Dust ion acoustic freak waves in a plasma with two temperature electrons featuring Tsallis distribution

Dust ion acoustic freak waves in a plasma with two temperature electrons featuring Tsallis distribution

Accepted Manuscript Dust ion acoustic freak waves in a plasma with two temperature electrons featuring Tsallis distribution Balwinder Singh Chahal, Ma...

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Accepted Manuscript Dust ion acoustic freak waves in a plasma with two temperature electrons featuring Tsallis distribution Balwinder Singh Chahal, Manpreet Singh, Shalini, N.S. Saini

PII: DOI: Reference:

S0378-4371(17)31012-9 https://doi.org/10.1016/j.physa.2017.10.004 PHYSA 18718

To appear in:

Physica A

Received date : 28 January 2017 Revised date : 2 October 2017 Please cite this article as: B.S. Chahal, M. Singh, Shalini Singh, N.S. Saini, Dust ion acoustic freak waves in a plasma with two temperature electrons featuring Tsallis distribution, Physica A (2017), https://doi.org/10.1016/j.physa.2017.10.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Highlights (for review)

Highlights  We have studied the dust ion acoustic freak waves in a multicomponent plasma with nonextensive distribution.  Modulated wave packets in the form bright envelope solitons are formed which give rise to freak waves.  It is observed that the nonextensivity of electrons, cool electronion density ratio and dust number density plays a significant role in modifying the nonlinear structures of freak waves.

*Manuscript Click here to view linked References

Dust ion acoustic freak waves in a plasma with two temperature electrons featuring Tsallis distribution Balwinder Singh Chahala,b , Manpreet Singha , Shalinia , N. S. Sainia,∗ a

Department of Physics, Guru Nanak Dev University, Amritsar-143005, India. b Department of Physics, Layallpur Khalsa College, Jalandhar-144001, India.

Abstract We present an investigation for the nonlinear dust ion acoustic wave modulation in a plasma composed of charged dust grains, two temperature (cold and hot) nonextensive electrons and ions. For this purpose, the multiscale reductive perturbation technique is used to obtain a nonlinear Schr¨ odinger equation. The critical wave number, which indicates where the modulational instability sets in, has been determined precisely for various regimes. The influence of plasma background nonextensivity on the growth rate of modulational instability is discussed. The modulated wavepackets in the form of either bright or dark type envelope solitons may exist. Formation of rogue waves from bright envelope solitons is also discussed. The investigation indicates that the structural characteristics of these envelope excitations (width, amplitude) are significantly affected by nonextensivity, dust concentration, cold electron-ion density ratio and temperature ratio. Keywords: Dust ion acoustic rogue waves, Nonlinear Schr¨ odinger equation, Modulational instability, Nonextensive distribution, Tsallis statistical mechanics



Corresponding author at : Department of Physics, Guru Nanak Dev University, Amritsar-143005, India. Email addresses: chahal [email protected] (Balwinder Singh Chahal), [email protected] (Manpreet Singh), [email protected] (Shalini), [email protected] (N. S. Saini)

Preprint submitted to Physica A: Statistical Mechanics and its Applications,October 2, 2017

1. Introduction The study of wave dynamics in different kinds of plasmas in the presence of dust has been a foremost area of research during the last more than five decades owing to salient applications of charged dust in industry as well as in different astrophysical environments [1–4]. Ion acoustic waves (IAWs) in an unmagnetized dusty plasma with phase velocity between the electron and ion thermal velocities were first studied theoretically by Shukla and Silin [5] and later on, their existence was confirmed experimentally by Barkan et al. [6]. Dust particles are often of micron to submicron size, with mass in the range of 106 -1012 proton masses and are usually found to have negative charge (possibly as large as ∼ 104 electron charges), depending upon the environment where they occur. On the other hand, smaller dust grains may be found to be positively charged. The presence of dust modifies the standard ion acoustic mode, giving rise to dust ion acoustic (DIA) wave. Over the last many years, after theoretical and experimental observations of dust ion acoustic waves (DIAWs), a large number of investigations to study the properties of such type of waves have been reported by a number of authors [4, 7–12]. For the last three decades, immense attention has been given to the nonextensive distribution (Tsallis distribution) due to its importance to model the systems with the long-range interactions (e.g. plasma and gravitational systems) for which Maxwellian distribution is inadequate [13–17]. Tsallis first time introduced nonextensive distribution which was used for various types of investigations in different systems, e.g., systems with long-time memory, long-range interactions and fractality of corresponding space-time etc. [18, 19]. A number of systems, e.g., plasmas, gravitational systems and longrange Hamiltonian systems, through different studies provide connection with Tsallis entropy [13, 20]. It was confirmed experimentally that q-nonextensive distribution plays an important role to describe the systems with long-range interactions which generally occur in astrophysical environments [13, 21]. A number of astrophysical/cosmological environments such as, stellar polytrops [22], solar neutrinos [23], galaxies with peculiar velocity distributions [24] etc. have been studied in the frame work of nonextensive statistics. Using Tsallis distribution, a large number of investigations have been reported by researchers to study the characteristics of ion-acoustic solitons (IASs), double layers (DLs), shocks, rogue waves, and amplitude modulation of solitary waves in different plasma systems [25–34]. 2

The propagation of nonlinear waves in dispersive media is governed by a generic nonlinear phenomenon (e.g. modulational instability). The nonlinear wave propagation in a dispersive media is generally subject to amplitude modulation due to carrier wave self-interaction or due to intrinsic medium nonlinearity. Paradigm used to study such mechanism is standard multiple scale method leading to derivation of nonlinear Schr¨ ondinger equation (NLSE) in one dimension. Most of theoretical efforts to investigate the modulational instability of IAWs are focused on two component plasma, whose constituents are singly charged positive ions and electrons. Modulational instability of different plasma modes in several plasma environments in the frame work of nonextensive distribution have been studied due to their relevance in stable wave propagation [35–39]. Modulation instability in the plasma can lead to the formation of rogue waves [40], that are short lived phenomenon appearing suddenly out of nowhere and disappearing, leaving no hint of their presence. The amplitude of the rogue waves can be two, three or more times the surrounding normal waves. In the past, rogue waves were observed in optical systems [41], superfluids [42], oceans [43], Bose-Einstein condensates [44], optical cavities [45], nonlinear fiber optics [46], atmosphere [47], plasmonics [48], astrophysical environments [49], capillary rogue waves [50] and also in finance [51]. During the evolution of the wave, small periodic perturbations in the medium develop into a highly periodic wave structure. Rogue waves are modeled by a rational solution to the nonlinear Schr¨odinger equation (NLSE). Over the last many years, existence of rogue waves has been reported theoretically and experimentally in different plasma systems in the frame work of different velocity distribution functions [52–56]. It is observed that high intensity laser irradiated plasmas, space plasmas (e.g. solar corona, Van Allen radiation belt, auroral zone of earth etc.) and many laboratory environments confirm the propagation of IAWs in the presence of two temperature electrons (cold and hot) [57, 58]. From the observations of the Voyager PLS [59, 60] and Cassini CAPS [61], it is confirmed that two temperature electrons featuring superthermal distribution (nonMaxwellian distribution) are present in Saturn’s Magnetosphere. During the last many years, numerous investigations on the study of characteristics of IAWs in a plasma with multi-temperature electrons have been reported by a number of researchers [63–68]. To the best of our knowledge, modulation instability of dust-ion acoustic waves in a plasma, comprising of positively charged ions, multi-temperature (hot and cold) non-Maxwellian electrons and charged stationary dust grains has not been studied so far. The relax3

ation times for both these species of electrons to reach thermal equilibrium states are different and much longer than the time scale of plasma oscillations. This type of plasma is present in the Saturn’s magnetosphere where non- Maxwellian hot and cold electrons are observed [59–62] along with positive ions and dust grains. Our plasma model can be applied to the E-ring of Saturn. This region comprises of dust [69–71] and is embedded inside the Saturn’s magnetospheric plasmas where hot and cold electrons as well as positive ions are observed. Aim of the present investigation is to derive the nonlinear Schr¨ odinger equation (NLSE) in a plasma comprising of ion fluid, immobile dust and two temperature electrons featuring nonextensive distribution. From the rational solution of NLSE, we have studied the characteristics of DIA freak waves. The manuscript is structured as, in Sec.2 basic fluid model equations are given, in Sec.3 these basic set of equations are reduced to NLSE by using reductive perturbation method, in Sec.4 MI of DIAWs is discussed and effect of different parameters on MI is discussed by performing a numerical analysis, in Sec.5 first and second order rogue wave solution of NLSE is given and effect of different parameters on rogue waves is described and in Sec.6 overall conclusions of this paper is drawn. 2. Fluid model equations The plasma model consists of ions, two temperature nonextensive electrons: hot (temperature Th , density nh ) and cold (temperature Tc , density nc ), and charged stationary dust grains (charge Zd , density nd0 ). The equilibrium charge neutrality condition yields nh0 = 1 − f + sδd , ni0

(1)

where s = +1(−1) for positive (negative) dust, f = nc0 /ni0 and δd = Znd ni0d0 . Number density of cold and hot electrons are determined from the nonextensive distribution function and is expressed as [72], nj = nj0

[

eϕ 1 + (qj − 1) Tj

(qj +1) ] 2(q −1) j

,

(2)

where the parameter qj (j = c for cold and h for hot electrons) measures the degree of nonextensivity of a system. In the limit qj → 1, Maxwellian 4

limit is recovered. Verheest [73] reported that in the superextensive range −1 < qj < 1, for the requirement of finite energy, allowed range shrinks to 1/3 < qj < 1. The author further emphasized that for hot species distribution with superthermal tails, the range 1/3 < q < 1 is more relevant. Thus, in the present investigation, we have considered the range 1/3 < qj < 1 for numerical analysis. The dynamics of DIAWs is governed by the set of fluid equations (continuity, momentum, Poisson) in normalized form : ∂Ni ∂(Ni Ui ) + = 0, ∂T ∂X

(3)

∂Ui ∂Ui ∂ψ + Ui + = 0, ∂T ∂X ∂X

(4)

∂ 2ψ = 1 + (c1 + d1 )ψ + (c2 + d2 )ψ 2 + (c3 + d3 )ψ 3 − Ni − sδd , ∂X 2

(5)

where c1 =

f (qc + 1) , 2g

c2 =

c1 (3 − qc ) , 4g

c3 =

c2 (5 − 3qc ) 6g

and d1 =

(1 − f + sδd )(qh + 1) , 2g

d1 σ(3 − qh ) , 4g

d2 =

d3 =

d2 σ(5 − 3qh ) , 6g

along with σ = TThc and g = f + (1 − f )σ. Eq.(5) is written in this form, by using the Taylor expansion of Eq.(2) for number densities of electrons when ψ ≪ 1, up to third order in ψ. Above equations are normalized by adopting the scaling parameters as, ψ = eϕ/Tef f , Nj=c,h = nj /nj0 , Ui = ui /Cs , X = Tc h0 +nc0 = f +(1−f , where Cs = (kB Tef f /mi )1/2 , x/λs , T = ωs t and Tef f = nnh0 n )σ + c0 Th

Tc

λs = (kB Tef f /4πni0 e2 )1/2 and ωs = (4πni0 e2 /mi )1/2 . 3. Perturbation Analysis We describe the stable variables (Ni , Ui , ψ) with position X and time T . (0) (0) At equilibrium state, Ni = 1 and Ui = ψ (0) = 0. A general description for the state variables can be written as S=S

(0)

+

∞ ∑

m=1

5

ϵm S (m) ,

(6)

where parameter ϵ is very small (ϵ << 1). The stretched slow space and time variables are scaled as Xm = ϵm X

and Tm = ϵm T.

(7)

A state variable is assumed as S (m) =

m ∑

(m)

Sl

l=−m (m)

(Xm , Tm )exp[il(kX − ωT )].

(8)

(m)∗

Sl = S−l represents a reality condition, where the asterisk denotes the complex conjugate. Substituting Eq.(8) into Eqs.(3-5) and using Eq.(7), we obtain the dispersion relation for DIAWs in the following form ω2 = or Vph

k2 k 2 + c1 + d1

ω = = k



k2

1 , + c1 + d1

(9)

(10)

which clearly shows the dependence of the phase velocity (Vph = ωk ) of DIAWs on the nonextensive parameters (via qc and qh ), cold electron to ion number density ratio (via f ), cold to hot electron temperature ratio (via σ) and dust density ratio (via δd ). It is easily seen that Vph decreases with increasing nonextensive parameters (via qc and qh ). It is noted that c1 is positive for different set of physical parameters and Eq.(9) can be used for linear analysis in the present case. In order of ϵ2 , we are able to find the relations for the group velocity Vg , and for two kinds of harmonics (zeroth and second orders). A compatibility condition is imposed in the following form for l = 1 and m = 2 (1)

(1)

∂ψ1 ∂ψ + Vg 1 = 0. ∂T1 ∂X1 The group velocity is defined as Vg (k) =

∂ω , ∂k

Vg = (c1 + d1 )

6

(11)

and is given as

ω3 . k3

(12)

In orders of ϵ2 , for first harmonics, we find the expressions for the amplitudes as (1)

(2)

(2)

N1 = (k 2 + c1 + d1 )ψ1 − 2ik

∂ψ1 ∂X1

(1)

(2)

and U1 =

k (2) ∂ψ ψ1 − iω 1 , (13) ω ∂X1

(1)

∂ψ (2) ψ1 = iS˜ 1 . ∂X1

(14)

We shall take S˜ equal to zero as its choice is arbitrary. The amplitudes of the second order harmonics can be determined from the evolution equations obtained for l = 2 and m = 2. The amplitudes (second harmonics) are expressed as ( )2 )2 ( ( )2 (1) (2) (22) (22) (1) (2) (22) (1) (2) , ψ1 and ψ2 = C3 ψ1 , U 2 = C2 N2 = C1 ψ1 (15) where ] ω [ (22) (22) (22) (22) 2 2 2 C1 − (k + c1 + d1 ) C1 = (c2 + d2 ) + (4k + c1 + d1 )C3 , C2 = k and

(c2 + d2 ) (k 2 + c1 + d1 )2 + . (16) =− 3k 2 2k 2 Finding the evolution equations for two different values of l and m i.e. l = 0, m = 2 and l = 0, m = 3, after some algebraic calculations, we obtain ( )2 ( )2 ( )2 (2) (20) (1) (2) (20) (1) (2) (20) (1) N0 = C1 ψ1 , U 0 = C2 ψ1 , ψ 0 = C3 ψ1 , (22) C3

(20)

C1

(20)

= [(c1 + d1 )C3

and (20)

C3

(20)

+ 2(c2 + d2 )],

=

C2

=

2ω 2 (20) (k + c1 + d1 ) + Vg C1 k

2(c2 + d2 )Vg2 − (k 2 + 3(c1 + d1 ) . 1 − (c1 + d1 )Vg2

All the above coefficients are dependent on the different physical parameters, viz qc,h , f and σ. In the order of ϵ3 , a new compatibility condition is obtained

7

from the reduced evolution equations for l = 1 and this provides the following equation [ ] (1) (1) (1) ∂ψ1 ∂ψ1 ∂ 2 ψ1 (1) (1) i + Vg +P + Q|ψ1 |2 ψ1 = 0. (17) ∂T2 ∂X2 ∂X12 Here P is the dispersion coefficient, expressed as 3 ω5 P = − (c1 + d1 ) 4 . 2 k

(18)

g The coefficient P is equal to 12 ∂V . Since c1 is positive, therefore the coeffi∂k cient P is always negative for the chosen set of parameters. The nonlinear coefficient Q is specified as { ] 3(c + d ) } [ ω3 3 3 (20) (22) + (19) Q = 2 (c2 + d2 ) C3 + C3 k 2 ] ] [ ω [ (20) (22) (20) (22) . − k C2 + C2 − C1 + C1 2

Now we shall consider the limit k << 1 (i.e. long wavelength case) and see the variation in P and Q. It is found that for very small k, P becomes zero (since P ≃ p0 k in the limit). The nonlinear coefficient is Q ≃ qk0 and it diverges for very small value of k. This provides us information regarding qualitative behaviour in accordance with earlier theoretical findings. Both the quantities, p0 and q0 are real and positive. These quantities are specified as ( )3/2 [ ] 1 3 1 2 p0 = and q = 2(c + d ) − 3(c + d ) . 0 2 2 1 1 2(c1 + d1 )3/2 12 c1 + d1 (20) The electrostatic potential solution for the first order may be obtained as [74] (1) ψ ≃ ψ1 = φ(X, T )ei(kX−ωT ) + φ∗ (X, T )e−i(kX−ωT ) . (21) It is assumed that φ(X, T ) varies slowly as compared to exp[i(kX − ωT )]. Using Eq.(21) in expression (17), we obtain NLSE in the following form [75] ) ( ∂φ ∂ 2φ ∂φ + Vg +P i + Q|φ|2 φ = 0. (22) 2 ∂T ∂X ∂X 8

Using a Galilean transformation in Eq.(22), the NLSE can be obtained as i

∂φ ∂ 2φ +P + Q|φ|2 φ = 0. ∂T ∂X 2

(23)

The NLSE in this standard form shall be used to study modulational instability and rogue waves in the present investigation. 4. Modulation Instability and Numerical Analysis P The wave amplitude stability occurs for Q < 0, on the other hand instaP bility occurs for Q > 0 (i.e. perturbation of amplitude grows and may lead to collapse of wave). It is known from Eq.(20) that the instability of wave packet is observed for k ≪ 1 (for large wavelength region). The quantity Q1 |ψ0 |2 is specified √as the maximum growth rate of the instability and is = k˜0 and window of the instability varies from k˜ = 0 found for k˜ = |ψ0 | Q P √ √ [75]. to 2|ψ0 | Q P

P It is noticed that dark solitons exist in the region for Q < 0 (i.e. where P no MI occurs) and bright solitons are observed in the region Q > 0 (where MI occurs). It has been observed from different investigations that MI is proper phenomenon that leads to the formation of envelope bright solitons. The bright type envelope solitons are widely observed in abundance in space/astrophysical plasmas [74]. The phase velocity and group velocity are modified by the composition of plasma and is manifested in modulational profile of wave packets. Now we shall study the MI and envelope dynamics by numerically analyzing the variation of various physical parameters, such as wave number (k), cold electronion density ratio (f ) and cold electron nonextensive parameter (qc ) which reflects the deviation from Maxwellian distribution.

4.1. Parametric Analysis We have studied the influence of various physical parameters on MI through product P Q, growth rate Γqc , ratio P/Q and observed the change in the value of kcr where MI sets in. In Fig.1, we have shown the behavior of kcr with qc (via PQ) in the form of a contour plot, for different set of values

9

of parameters (f , σ, δd ). It is observed that there is a significant concentration of energy leading to the formation of higher amplitude pulses in the region P Q > 0, due to enhanced non-linearity. As the value of qc increases, the critical wave number reduces significantly. With the increase in f , the area of the unstable region increases. At f = 0.5, the value of critical wave number kcr first starts decreasing with increase in qc , reaches the minimum at about qc = 0.6, and then starts increasing with increase in qc . For increase in σ, area of the unstable region remains almost same but the critical wave number increases for higher qc . As the value of δd increases, the area of the unstable region decreases slightly. Similar behavior is observed for positive dust plasma (figure not shown) on variation of same physical parameters, except for increase in δd , area of unstable region slightly increases.

Figure 1: (Color online) Contour plot of P Q = 0 (for negative dust) for critical wave number (kcr ) versus qc for different values of the parameters. Below (Above) different curve region is stable (unstable) for a particular parametric values. Here, k = 0.5, s = −1, qc = 0.40, qh = 0.35, f = 0.2, σ = 0.1, δd = 0.1 for blue (solid) curve. Red (dashed) curve is for f = 0.5, black (dotted) curve is for σ = 0.4 and black (dot-dashed) curve is for δd = 0.4.

10

4.1.1. Effect of nonextensivity (via qc ) and effect of cool electron-ion density ratio (via f ) The growth rate of MI for negative dust for different values of qc and f is illustrated in Fig.2(a). It is noticed that the growth rate enhances greatly with increase in nonextensivity of cold electrons (via qc ) and ratio of cool electron to ion density (via f ) as illustrated by black (dashed and dotted) and red (dotted-dashed and long-dashed) curve respectively. For P 1/3 < qc < 1, the stability profile has been analyzed numerically from Q versus k for different values of qc and f , keeping other parameters fixed (see Fig.2(b)). Dark or grey solitons are observed in the case of large wavelengths, while bright envelope solitons occur in the case of shorter wavelengths. The P absolute value of Q is reduced as qc and f is minimized for given k and in the region for which k < kcr is also reduced with k for given qc (or f ) value. The critical wave number kcr is reduced with increase in qc and f . It is emphasized that the nonextensivity and concentration of cold electrons has remarkable influence on kcr and growth rate of MI.

(a)

(b)

Figure 2: (Color online) Variation of (a) the growth rate Γqc vs y and (b) NLSE coefficient P ratio Q vs k (for negative dust) for different values of qc and f , with s = −1, k = 0.9, qc = 0.40, qh = 0.35, f = 0.2, σ = 0.1, δd = 0.3 for solid (blue) curve. In Figs.(a) and (b), qc = 0.40 for dashed (black) curve, 0.50 for dotted (black) and f = 0.3 for dotted-dashed (red) curve, f = 0.4 for long-dashed (red) curve.

We have also studied numerically the influence of various physical parameters on MI through product P Q, growth rate Γqc , ratio P/Q for positive dust plasma and observed the change in the value of critical wave number kcr . It is observed that with the increase in qc , f and σ, similar kind of 11

variation in stable and unstable regions is observed as seen for the case of negative dust. It is stressed that with the change in polarity of dust (from negative to positive), there is a significant increase in the growth rate as well as kcr where MI sets in for the variation of physical parameters f , σ, qc in different cases for numerical analysis. 4.1.2. Effect of negative/positive dust concentration (via δd ) Fig.3(a) shows the variation of the growth rate of MI for negative as well as positive dust for different values of δd (i.e., concentration of negative/positive dust). It is seen that the increase in concentration of negative dust reduces the growth rate of MI while increase in concentration of positive dust enhances the growth rate. Further, it is observed that the critical wave number kcr is enhanced with increase in the value of δd (see Fig.3(b)) for negative dust, but it reduces in the case of positive dust. This shows that modulational instability region shrinks with increase in negative dust concentration and expands with increase in positive dust concentration. It is also observed that growth rate of MI is significantly higher for a positive dust plasma as compared to negative dust plasma for same values of physical parameters. It is pertinent to mention that the dust concentration (negative and positive cases) has a profound influence on the growth rate as well as modulational instability. 5. Dust ion acoustic rogue (freak) waves (DIARWs) In this section, we have investigated the characteristics of DIARWs from first order and higher order rogue wave solutions of NLSE. Since various physical parameters are very decisive for the formation of rogue waves, so it is of remarkable importance to study the dynamics of rogue waves under the influence of such parameters. Using standard procedure, the first order rational rogue wave solution of NLSE (23) is given as √ [ ] P 4(1 + 2ιP T ) φ1 = − 1 exp(ιP T ) (24) Q 1 + 4X 2 + 4P 2 T 2 Higher-order rogue waves were theoretically predicted in finance [51] as well as in plasmas [52, 76–78] and were experimentally observed in water [79]. The nonlinear superposition of two or more first order rogue waves gives rise 12

(a)

(b)

Figure 3: (Color online) Variation of (a) the growth rate Γqc vs y and (b) NLSE coefficient P ratio Q vs k for different values of δd with k = 0.9, qc = 0.40, qh = 0.35 and σ = 0.1. For Figs.(a) and (b), δd = 0.3 for solid (blue) curve and broken (blue), 0.4 for dashed (black) and dotted-dashed (red) curve, and 0.5 for dotted (black) curve and long-dashed (red) curve.

to higher-order rogue waves and form a higher amplitude more complicated nonlinear structure. The second-order rogue wave solution (localized on both the T and X directions) is expressed as √ ( ) P R2 + ιS2 1+ exp (ιP T ), (25) φ2 = Q T2 where R2 =

3 1 4 3 2 − X − X − 6(P XT )2 − 10(P T )4 − 9(P T )2 , 8 2 2 [

] 15 4 2 2 4 2 S2 = −P T − + X − 3X + 4(P XT ) + 4(P T ) + 2(P T ) , 4

(26)

(27)

and T2 =

1 1 1 9 3 + X 6 + X 4 + X 4 (P T 2 + X 2 + X 2 (P T )4 32 12 8 2 16 2 9 33 3 − (P XT )2 + (P T )6 + (P T )4 + (P T )2 . 2 3 2 8 13

(28)

5.1. Effects of nonextensivity (qc ) and dust density (via δd ) For a negative dust plasma, considering the parametric values of unstable region (P Q > 0) where MI occurs, we have analyzed numerically the role of nonextensivity of cold electrons and dust concentration on the properties of rogue waves. In Fig.4, we have shown the effect of nonextensivity of cold electrons via qc on the characteristics of first order rogue waves (φ1 ) as well as higher order rogue waves (φ2 ). It is seen that as the value of qc increases from qc = 0.35 (lower solid curve for |φ1 | and upper solid curve for |φ2 | ) to qc = 0.80 (lower dashed curve for |φ1 | and upper dashed curve for |φ2 |), the amplitude of the rogue wave pulses is reduced. However, amplitude of higher order rogue waves are significantly larger than first order rogue waves. This change in amplitude takes place due to variation in effect of nonlinearity. Figs.5 and 6 depict the 3D variation of DIARWs with change in qc for negative dust plasma for first order and second order rogue waves respectively. These figures confirm the variation of amplitude of rogue waves as seen in earlier figures. Dust density has a significant influence on amplitude of rogue waves. For 1/3 < qc < 1, it can be seen from Fig.4, as the value of dust density (via δd ) is increased from δd = 0.1 (lower solid curve for |φ1 | and upper solid curve for |φ2 | ) to δd = 0.6 (lower dotted curve for |φ1 | and upper dotted curve for |φ2 |), amplitude of both the first and second order rogue waves are significantly enhanced. This implies that with higher concentration of dust grains, nonlinear effects are more dominant, that leads to enhancement in amplitude of both first order and higher order rogue waves. The nonextensivity of cold electrons (via qc ) and dust concentration (via δd ) have a profound effect on the amplitude of rogue waves.

14

Èj1 È,Èj2 È 7

6

5

4

3

2

1

X -2

1

-1

2

Figure 4: (Color online) Variations of the pulse profile |φ1 |, |φ2 | (for negative dust) with X for different values of qc and δd with s = −1, k = 0.9 qh = 0.35, f = 0.2 σ = 0.1, qc = 0.35 and δd = 0.1. Lower solid (blue) and upper solid (black) curves are for |φ1 | and |φ2 | respectively. Here qc = 0.80 for lower dashed (blue) curve for |φ1 | and upper dashed (black) curve for |φ2 | and δd = 0.6 for lower dotted (blue) curve for |φ1 | and upper dotted (black) curve for |φ2 |.

5.2. Effects of concentration of cold electrons (via f ) and temperature of hot electrons (via σ) Since nonlinear coefficient Q and dispersion coefficient P are modified with change in the values of f and σ in the form of concentration of cold electrons and temperature of hot electrons respectively, it motivates us to study numerically the variation in amplitude of rogue waves. We have illustrated in Fig.7 the effect of concentration of cold electrons (via f ) on both the first order (φ1 ) and higher order (φ2 ) rogue waves. The amplitude of the rogue waves again reduces as the value of f increases from f = 0.2 (lower solid curve for |φ1 | and upper solid curve for |φ2 | ) to f = 0.3 (lower dashed curve for |φ1 | and upper dashed curve for |φ2 |). It is emphasized that for the variation of cold electron number density (via f ), the nonlinear effects are less dominant, that are responsible for shrinking of the amplitude of first order as well as higher order rogue waves. Further, it is observed that as the temperature of cold species of electrons increases (via σ), amplitude of both first as well as second order rogue waves is reduced. This is due to the fact that as the two species of electrons tends to approach to the same temperature (i.e. σ → 1), these electron species tends to behave similarly. 15

T

T

5

0

5

0

-5

-5

2.0

2.0

1.5

1.5

Èj1 È 1.0

Èj1 È 1.0

0.5

0.5

0.0

0.0 -2

-2 0

X

0

X

2

2

(a) T

(b)

5

0 -5 2.0 1.5

Èj1 È 1.0 0.5 0.0 -2 0

X

2

(c)

Figure 5: (Color online) Variations of the rogue wave profile |φ1 | (for negative dust) with X and T for different values of qc with s = −1, k = 0.9 qh = 0.35, δd = 0.1, σ = 0.1 and f at (a) qc = 0.35 , (b) qc = 0.55 and (c) qc = 0.75.

The nonlinearity in the system decreases which results in smaller amplitude pulse of rogue waves (see Fig.7). As the value of σ increases from σ = 0.10 (lower solid curve for |φ1 | and upper solid curve for |φ2 | ) to σ = 0.11 (lower dotted curve for |φ1 | and upper dotted curve for |φ2 |) amplitude of the rogue waves is minimized. Finally it is realized that concentration of cold electrons and temperature of hot electrons are playing very significant role for the formation of rogue waves.

16

3

3

Èj2 È

2

2

Èj2 È 1

1 0

0

-2

5

-2

5

-1 0 1

X

-1

0

0

T

X

0 1

-5 2

T

-5 2

(a)

(b)

3 2

Èj2 È 1 0

-2

5 -1 0

0 1

X

T

-5 2

(c)

Figure 6: (Color online) Variations of the rogue wave profile |φ2 | (for negative dust) with X and T for different values of qc with s = −1, k = 0.9 qh = 0.35, δd = 0.1, σ = 0.1 and f at (a) qc = 0.35 , (b) qc = 0.55 and (c) qc = 0.75.

6. Conclusions We have studied the modulational instability of DIAWs and characteristics of dust ion acoustic rogue waves (DIARWs) in a plasma having ion fluid, immobile charged dust particles and two temperature electrons featuring Tsallis distribution. Using multiple scale perturbation technique, NLSE has been derived and its rational solution is determined to study the propagation properties of DIA wave packets and DIARWs. From the dispersion and non-linear coefficients of NLSE, we have determined different stable and unstable regions of modulation w.r.t. the range of the wave number k by varying the different plasma parameters, viz, nonextensivity of electrons (via qc,h ), dust concentration (via δd ) and the ratio of cold electrons to ion number density (via f ). It is observed that modulated DIAWs can propagate in the form of bright envelope solitons or high amplitude rogues in the presence of MI and dark envelope solitons in the absence of MI. We have also illustrated the growth rate of MI and properties of rogue waves with variation of the plasma parameters. 17

Èj1 È,Èj2 È 10

8

6

4

2

X -2

1

-1

2

Figure 7: (Color online) Variations of the pulse profile |φ1 |, |φ2 | (for negative dust) with X for different values of f and σ with s = −1, k = 0.9 qh = 0.35, f = 0.2 σ = 0.10, qc = 0.35 and δd = 0.7. Lower solid (blue) and upper solid (black) curves are for |φ1 | and |φ2 | respectively. Here f = 0.3 for lower dashed (blue) curve for |φ1 | and upper dashed (black) curve for |φ2 | and σ = 0.11 for lower dotted ( blue) curve for |φ1 | and upper dotted (black) curve for |φ2 |.

We have determined the parametric regime for kcr (critical wave number) where MI sets in by considering P Q = 0. For negative dust plasma, the critical wave number (kcr ) decreases with increase in qc as well as f , but it increases with increase in δd . For positive dust, similar behaviour is observed for qc and f , but it is opposite with dust density δd . The stable and unstable region are significantly modified with change in the plasma parameters. The critical wave number (kcr ) has higher value for negative dust than for positive dust. Modulational instability region and growth rate are enhanced with increase in nonextensivity of electrons (via qc ) for both negative as well as positive dust. The amplitude of the first order and higher order DIA rogue waves are significantly reduced (enhanced) for the increase in value of f and qc (δd ) parameters. The physical parameters such as dust concentration, cold electron to ion number density ratio (via f ), nonextensivity of electrons (via qc,h ) and dust concentration (via δd ) have profound effect on the MI of DIA waves and amplitude of DIA rogue waves. This investigation can be extended to study 3-D rogue waves in different kind of plasma systems where magnetized plasma is considered for the propagation of solitary waves. 18

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