written as n e + Z d n d = Z i n i ,where n e , n d , and n i are the equilibrium electron, dust, andion number densities respectively, and Z d ( Z i ) is the charge stateof the negative (positive) dust grain (ion). The normalized equa-tions describing the system can be written as ∂ n d ∂ t + ∂∂ x ( n d u d ) = , (1) ∂ u d ∂ t + u d ∂ u d ∂ x = µ ∂φ∂ x , (2) ∂ n i ∂ t + ∂∂ x ( n i u i ) = , (3) ∂ u i ∂ t + u i ∂ u i ∂ x + µ n i ∂ n i ∂ x = − ∂φ∂ x , (4) ∂ φ∂ x + n i = (1 − µ ) n e + µ n d , (5)where n d ( n i ) is the dust (ion) number density normalized bythe equilibrium value n d ( n i ); u d ( u i ) is the dust (ion) fluidspeed normalized by the ion sound speed C i = ( Z i k B T e / m i ) / (where T e being the super-thermal electron temperature, m i isthe ion rest mass, and k B is the Boltzmann constant); φ is theelectrostatic wave potential normalized by k B T e / e (with e beingthe electron charge). The time variable t is normalized by ω − P i = [ m i / (4 π e Z i n i )] / and the space variable x is normalized by λ D i = ( k B T e / π e Z i n i ) / . The pressure of the ion is expressedas P i = P i ( N i / N i ) γ , where P i = n i k B T i being the equilibriumpressure of the ion, and T i being the warm ion temperature, and γ = ( N + / N (where N be the degree of freedom and for onedimensional case N =
1, then γ = µ = ρµ , ρ = Z d / Z i , µ = m i / m d , µ = T i / Z i T e ,and µ = Z d n d / Z i n i . The expression for the number density of the super-thermal electrons (following the κ -distribution) canbe expressed as n e = h − φκ − / i − κ + = + n φ + n φ + n φ + · · · , (6)where n = (2 κ − / (2 κ − n = [(2 κ − κ + / κ − ,and n = [(2 κ − κ + κ + / κ − . The parame-ter κ , generally stands for super-thermality, which measures thedeviation of the plasma particles from Maxwellian distribution.Now, by substituting Eq. (6) into (5) and expanding up to thirdorder in φ , we get ∂ φ∂ x + n i + µ = + µ n d + A φ + A φ + A φ + · · · , (7)where A = n (1 − µ ), A = n (1 − µ ), and A = n (1 − µ ).
3. Derivation of the NLSE
To study the MI of the DIAWs, we want to derive the NLSEby employing the RPM. First, we can write the stretched co-ordinates in the following form [20, 21, 22, 23, 24] ξ = ǫ ( x − v g t ) , (8) τ = ǫ t , (9)where v g is the group velocity and ǫ (0 < ǫ <
1) is a smallparameter. We can write the dependent variables ( n d , u d , n i , u i ,and φ ) as n d = + ∞ X m = ǫ m ∞ X l = −∞ n ( m ) dl ( ξ, τ ) e il ( kx − ω t ) , (10) u d = ∞ X m = ǫ m ∞ X l = −∞ u ( m ) dl ( ξ, τ ) e il ( kx − ω t ) , (11) n i = + ∞ X m = ǫ m ∞ X l = −∞ n ( m ) il ( ξ, τ ) e il ( kx − ω t ) , (12) u i = ∞ X m = ǫ m ∞ X l = −∞ u ( m ) il ( ξ, τ ) e il ( kx − ω t ) , (13) φ = ∞ X m = ǫ m ∞ X l = −∞ φ ( m ) l ( ξ, τ ) e il ( kx − ω t ) , (14)where k and ω are the real variables representing the carrierwave number and frequency, respectively. The derivative oper-ators can be written as ∂∂ x → ∂∂ x + ǫ ∂∂ξ , (15) ∂∂ t → ∂∂ t − ǫ v g ∂∂ξ + ǫ ∂∂τ . (16)Now, by substituting Eqs, (8)-(16) into Eqs. (1)-(4), and (7),and collecting the terms containing ǫ , the first order ( m = igure 1: The variation of ω f vs k (left panel) and ω s vs k (right panel) whenother plasma parameters are κ = . ρ = × , µ = × − , µ = .
3, and µ = . with l =
1) reduced equations can be written as u (1) d = − k µ ω φ (1)1 , (17) n (1) d = − µ k ω φ (1)1 , (18) u (1) i = − k ωµ k − ω φ (1)1 , (19) n (1) i = − k µ k − ω φ (1)1 , (20)these relations provide the dispersion relations of DIAWs. Now,the dispersion relations of DIAWs are ω ≡ ω f = k M + k √ M − GH G , (21)and ω ≡ ω s = k M − k √ M − GH G , (22)where M = + µ µ + µ k + A µ , G = A + k , and H = µ µ µ . In Eqs. (21) and (22), to get the positive value of ω ,the condition M > GH must be satisfied. In the fast ( ω f )DIA mode, both inertial ion and dust oscillate in phase withthe inertialess electrons. While in the slow ( ω s ) DIA mode,one of the inertial elements ion (dust) oscillates in phase withthe inertialess electrons and other inertial element dust (ion)oscillates in anti-phase with them [25, 26]. Both the fast ( ω f )and slow ( ω s ) DIA modes have been analyzed numerically inFig. 1 in the presence of super-thermal electrons. The second-order (when m = l =
1) equations are given by u (2) d = − k µ ω φ (2)1 − µ i ω ∂φ (1)1 ∂ξ + kv g µ i ω ∂φ (1)1 ∂ξ , (23) n (2) d = − µ k ω φ (2)1 + k µ ( kv g − ω ) i ω ∂φ (1)1 ∂ξ , (24) u (2) i = k ωω − µ k φ (2)1 − i ( ω − kv g )( ω + µ k )( ω − µ k ) ∂φ (1)1 ∂ξ , (25) n (2) i = k ω − µ k φ (2)1 − ik ω ( ω − kv g )( ω − µ k ) ∂φ (1)1 ∂ξ , (26)with the compatibility condition v g = µ µ k ω ( ω − µ k ) + k ω − k ω ( ω − µ k ) k µ µ ( ω − µ k ) + k ω . (27) The coe ffi cients of the ǫ when m = l = | φ (1)1 | u (2) d = A | φ (1)1 | , (28) n (2) d = A | φ (1)1 | , (29) u (2) i = A | φ (1)1 | , (30) n (2) i = A | φ (1)1 | , (31) φ (2)2 = A | φ (1)1 | , (32)where A = k µ − A µ k ω ω , A = k µ − A µ k ω ω , A = − A k ω ( µ k − ω ) + µ ω k + k ω µ k − ω ) , A = − A k ( µ k − ω ) + ω k + µ k µ k − ω ) , A = (2 A ω + µ µ k ) + ω k + µ k ω ω (cid:2) ( k µ µ − k ω − A ω ) − k ω ( µ k − ω ) − (cid:3) . When m = l = m = l = u (2) d = A | φ (1)1 | , (33) n (2) d = A | φ (1)1 | , (34) u (2) i = A | φ (1)1 | , (35) n (2) i = A | φ (1)1 | , (36) φ (2)0 = A | φ (1)1 | , (37)where A = k µ − µ ω A ω v g , A = k µ v g + k µ ω − µ ω A v g ω , A = v g A ( ω − µ k ) + µ ω k + µ k v g + k ω v g ( ω − µ k ) ( v g − µ ) , A = A ( ω − µ k ) + ω k v g + µ k + k ω ( ω − µ k ) ( v g − µ ) , A = ( ω − µ k ) × F − v g ω (2 ω k v g + µ k + k ω ) ω ( ω − µ k ) (cid:2) µ µ ( v g − µ ) + v g − A v g ( v g − µ ) (cid:3) , where F = ( v g − µ )(2 A v g ω + k µ µ v g + k ωµ µ ). Finally,the third-order harmonic modes (when m = l =
1) andwith the help of Eqs. (17)-(37), given a set of equations whichcan be reduced to the standard NLSE: i ∂ Φ ∂τ + P ∂ Φ ∂ξ + Q | Φ | Φ = , (38)3 igure 2: The variation of P / Q with k for di ff erent values of κ when otherplasma parameters are ρ = × , µ = × − , µ = . µ = .
05, and ω ≡ ω f .Figure 3: The variation of P / Q with k for di ff erent values of κ when otherplasma parameters are ρ = × , µ = × − , µ = . µ = .
05, and ω ≡ ω s . where Φ = φ (1)1 for simplicity. In Eq. (38), the dispersion co-e ffi cients ( P ) and non-linear coe ffi cients ( Q ) can be written, re-spectively, as P = − ( ω − kv g )( ω − µ k ) (3 kv g µ µ − µ µ ω ) + F ω ( ω − µ k ) (cid:2) µ µ k ( ω − µ k ) + k ω (cid:3) , Q = ω ( ω − µ k ) [3 A + A ( A + A ) − F ]2 µ µ k ( ω − µ k ) + k ω , where F = (2 k ω v g − µ k ω )( ω − kv g ) + ( ω kv g − ω )( ω − kv g )( ω + µ k ) + ω ( ω − µ k ) and F = { ( k ω + µ k )( A + A ) + k ω ( A + A ) } / ( ω − µ k ) + { k µ µ ( A + A ) + k ωµ µ ( A + A ) } /ω . It is interesting that P and Q of the Eq.(38) are function of various plasma parameters such as carrierwave number ( k ), ratio of ion mass to dust mass ( µ ), ratio of dustcharge state to ion charge state ( ρ ), and super-thermal parameter( κ ), etc.
4. Modulational instability and rogue waves
The stable and unstable parametric regimes of the DIAWsare organized by the sign of the dispersion ( P ) and nonlinear( Q ) coe ffi cients of the standard NLSE [27, 28, 29]. The stabil-ity of DIAWs in a three-component DPM is governed by thesign of P and Q [27, 28, 29]. When P and Q have same sign Figure 4: The variation of Γ g with ˜ k for di ff erent values of µ when other plasmaparameters are k = .
0, ˜ Φ = . κ = . ρ = × , µ = × − , µ = . ω ≡ ω f .Figure 5: The variation of | Φ | with ξ for di ff erent values of µ when otherplasma parameters are τ = k = .
0, ˜ Φ = . κ = . ρ = × , µ = × − , µ = .
05, and ω ≡ ω f .Figure 6: The variation of | Φ | with ξ for di ff erent values of µ when otherplasma parameters are τ = k = .
0, ˜ Φ = . κ = . ρ = × , µ = × − , µ = .
05, and ω ≡ ω f . (i.e., P / Q > P and Q haveopposite sign (i.e., P / Q < P / Q against k yields stable and unstable parametric regimes ofDIAWs. The point, at which transition of P / Q curve intersectswith k -axis, is known as threshold or critical wave number k igure 7: The variation of first-order (dashed green curve) and second-order(solid blue curve) rational solutions of NLSE at k = . τ = ( = k c ) [27, 28, 29, 30, 31, 32]. When P / Q > k < k c , theMI growth rate ( Γ g ) is given by Γ g = | P | ˜ k s ˜ k c ˜ k − . (39)The first-order rational solution, which can predict the concen-tration of the large amount of energy in a small region in themodulationally unstable parametric regime ( P / Q >
0) of DI-AWs, of Eq. (38) can be written as [33, 34] Φ ( ξ, τ ) = s PQ h + i τ P + ξ + τ P − i exp(2 i τ P ) . (40)and the second-order rational solution is Φ ( ξ, τ ) = r PQ h + G ( ξ, τ ) + iM ( ξ, τ ) D ( ξ, τ ) i exp( i τ P ) , (41)where 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 ) + . The nonlinear behavior of the plasma medium is considered tobe responsible for the concentration of large amount of energyinto tiny region.
5. Results and discussion
First, we are interested to observe numerically the stable andunstable parametric regimes of DIAWs in the presence of super-thermal electrons by depicting the variation of P / Q with k for di ff erent plasma parameters. In our present analysis, we haveconsidered that m d = m i , Z d = Z i , and T e = T i .We have graphically shown the variation of P / Q with k incase of both fast ( ω f ) and slow ( ω s ) DIA modes for di ff erentvalues of κ in Figs. 2 and 3, respectively. From these two fig-ures, it can be seen that (a) under consideration ω ≡ ω f and ω ≡ ω s , possible stable and unstable parametric regimes canbe occurred for DIAWs; (b) the DIAWs become unstable forsmall value of k (i.e., k ≃ .
6) in first mode while in slow modethe DIAWs become unstable for large value of k (i.e., k ≃ k c decreases with anincrease in the value of κ .We have numerically analyzed the MI growth rate of DIAWsunder consideration fast mode in Fig. 4 by using these plasmaparameters: k = .
0, ˜ Φ = . ρ = × , µ = × − , and µ = .
3. It is clear that the maximum value of the Γ g increases(decreases) with the increase in the value of ion (dust) num-ber density for a constant value of their charge state (via µ ).The nonlinearity as well as the Γ g increases (decreases) withion (dust) charge state when other plasma parameters remainconstant.We have presented the evaluation of first and second-orderDIARWs with ξ for di ff erent values of µ in Figs. 5 and 6, re-spectively, and from these figures it is observed that both thefirst and second-order DIARW solutions can concentrate largeamount of energy into a small region. It is clear from thesetwo figures that (a) the amplitude of the first and second-orderrogue waves decreases (increases) with increasing the temper-ature of the ion (electron) for a fixed ion charge state; (b) thenonlinearity as well as the amplitude and width of the first andsecond-order DIARWs increases with ion charge state.Figure 7 shows a comparison between the first and second-order DIARWs and it can be seen from this figure that (a) theamplitude of the second-order DIARWs is always higher thanthe first-order DIARWs for same plasma parameters, whichmeans that the second-order DIARWs can concentrate more en-ergy than the first-order DIARWs; (b) the first-order DIARWshas two zeros symmetrically located on its ξ -axis, where thesecond-order DIARWs has four zeros symmetrically located onits ξ -axis.
6. Conclusion
In this paper, we have considered a realistic DPM havingnegatively charged dust grains, ions and electrons. A standardNLSE is derived by using RPM, and this three-component DPMcan generate DIAWs in which the moment of inertia is providedby the warm ions and dust grains, and the restoring force is pro-vided by the thermal pressure of inertialess super-thermal elec-trons. The interaction of the nonlinear ( Q ) and dispersive ( P )coe ffi cients of NLSE can easily divide the modulationally sta-ble and unstable parametric regimes, and the unstable paramet-ric regime also allows to generate highly energetic DIARWs.The outcomes of present investigation can be useful in explain-ing the DIARWs cometary tails [1, 2, 3, 4], magnetosphere [3],ionosphere [3], aerosols in the astrosphere [2], planetary rings[1], Earth’s ionospheres [1], and interstellar medium [4].5 eferences [1] P. Eslami, et al. , IEEE Trans. Plasma Sci. , 3589 (2013).[2] N.S. Saini and K. Singh, Phys. Plasmas , 103701 (2016).[3] M. Shahmansouri and M. Tribeche, Astrophys. Space Sci. , 87 (2012).[4] P.K. Shukla and A.A. Mamun,
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