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Received: 26 July 2017 Revised: 24 October 2017 Accepted: 5 February 2018

DOI: 10.1002/ctpp.201700069

ORIGINAL ARTICLE

Modulational instability and generation of envelope solitons in

four-component space plasmas

N.A. Chowdhury1A. Mannan1M.R. Hossen2A.A. Mamun1

1Department of Physics, Jahangirnagar University,

Dhaka, Bangladesh

2Department of General Educational Development,

Daffodil International University, Dhaka,

Bangladesh

*Correspondence

N.A. Chowdhury, Department of Physics,

Jahangirnagar University, Savar, Dhaka,

Bangladesh.

Email: nurealam1743phy@gamil.com

The four-component space plasma (containing immobile positive ions, inertial cold

positrons, and inertia-less hot electrons and positrons following Cairns’ non-thermal

distribution function) is considered. The modulational instability and the generation

of positron-acoustic (PA) envelope solitons in such a space plasma system are inves-

tigated by deriving the non-linear Schrödinger (NLS) equation. It is found from the

numerical analysis of the NLS equation that the plasma system under consideration

supports the existence of both dark and bright envelope solitons associated with the

PA waves. It is also observed that the instability criterion and the growth rate of the

unstable PA waves are significantly modified by the plasma parameters (i.e. tem-

perature and number density of different plasma species). The implications of the

results to space plasma research are briefly discussed.

KEYWORDS

envelope solitons, modulational instability, positron-acoustic waves

1INTRODUCTION

Nowadays, physicists are mesmerized by the natural beauty of electron–positron–ion (e-p-i) plasmas because of many painstak-

ing observations have disclosed the existence of e-p-i plasmas in various regions of our universe, viz. supernovae, pulsar

environments, cluster explosions,[1,2]polar regions of neutron stars,[3]white dwarfs,[4,5]early universe,[6]inner regions of the

accretion disc surrounding black holes,[7]pulsar magnetosphere,[8,9]centre of our galaxy,[10]and solar atmospheres.[11,12]

To understand the physics of the collective processes in such plasmas, many researchers have studied the ion-acoustic

waves (IAWs)[13–17]and electron-acoustic waves (EAWs)[18,19]in electron-ion (e-i) plasmas by considering Maxwellian

distribution[20–24]for the plasma species. But in astrophysical environments, generally, a non-thermal plasma system (i.e. there

is an excess of non-Maxwellian particles such as electrons and positrons which means that at least one of the components of such

plasmas does not follow the prominent Maxwell–Boltzmann distribution) is characterized by the long tail in the high-energy

region of space plasmas,[25]viz. around the Earth’s bow shock,[26]lower part of the magnetosphere,[27]and the upper Mar-

tin ionosphere.[28]The theoretically investigated results, which obtain by considering non-thermal distribution of the plasma

species instead of Maxwellian distribution in the highly populated region of non-thermal particles, are rigorously justified by

the observational data from Freja and Viking Satellites[27,29]. So, for a better understanding of these highly energetic space

plasmas, a non-thermal distribution can be used to model such kinds of space plasma systems.

A number of researchers have used non-thermal distribution to study the linear and non-linear structure of e-p-i plasmas.

For example, Cairns et al.[30]used a non-thermal distribution of electrons to understand how the presence of a population of

energetic electrons changes the nature of ion-sound solitary waves. Pakzad[31]studied the formation of solitons and the effect of

non-thermal electrons on solitons in e-p-i plasmas based on certain criteria. Sahu[32]analysed the effects of the ion kinematic

viscosity on the properties of positron-acoustic (PA) shock waves. Messekher et al.[33]examined the influence of quantum effects

on solitary structures as well as on double layers by deriving the Korteweg–de Vries equation in unmagnetized four-component

plasmas. Eslami et al.[34]investigated the modulational instability (MI) of IAWs in q-non-extensive e-p-i plasmas. Sultana and

Kaurakis[18]derived the non-linear Schrödinger (NLS) equation by using of a multi-scale perturbation technique to examine the

Contrib. Plasma Phys. 2018;n/a www.cpp-journal.org © 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1

2CHOWDHURY ET AL.

stability of the EAWs and the formation of envelope solitons under certain conditions in e-i plasmas. Zhang et al.[35]studied

the MI of an e-p-i plasma system and observed that the amplitude of dark and bright envelope solitons significantly depends

on the non-thermal parameter, concentration of positrons, and the ion temperature. To the best of our knowledge, no theoretical

investigation has been made on the non-linear properties of PA waves (PAWs) in unmagnetized plasmas with immobile ions,

inertial cold positrons, and non-thermally-distributed hot electrons and positrons. The existence of positrons of two temperatures

is found in many space plasma situations and plays a very significant role in modifying some basic non-linear features of

PAWs.[36–40]Therefore, in this paper, we will study the MI of PAWs and the formation of envelope solitons by deriving the NLS

equation in a plasma system having an excess of non-thermally-distributed hot electrons and positrons in a “non-Maxwellian

tail.”

The rest of the paper is organized as follows. The governing equations describing our plasma model are presented in Section

2. The NLS equation is derived by using the reductive perturbation technique in Section 3. Stability analysis is given in Section

4. The basic features of envelope solitons are examined in Section 5. A brief discussion is finally provided in Section 6.

2GOVERNING EQUATIONS

We consider an unmagnetized, four-component plasma system consisting of immobile positive ions, inertial cold positrons,

and non-thermally-distributed hot electrons and hot positrons. At equilibrium, the quasi-neutrality condition can be expressed

as ncp0 +nhp0 +ni0 =nhe0,wherencp0 ,nhp0,ni0 ,andnhe0 are the unperturbed number densities of cold positrons, hot positrons,

immobile ions, and hot electrons, respectively. The dynamics of the PAWs in four component plasma system is described by

normalized equations of the form

𝜕ncp

𝜕t+𝜕

𝜕x(ncpucp )=0,(1)

𝜕ucp

𝜕t+ucp

𝜕ucp

𝜕x=−

𝜕𝜙

𝜕x,(2)

𝜕2𝜙

𝜕x2=𝜇enhe −𝜇pnhp −ncp −(𝜇e−𝜇p−1),(3)

where ncp is the cold positron number density normalized by its equilibrium quantity (ncp0); ucp is the cold positron fluid

speed normalized by the PA speed Ccp =(kBThe/mp)1/2 with kBthe Boltzmann constant, The (Thp) the hot electron (positron)

temperature, and mpthe positron rest mass; 𝜙is the electrostatic wave potential normalized by kBThe/e,withethe magnitude of

the electronic charge; some parameters can be recognized as 𝜇e=nhe0/ncp0 and 𝜇p=nhp0/ncp0 . The time variable tis normalized

by 𝜔−1

cp =(mp∕4𝜋e2ncp0 )1∕2, and the space variable xis normalized 𝜆Dm =(kBThe/4𝜋e2ncp0 )1/2 . We note that PAWs are similar to

EAWs[41,42]in the sense that, instead of cold (hot) electron species (in EAWs) the cold (hot) positron species (in PAWs) provide

the inertia (restoring force). This is, in fact, the concept of considering inertial cold positrons and inertia-less hot positrons in

the study of PAWs.[36–40]

The number densities (nhp and nhp) of hot positrons and electrons (following the Cairns’ non-thermal distribution[30,43–46])

can be expressed as

nhp =(1+𝛽𝜎𝜙 +𝛽𝜎2𝜙2)exp(−𝜎𝜙),(4)

nhe =(1−𝛽𝜙 +𝛽𝜙2)exp(𝜙),(5)

where 𝜎=The/Thp and 𝛽=4𝛼/(1 +3𝛼), with 𝛼being the parameter[30,43–46]determining the fast particles present in our plasma

model. We note that many space plasma systems contain a fraction of energetic/fast plasma particles in addition to thermal ones.

To explain the distribution of particles in a plasma system with energetic/fast particles, one should consider Cairns’ non-thermal

distribution[30]instead of the Maxwellian one. We further note that for 𝛼=0, the Cairns’ non-thermal distribution function

reduces to the Maxwellian distribution function. There are other types of non-Maxwellian distribution, e.g. 𝜅-distribution[47]

and non-extensive q-distribution[48]which arise from the deviation from thermal equilibrium of the plasma species and can

be used for other types of space and laboratory plasma systems. Now, by substituting Equations 4 and 5 into Equation 3, and

expanding up to third order in 𝜙,weget

𝜕2𝜙

𝜕x2+ncp −(1+𝛾1𝜙+𝛾2𝜙2+𝛾3𝜙3+···)=0,(6)

where

𝛾1=(1−𝛽)(𝜇e+𝜇p𝜎),𝛾

2=(𝜇e−𝜇p𝜎2)∕2,𝛾

3=(1+3𝛽)(𝜇e+𝜇p𝜎3)∕6.

CHOWDHURY ET AL.3

We note that the last term on the left-hand side of Equation 6 is the contribution of immobile ions, hot electrons, and hot

positrons.

3DERIVATION OF THE NLS EQUATION

To study the MI of the PAWs, we will derive the NLS equation by employing the reductive perturbation method.[49]So, we first

introduce the stretched coordinates

𝜉=𝜀(x−vgt),(7)

𝜏=𝜀2t,(8)

where vgis the envelope group speed to be determined later, and 𝜀(0 <𝜀<1) is a small parameter. Then, we can write the

dependent variables as

ncp =1+

∞

m=1

𝜀(m)

∞

l=−∞

n(m)

l(𝜉,𝜏)exp[𝑖𝑙(𝑘𝑥 −𝜔𝑡)],(9)

ucp =

∞

m=1

𝜀(m)

∞

l=−∞

u(m)

l(𝜉,𝜏)exp[𝑖𝑙(𝑘𝑥 −𝜔𝑡)],(10)

𝜙=

∞

m=1

𝜀(m)

∞

l=−∞

𝜙(m)

l(𝜉,𝜏)exp[𝑖𝑙(𝑘𝑥 −𝜔𝑡)],(11)

where kand 𝜔are real variables representing the carrier wave number and frequency, respectively. The derivative operators in

the above equations are treated as follows:

𝜕

𝜕t

→𝜕

𝜕t−𝜀vg

𝜕

𝜕𝜉 +𝜀2𝜕

𝜕𝜏 ,(12)

𝜕

𝜕x

→𝜕

𝜕x+𝜀𝜕

𝜕𝜉 .(13)

By substituting Equations 9–13 into Equations 1, 2, and 6, and collecting the terms containing 𝜀, the first-order (m=1with

l=1) equations can be expressed as

𝑖𝑘𝑢(1)

1−𝑖𝜔n(1)

1=0,(14)

𝑖𝑘𝜙(1)

1−𝑖𝜔u(1)

1=0,(15)

n(1)

1−k2𝜙(1)

1−𝛾1𝜙(1)

1=0.(16)

These equations reduce to

n(1)

1=k2

𝜔2𝜙(1)

1,(17)

u(1)

1=k

𝜔𝜙(1)

1.(18)

We thus obtain the dispersion relation for PAWs as

𝜔2=k2

k2+𝛾1

.(19)

We have numerically analysed Equation 19 to examine the linear dispersion properties of PAWs for different values of the

non-thermal parameter 𝛽. The results are displayed in Figure 1a, which shows that (a) in the short wavelength limit (k2≫𝛾1),

the dispersion curves become saturated and the maximum frequency of the PAWs is equal to the cold positron plasma frequency

(𝜔cp); (b) in the long wavelength limit (k2≪𝛾1), the angular frequency of the PAWs linearly increases with the wavenumber k;

(c) the nature of the PAWs are, therefore, similar to other kinds of acoustic-type waves (i.e. EAWs and IAWs) but with different

times and scale lengths; and (d) as the non-thermal parameter is increased, the angular frequency increases.

4CHOWDHURY ET AL.

0.2

0.5

0.7

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

k

𝝎

0.1

0.3

0.5

0 1 2 3 4

200

150

100

50

0

50

100

k

Q/P

(a) (b)

FIGURE 1 (a) Variation of 𝜔with the carrier wave number (k)for𝛽. (b) Variation of Q/Pwith kfor 𝛽. Along with 𝜇e=1.5, 𝜇p=0.3, and 𝜎=2

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

n(2)

1=k2

𝜔2𝜙(2)

1+2𝑖𝑘(vgk−𝜔)

𝜔3

𝜕𝜙(1)

1

𝜕𝜉 ,(20)

u(2)

1=k

𝜔𝜙(2)

1+i(vgk−𝜔)

𝜔2

𝜕𝜙(1)

1

𝜕𝜉 ,(21)

with the compatibility condition

vg=𝜕𝜔

𝜕k=𝜔(1−𝜔2)

k.(22)

The amplitude of the second-order harmonics is found to be proportional to 𝜙(1)

12:

n(2)

2=C1𝜙(1)

12,u(2)

2=C2𝜙(1)

12,𝜙

(2)

2=C3𝜙(1)

12,

n(2)

0=C4𝜙(1)

12,u(2)

0=C5𝜙(1)

12,𝜙

(2)

0=C6𝜙(1)

12,(23)

where

C1=3k4

2𝜔4+C3k2

𝜔2,C2=k3

2𝜔3+C3k

𝜔,C3=3k4−2𝛾2𝜔4

2𝜔4(4k2+𝛾1)−2𝜔2k2,

C4=2vgk3+𝜔k2+C6𝜔3

v2

g𝜔3,C5=k2+C6𝜔2

vg𝜔2,C6=2vgk3+𝜔k2−2𝛾2v2

g𝜔3

𝛾1v2

g𝜔3−𝜔3.

Finally, the third-harmonic modes (m=3) and (l=1), with the help of Equations 17–23, give a system of equations that can

be reduced to the following NLS equation:

i𝜕Φ

𝜕𝜏 +P𝜕2Φ

𝜕𝜉2+QΦ2Φ=0,(24)

where Φ=𝜙(1)

1for simplicity. The dispersion coefficient Pis

P=1

2

𝜕vg

𝜕k=−

3

2

𝜔2

kvg,(25)

and the non-linear coefficient Qis

Q=𝜔3

2k22𝛾2(C3+C6)+3𝛾3−k2(C1+C4)

𝜔2−2k3(C2+C5)

𝜔3.(26)

4STABILITY ANALYSIS

Let us consider the harmonic, modulated amplitude solution Φ= ̂

Φe𝑖𝑄̂

Φ2𝜏+c.c.,wherê

Φ=̂

Φ0+𝜀̂

Φ1and ̂

Φ1=̂

Φ1,0exp[i(̃

k𝜉−

̃𝜔𝜏)] + c.c.(with perturbation wave number ̃

kand frequency ̃𝜔). Hence, the non-linear dispersion relation for the amplitude

CHOWDHURY ET AL.5

𝜇e1.5

𝜇e2.5

𝜇e3.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0

60

40

20

0

20

40

60

80

k

𝜇p0.2

𝜇p0.3

𝜇p0.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0

40

20

0

20

40

60

80

k

(a) (b)

Q/P

Q/P

FIGURE 2 Variation of Q/Pwith kfor (a) 𝜇eand 𝜇p=0.3; (b) 𝜇pand 𝜇e=1.5. Along with 𝜎=2and𝛽=0.7

2.0

2.3

2.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

60

40

20

0

20

40

60

80

k

2.0

2.3

2.6

0 1 2 3 4 5

0.00

0.02

0.04

0.06

0.08

0.10

0.12

k

g

(a) (b)

Q/P

FIGURE 3 (a) Variation of Q/Pwith kfor 𝜎. (b) Variation of MI growth rate (Γg)with ̃

kfor 𝜎. Along with 𝜇e=1.5, 𝜇p=0.3, 𝛽=0.7, k=3, and ̂

Φ0=0.4

modulation is given by[50–53]

̃𝜔2=P2̃

k2̃

k2−2Q

P̂

Φo2.(27)

Clearly, if Q/P<0, ̃𝜔 is always real for all values of ̃

k, and hence in this region the PAWs are stable in the presence of

small perturbations. On the other hand, when Q/P>0, MI would set in as ̃𝜔 becomes imaginary and the PAWs are unstable for

̃

k<kc=2Q̂

Φo2∕P,wherekcis the critical value of the wave number of modulation, and ̂

Φois the amplitude of the carrier

waves. The growth rate (Γg) of MI (within conditions, when Q/P>0 and simultaneously ̃

k<kc)isgivenby

Γg=∣ P∣̃

k2k2

c

̃

k2−1.(28)

The maximum value Γg(max) of Γgis obtained at ̃

k=kc∕2 and is given by Γg(max)=Q̂

Φ02. The coefficients of the

dispersion term Pand the non-linear term Qare dependent on various physical plasma parameters such as 𝜇e,𝜇p,𝜎,and𝛽. Thus,

these parameters may sensitively to change the stability conditions of the PAWs. One can recognize the stability conditions of

PAWs by depicting Q/Pagainst kfor different physical plasma parameters. When the sign of the ratio Q/Pis negative, the PAWs

are modulationally stable, and when the sign of the ratio Q/Pis positive, the PAWs will be modulationally unstable against

external perturbations. It is clear that both stable and unstable region for PAWs are obtained from the Figures 1b, 2a,b, and 3a.

When Q/P=0, the corresponding value of k(=kc) is called the critical or the threshold wave number for the onset of MI. This

critical value separates the unstable (Q/P>0) region from the stable region (Q/P<0). The non-thermal parameter 𝛽plays a

significant role in changing the stability of the PAWs. With increasing values of 𝛽, the critical value kcis shifted to the lower

value (see Figure 1b). It is also found that the absolute value of the ratio Q/Pdecreases with the increasing values of 𝛽.

6CHOWDHURY ET AL.

𝜇e1.5

𝜇e2.5

𝜇e3.5

0 1 2 3 4 5 6

0.00

0.05

0.10

0.15

0.20

0.25

0.30

g

𝜇p0.1

𝜇p0.2

𝜇p0.3

0 1 2 3 4 5 6 7

0.00

0.05

0.10

0.15

0.20

g

(a) (b)

kk

FIGURE 4 Variation of MI growth rate (Γg)with̃

kfor (a) 𝜇eand 𝜇p=0.3, and (b) 𝜇pand 𝜇e=1.5. Along with 𝜎=2, 𝛽=0.7, k=3, and ̂

Φ0=0.4

The variation of Q/Pwith kfor different values of the ratio of the hot electron to cold positron concentration (via 𝜇e)with

fixed values of other physical parameters is depicted in Figure 2a. It is seen that the kcvalue decreases with increasing values

of nhe0 for a fixed ncp0. So, an excess number of electrons of the system lead to the minimization of the stable domain of the

wave profile. The variation of Q/Pwith kfor different values of the ratio of hot to cold positron concentration (via 𝜇p)with

fixed values of the other physical parameters is shown in Figure 2b. The kcvalue at which the instability sets increases with

the increasing values of 𝜇p. Actually, increasing the values of nhp0 for a fixed ncp0 is responsible for the increase in the stable

domain of the PAWs. On the other hand, the absolute value of the ratio Q/Pincreases with increasing values of 𝜇p.

We have also analysed the effect of the ratio of the hot electron temperature to the hot positron temperature (via 𝜎)onthe

stability of the wave profiles (see Figure 3a). It is seen that as the value of The (Thp) is increased, kcis shifted to a higher (lower)

value (via 𝜎). So, increasing the hot electron or hot positron temperature plays simultaneously opposite roles to recognize the

stability region of the PAWs.

The variation of the growth rate (Γg) of MI versus the MI wave number (̃

k) is depicted in Figures 3b and 4a,b. It is obvious from

these figures that (a) Γgdecreases (increases) with the increase in the values of The (Thp); (b) as the value of nhe0 is increased

for a fixed value of ncp0,Γgincreases (via 𝜇e). This also implies that with larger values of the hot electron concentration, the

non-linearity of the PAWs is stimulated, which manifests via the maximum value of the growth rate of MI; (c) Γgdecreases

with the increase in the value of nhp0 for a fixed value of ncp0 (via 𝜇p).

5ENVELOPE SOLITONS

If PQ <0, the modulated envelope pulse is stable (in this region, dark envelope solitons exist), and when PQ >0, the modulated

envelope pulse is unstable against external perturbations and leads to the generation of bright envelope solitons. A solution of

Equation 24 may be sought in the form

Φ=𝜓exp(𝑖𝜃),(29)

where 𝜓and 𝜃are real variables that are determined by substituting into the NLS equation and separating the real and imag-

inary parts. Interested readers are referred to refs. [49,54–57]for details. The different types of solution thus obtained are clearly

summarized in the following paragraphs.

5.1 Bright solitons

When PQ >0, we find bright envelope solitons. The general analytical form of the bright solitons reads

𝜓=𝜓0sech2𝜉−𝑈𝜏

W,(30)

𝜃=1

2P𝑈𝜉 +Ω0−U2

2𝜏.(31)

where Uis the propagation speed, Wis the soliton width, and Ω0is the oscillating frequency for U=0. The bright envelope

soliton is depicted in Figure 5a.

CHOWDHURY ET AL.7

0 10 20

0.02

0.01

0.00

0.01

0.02

𝜉

Re

Re

0 100 200 300

0.02

0.01

0.00

0.01

0.02

𝜉

–300 –200 –100–10–20

(a) (b)

FIGURE 5 (a) Bright envelope solitons for k=3. (b) Dark envelope solitons for k=0.1. Along with 𝜇e=1.5, 𝜇p=0.3, 𝜎=2, 𝛽=0.7, 𝜓0=0.0005, U=0.1,

𝜏=0, and Ω0=0.4

5.2 Dark solitons

When PQ <0, we find dark envelope solitons whose general analytical form reads

𝜓=𝜓0tanh2𝜉−𝑈𝜏

W,(32)

𝜃=1

2P𝑈𝜉 −U2

2−2𝑃𝑄𝜓

0𝜏.(33)

Interestingly, in both bright and dark envelope solitons, the relation between the soliton width Wand the constant maximum

amplitude 𝜓0is given by

W=2∣P∕Q∣

𝜓0

.(34)

The dark envelope soliton is depicted in Figure 5b.

6DISCUSSION

We have considered an unmagnetized, four-component e-p-i plasma system consisting of immobile positive ions, inertial mobile

cold positrons, and non-thermally-distributed hot positrons and electrons. We used the well-known reductive perturbation

method to derive the NLS equation, which is valid for a small but finite amplitude limit. The results that have been obtained

from this theoretical investigation can be summarized as follows:

1. PAWs will be stable (unstable) for the range of values of kin which the ratio Q/Pis negative (positive), i.e. Q/P<0(Q/P>0).

With increasing values of 𝛽(non-thermal case), kcis shifted to lower values.

2. The value (kc) at which the instability sets decreases (increases) with increasing values of nhe0 (nhp0) for a fixed value of ncp0

(via 𝜇e(𝜇p)).

3. As the value of The is increased for a fixed value of Thp,kcis shifted to higher values (via 𝜎).

4. Γgdecreases (increases) with increase in the values of The (Thp). On the other hand, Γgincreases (decreases) with the increase

of nhe0 (nhp0) for a fixed value of ncp0 (via 𝜇e(𝜇p)).

We highlight here that the findings of our investigation should be useful for understanding the striking features (MI and

envelope solitons) of space environments, i.e. cluster explosions,[1,2]active galactic nuclei, auroral acceleration regions, lower

part of the magnetosphere, and ionosphere, etc.

ORCID

N.A. Chowdhury http://orcid.org/0000-0003-3770-168X

ACKNOWLEDGEMENT

N.A.C. is grateful to the Bangladesh Ministry of Science and Technology for the award of a National Science and Technology

(NST) Fellowship.

8CHOWDHURY ET AL.

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How to cite this article: Chowdhury N.A, Mannan A, Hossen M.R, Mamun A.A. Modulational instability and gener-

ation of envelope solitons in four-component space plasmas, Contributions to Plasma Physics 2018. https://doi.org/10.

1002/ctpp.201700069.