Received: 26 July 2017 Revised: 24 October 2017 Accepted: 5 February 2018
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,
2Department of General Educational Development,
Daffodil International University, Dhaka,
N.A. Chowdhury, Department of Physics,
Jahangirnagar University, Savar, Dhaka,
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.
envelope solitons, modulational instability, positron-acoustic waves
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,white dwarfs,[4,5]early universe,inner regions of the
accretion disc surrounding black holes,pulsar magnetosphere,[8,9]centre of our galaxy,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,viz. around the Earth’s bow shock,lower part of the magnetosphere,and the upper Mar-
tin ionosphere.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.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. Pakzadstudied the formation of solitons and the effect of
non-thermal electrons on solitons in e-p-i plasmas based on certain criteria. Sahuanalysed the effects of the ion kinematic
viscosity on the properties of positron-acoustic (PA) shock waves. Messekher et al.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.investigated the modulational instability (MI) of IAWs in q-non-extensive e-p-i plasmas. Sultana and
Kaurakisderived 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.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
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.
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
𝜕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
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
distributioninstead 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
and non-extensive q-distributionwhich 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
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
3DERIVATION OF THE NLS EQUATION
To study the MI of the PAWs, we will derive the NLS equation by employing the reductive perturbation method.So, we first
introduce the stretched coordinates
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
where kand 𝜔are real variables representing the carrier wave number and frequency, respectively. The derivative operators in
the above equations are treated as follows:
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
These equations reduce to
We thus obtain the dispersion relation for PAWs as
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 4 6 8 10
0 1 2 3 4
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
with the compatibility condition
The amplitude of the second-order harmonics is found to be proportional to 𝜙(1)
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:
1for simplicity. The dispersion coefficient Pis
and the non-linear coefficient Qis
Let us consider the harmonic, modulated amplitude solution Φ= ̂
̃𝜔𝜏)] + c.c.(with perturbation wave number ̃
kand frequency ̃𝜔). Hence, the non-linear dispersion relation for the amplitude
CHOWDHURY ET AL.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
FIGURE 2 Variation of Q/Pwith kfor (a) 𝜇eand 𝜇p=0.3; (b) 𝜇pand 𝜇e=1.5. Along with 𝜎=2and𝛽=0.7
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0 1 2 3 4 5
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 ̂
modulation is given by[50–53]
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
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 ̃
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.
0 1 2 3 4 5 6
0 1 2 3 4 5 6 7
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 ̂
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).
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
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
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
0 100 200 300
–300 –200 –100–10–20
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
Interestingly, in both bright and dark envelope solitons, the relation between the soliton width Wand the constant maximum
amplitude 𝜓0is given by
The dark envelope soliton is depicted in Figure 5b.
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
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.
N.A. Chowdhury http://orcid.org/0000-0003-3770-168X
N.A.C. is grateful to the Bangladesh Ministry of Science and Technology for the award of a National Science and Technology
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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.