Modulational Instability and Generation of Envelope Solitons in Four Component Space Plasmas

Abstract and Figures

A four component space plasma system (consisting of immobile positive ions, inertial cold positrons as well as hot electrons and positrons following Cairns' nonthermal distribution function is considered. The nonlinear propagation of the positron-acoustic (PA) waves, in which the inertia (restoring force) is provided by the cold positron species (nonthermal pressure of both hot electron and positron species) has been theoretically investigated by deriving the nonlinear Schr\"odinger (NLS) equation. It is found from the numerical analysis of this NLS equation that the space plasma system under consideration supports the existence of both dark and bright envelope solitons associated with PA waves, and that the dark (bright) envelope solitons are modulationally stable (unstable). It is also observed that the basic properties (viz. stable regime and unstable regime with growth rate) of the PA envelope solitions are significantly modified by related plasma parameters (viz. number densities and temperature of plasma species), which correspond to different realistic space plasma situations.
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Received: 26 July 2017 Revised: 24 October 2017 Accepted: 5 February 2018
DOI: 10.1002/ctpp.201700069
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,
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,[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 © 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1
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
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
𝜕x(ncpucp )=0,(1)
𝜕x2=𝜇enhe 𝜇pnhp ncp −(𝜇e𝜇p1),(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 𝜔1
cp =(mp4𝜋e2ncp0 )12, 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 =(1𝛽𝜙 +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
𝜕x2+ncp −(1+𝛾1𝜙+𝛾2𝜙2+𝛾3𝜙3+···)=0,(6)
We note that the last term on the left-hand side of Equation 6 is the contribution of immobile ions, hot electrons, and hot
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
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+
l(𝜉,𝜏)exp[𝑖𝑙(𝑘𝑥 𝜔𝑡)],(9)
ucp =
l(𝜉,𝜏)exp[𝑖𝑙(𝑘𝑥 𝜔𝑡)],(10)
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:
𝜕𝜉 +𝜀2𝜕
𝜕𝜏 ,(12)
𝜕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
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.
0 2 4 6 8 10
0 1 2 3 4
(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
𝜕𝜉 ,(20)
𝜕𝜉 ,(21)
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:
𝜕𝜏 +P𝜕2Φ
where Φ=𝜙(1)
1for simplicity. The dispersion coefficient Pis
and the non-linear coefficient Qis
Let us consider the harmonic, modulated amplitude solution Φ= ̂
Φ1and ̂
̃𝜔𝜏)] + c.c.(with perturbation wave number ̃
kand frequency ̃𝜔). Hence, the non-linear dispersion relation for the amplitude
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
(a) (b)
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
(a) (b)
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
Φo2P,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 ̃
Γg=∣ P̃
The maximum value Γg(max) of Γgis obtained at ̃
k=kc2 and is given by Γg(max)=Q̂
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.
0 1 2 3 4 5 6
0 1 2 3 4 5 6 7
(a) (b)
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
2P𝑈𝜉 +Ω0U2
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.
0 10 20
0 100 200 300
–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
2P𝑈𝜉 U2
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
(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.
N.A. Chowdhury
N.A.C. is grateful to the Bangladesh Ministry of Science and Technology for the award of a National Science and Technology
(NST) Fellowship.
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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.
... MI of IAWs have been theoretically investigated [47] in a plasma system composed of inertial warm adiabatic ions, isothermal positrons and two-temperature super-thermal electrons (cool and hot) which is useful for understanding different nonlinear phenomena in both space (viz., Saturn's magnetosphere and interplanetary medium) and laboratory plasmas (viz., hot-cathode discharge and high-intensity laser irradiation). In addition, MI has been also investigated [48,49] in multi-component plasmas relevant in both space and laboratory plasmas. ...
Face-to-face collisional effects of ion acoustic waves (IAWs) in multi-component unmagnetized plasma containing pair-ions, nonthermal electrons and isothermal positrons are investigated theoretically. The nonlinear Korteweg–de Vries (KdV) equations are derived employing the extended Poincaré–Lighthill–Kuo (ePLK) method to perceive the consequences of collisions of IAWs on amplitude, width and phase shift taking into account the concerned plasma parameters. This study reveals that the compressive (hump shape) and rarefactive (dip shape) solitons as well as the negative phase shifts are produced depending on plasma species densities, temperatures and collisions. The nonlinear coefficient of IAWs is increasing due to the changes in the mass ratio of positive to negative ions, density ratio of electron to positive ion and temperature ratio of electron to positron. The results of this study can be useful understanding in different regions of space (viz., upper region of Titan’s atmosphere, cometary comae and Earth’s ionosphere, etc.) and laboratory (viz., plasma processing reactor and neutral beam sources, etc.) plasmas.
In this paper, the collision dynamics of positron acoustic waves (PAWs) in an unmagnetized plasma containing immobile ions, inertial cold positron, non-thermally distributed hot electrons and positrons is investigated theoretically. The study of soliton collisions represent one of the marvellous phenomena in nonlinear wave dynamics. The dynamics of the system is governed by the well established nonlinear Schrödinger equation (NLSE). By employing Hirota bilinearization technique the collision between two solitary wave solutions is obtained. It is found that the effect of the ratio of the hot electron temperature to the hot positron temperature (σ) and non-thermal parameter (β) plays a significant role in changing the amplitude of the PAWs. From this result it is perceived that the fascinating inelastic collisions of soliton supports energy redistribution among the modes including amplitude change of the soliton. It is observed that the profile of the solitons is strongly influenced by the relevant physical parameters. Finally the result of the investigation may be useful to analyze the collective phenomena related to PAWs collisions that are of considerable interest in space and laboratory plasmas as well as in plasma applications.
Slow and fast modulation instability and envelope soliton of ion-acoustic wave (IAW) are studied in an electron–ion–positron plasma when the ions are warm and have constant stream velocity, positrons are isothermal and electrons are superthermal. The non-linear Schrodinger (NLS) equation has been derived using the Fried and Ichikawa method and the solution of envelope soliton is obtained. It is seen that IAW propagates with two modes (slow and fast) in the presence of an ion stream in the plasma which results in slow and fast modulation instability of the wave. The stability criteria for slow mode (SM) and fast mode (FM) of the wave are established and studied graphically for different values of the ion temperature, positron density, positron temperature and kappa factor of electrons. The bright envelope soliton and dark envelope soliton of SM and FMs of the wave in electron–ion–positron plasma are also studied. The solution of rogue waves (RWs) is obtained from the NLS equation and the nature of RWs of the SM and FM of the wave is discussed graphically. Our results are new and no author has reported this kind of results of slow and fast modulation instability of IAW till now.
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The modulational instability (MI) criteria of dust‐ion‐acoustic (DIA) waves (DIAWs) have been 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 nonlinear Schrödinger equation (NLSE) is derived by using reductive perturbation method. The nonlinear and dispersive coefficients of the NLSE can predict the modulationally stable and unstable 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 is found that the MI growth rate increases (decreases) with increasing the number density of the negatively charged dust grains in the presence (absence) of the negative ions. It is also observed that the amplitude and width of the DIARWs increase (decrease) with the negative (positive) ion mass. The implications of the results to laboratory and space plasmas are briefly discussed.
Dynamics of the positron acoustic waves (PAWs) in magnetoplasmas following Cairns non-thermal distribution is studied on the frameworks of the Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations. The reductive perturbation technique is used to derive the KdV and mKdV equations. Bifurcations of positron acoustic traveling waves of these equations are addressed by employing the bifurcation theory of planar dynamical systems. It is found that the KdV equation supports compressive positron acoustic solitary waves (PASWs), while the mKdV equation supports both compressive and rarefactive PASWs. Using numerical simulations, effect of the nonthermal parameter (β), temperature ratio of hot electron to hot positron (σ), magnetic field (ωc1), ratio of hot electron to cold positron concentration (μe), and ratio of hot to cold positron concentration (μp) are discussed on the PASWs solutions of the KdV and mKdV equations. The criterion of chaos for these perturbed equations under the external periodic perturbation are obtained through quasi-periodic route to chaos. It is in fact shown that transition to chaos in our system depends on the frequency ω and the strength of the external periodic perturbation f0. These parameters control the dynamic behavior of the PAWs. The relevance of this work may be useful to understand the qualitative changes in the dynamics of perturbed PAWs appearing in auroral acceleration region as well as the astrophysical and laboratory plasma, where static external magnetic field and nonthermal parameter are present.
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A nonlinear Schrödinger equation (NLSE) has been derived by employing reductive perturbation method for investigating the modulational instability of dust-ion-acoustic waves (DIAWs) in a four-component plasma having stationary negatively charged dust grains, inertial warm ions, and inertialess non-thermal electrons, and positrons. It is observed that under consideration, the plasma system supports both modulationally stable and unstable domains, which are determined by the sign of the dispersive and nonlinear coefficients of NLSE, of the DIAWs. It is also found that the nonlinearity as well as the height and width of the first and second-order rogue waves increases with the non-thermality of electron and positron. The relevancy of our present investigation to the observations in space plasmas is pinpointed.
A more general and realistic four‐component magnetized plasma medium consisting of opposite polarity ions and nonthermal distributed positrons and electrons is considered to investigate the stable/unstable frequency regimes of modulated ion‐acoustic waves (IAWs) in the D‐F regions of Earth's ionosphere. A (3 + 1)‐dimensional nonlinear Schrödinger equation, which leads to the modulation instability (MI) of IAWs, is derived. The parametric regimes for the existence of the MI, first‐ and second‐order rogue waves, and also their basic features (viz., amplitude, width, and speed) are found to be significantly modified by the effect of physical plasma parameters and external magnetic field. It is found that the nonlinearity of the different types of electronegative plasma system depends on the positive to negative ion mass ratio. It is also shown that the presence of nonthermal distributed electrons and positrons modifies the nature of the MI of the modulated IAWs. The implication of our results for the laboratory plasma [e.g., (Ar+, F−) electronegative plasma] and space plasma [e.g., (H+, H−), (H+,O2-) electronegative plasma in D‐F regions of Earth's ionosphere] are briefly discussed.
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A theoretical and numerical study on amplitude modulated positron-acoustic waves (PAWs) in a magnetized four-component space plasma (containing immobile positive ions, inertial cold positrons, and inertia-less hot electrons and positrons following Cairn’s non-thermal distribution function) has been carried out. The reductive perturbation method have been applied to derive the corresponding nonlinear Schrödinger (NLS) equation, whose the nonlinear and dispersion coefficients \(Q\) and \(P\) are function of the external magnetic field. The criteria for the occurrence of modulational instability (MI) of PAWs is addressed. It is shown that the plasma parameters contribute to enhance substantially the growth rate and the bandwidth of the MI. It is also found from the analysis of the NLS equation that the plasma system under assumption supports the existence of Peregrine solitons and super-rogue waves, whose amplitude are significantly modified by the effects of the external magnetic field, the density ratio of hot positron and cold positron, the density ratio of electron and cold positron, and the non-thermal parameter. Moreover, the various types of localized positron-acoustic excitations exist in the form of bright envelope soliton and dark envelope soliton. It is found that the localized structures’s properties (width and amplitude) are influenced by the presence of magnetic field. The relevance of present study can help researchers to explain the various localized structures and the basic features of PAWs in a magnetized plasmas environments.
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The nonlinear propagation of dust-acoustic waves (DAWs) and associated dust-acoustic rogue waves (DARWs), which are governed by the nonlinear Schrödinger equation, is theoretically investigated in a four-component plasma medium containing inertial warm negatively charged dust grains and inertialess non-thermal distributed electrons as well as isothermal positrons and ions. The modulationally stable and unstable parametric regimes of DAWs are numerically studied for the plasma parameters. Furthermore, the effects of temperature ratios of ion-to-electron and ion-to-positron, and the number density of ion and dust grains on the DARWs are investigated. It is observed that physical parameters play very crucial roles in the formation of DARWs. These results may be useful in understanding the electrostatic excitations in dusty plasmas in space and laboratory situations.
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The nonlinear propagation of heavy-ion-acoustic (HIA) waves (HIAWs) in a four component multi-ion plasma (containing inertial heavy negative ions and light positive ions, as well as inertialess nonextensive electrons and positrons) has been theoretically investigated. The nonlinear Schr\"{o}dinger (NLS) equation is derived by employing the reductive perturbation method. It is found that the NLS equation leads to the modulational instability (MI) of HIAWs, and to the formation of HIA rogue waves (HIARWs), which are due to the effects of nonlinearity and dispersion in the propagation of HIAWs. The conditions for MI of HIAWs, and the basic properties of the generated HIARWs are identified. It is observed that the striking features (viz. instability criteria, growth rate of MI, amplitude and width of HIARWs, etc.) of the HIAWs are significantly modified by effects of nonextensivity of electrons and positrons, ratio of light positive ion mass to heavy negative ion mass, ratio of electron number density to light positive ion number density, and ratio of electron temperature to positron temperature, etc. The relevancy of our present investigation to the observations in the space (viz. cometary comae and earth's ionosphere) and laboratory (laser plasma interaction experimental devices) plasmas is pointed out.
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An investigation has been made on heavy ion-acoustic (HIA) nonplanar shocks and solitons in an unmagnetized, collisionless, strongly coupled plasma whose constituents are strongly correlated adiabatic inertial heavy ions, weakly correlated nonextensive distributed electrons and Maxwellian light ions. By using appropriate nonlinear equations for our strongly coupled plasma system and the well-known reductive perturbation technique, a modified Burgers (mB) equation and a modified Korteweg-de Vries (mK-dV) equation have been derived. They are also numerically solved in order to investigate the basic features (viz. polarity, amplitude, width, etc.) of cylindrical and spherical shock/solitary waves in such a strongly coupled plasma system. The roles of heavy ion dynamics, nonextensivity of electrons, and other plasma parameters arised in this investigation have significantly modified the basic features of the cylindrical and spherical HIA solitary and shock waves. The findings of our results obtained from this theoretical investigation may be useful in understanding the nonlinear phenomena associated with the cylindrical and spherical HIA waves both in space and laboratory plasmas.
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Modulation instability of ion-acoustic waves (IAWs) is investigated in a collisionless unmagnetized one dimensional plasma, containing positive ions and electrons following the mixed nonextensive nonthermal distribution [Tribeche et al., Phys. Rev. E 85, 037401 (2012)]. Using the reductive perturbation technique, a nonlinear Schrödinger equation which governs the modulation instability of the IAWs is obtained. Valid range of plasma parameters has been fixed and their effects on the modulational instability discussed in detail. We find that the plasma supports both bright and dark solutions. The valid domain for the wave number k where instabilities set in varies with both nonextensive parameter q as well as non thermal parameter α. Moreover, the analysis is extended for the rational solutions of IAWs in the instability regime. Present study is useful for the understanding of IAWs in the region where such mixed distribution may exist.
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We study the amplitude modulation of ion-acoustic wave (IAW) packets in an unmagnetized electron-ion plasma with two-temperature (cool and hot) electrons in the context of the Tsallis' nonextensive statistics. Using the multiple-scale technique, a nonlinear Schr{\"o}dinger (NLS) equation is derived which governs the dynamics of modulated wave packets. It is shown that in nonextensive plasmas, the IAW envelope is always stable for long-wavelength modes $(k\rightarrow0)$ and unstable for short-wavelengths with $k \gtrsim1$. However, the envelope can be unstable at an intermediate scale of perturbations with $0<k<1$. Thus, the modulated IAW packets can propagate in the form of bright envelope solitons or rogons (at small- and medium scale perturbations) as well as dark envelope solitons (at large scale). The stable and unstable regions are obtained for different values of temperature and density ratios as well as the nonextensive parameters $q_c$ and $q_h$ for cool and hot electrons. It is found that the more (less) the population of superthermal cool (hot) electrons, the smaller is the growth rate of instability with cutoffs at smaller wave numbers of modulation.
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A nonlinear propagation of cylindrical and spherical modified ion-acoustic (mIA) waves in an unmagnetized, collisionless, relativistic, degenerate multi-species plasma has been investigated theoretically. This plasma system is assumed to contain non-relativistic degenerate light ions, both non-relativistic and ultra-relativistic degenerate electron and positron fluids, and arbitrarily charged static heavy ions. The restoring force is provided by the degenerate pressures of the electrons and positrons, whereas the inertia is provided by the mass of ions. The arbitrarily charged static heavy ions participate only in maintaining the quasi-neutrality condition at equilibrium. The modified Burgers (mB) equation is derived by using reductive perturbation technique and numerically analyzed to identify the basic features of mIA shock structures. The basic characteristics of mIA shock waves are found to be significantly modified by the effects of degenerate pressures of electron, positron, and ion fluids, their number densities, and various charge state of heavy ions. The implications of our results to dense plasmas in astrophysical compact objects (e.g., non-rotating white dwarfs, neutron stars, etc.) are briefly discussed.
Positron acoustic shock waves (PASHWs) in unmagnetized electron-positron-ion (e-p-i) plasmas consisting of mobile cold positrons, immobile positive ions, q-nonextensive distributed electrons, and hot positrons are studied. The cold positron kinematic viscosity is considered and the reductive perturbation technique is used to derive the Burgers equation. Applying traveling wave transformation, the Burgers equation is transformed to a one dimensional dynamical system. All possible vector fields corresponding to the dynamical system are presented. We have analyzed the dynamical system with the help of potential energy, which helps to identify the stability and instability of the equilibrium points. It is found that the viscous force acting on cold mobile positron fluid is a source of dissipation and is responsible for the formation of the PASHWs. Furthermore, fully nonlinear arbitrary amplitude positron acoustic waves are also studied applying the theory of planar dynamical systems. It is also observed that the fundamental features of the small amplitude and arbitrary amplitude PASHWs are significantly affected by the effect of the physical parameters qe ; qh ; le ; lh ; r; g, and U. This work can be useful to understand the qualitative changes in the dynamics of nonlinear small amplitude and fully nonlinear arbitrary amplitude PASHWs in solar wind, ionosphere, lower part of magnetosphere, and auroral acceleration regions.
In the present work, employing a one dimensional model of an unmagnetized collisionless plasma consisting of a cold electron fluid, hot electrons obeying κ velocity distribution, and stationary ions, we study the amplitude modulation of an electron-acoustic waves by use of the conventional reductive perturbation method. Employing the field equations of such a plasma, we obtained the nonlinear Schrödinger equation as the evolution equation. Seeking a harmonic wave solution with progressive wave amplitude to the evolution equation, as opposed to the plasma with vortex distribution, the amplitude wave assumes a shock wave type of solution. Finally, the modulational stability of the wave is studied and it is observed that the wave is modulationally stable for all admissible wave numbers.
The problem of nonlinear quantum positron-acoustic waves (QPAW's) is addressed in a dense astrophysical plasma. The latter is composed of four different species. Using the quantum hydrodynamic model and carrying out a weakly nonlinear analysis, Korteweg-de Vries (K-dV) and generalized K-dV equations are derived. The influence of quantum effects on solitary structures as well as double-layers is then examined. Due to quantum effects, the QPA soliton experiences a compression while the double-layers enlarge. Our results may aid to interpret and understand the QPAWs that may occur in dense plasmas.
The ion–acoustic solitons in collisionless plasma consisting of warm adiabatic ions, isothermal positrons, and two temperature distribution of electrons have been studied. Using reductive perturbation method, Korteweg-de Vries (K-dV), the modified K-dV (m-KdV), and Gardner equations are derived for the system. The solitonsolution of the Gardner equation is discussed in detail. It is found that for a given set of parameter values, there exists a critical value of β=Tc/Th, (ratio of cold to hot electrontemperature) below which only rarefactive KdV solitons exist and above it compressive KdV solitons exist. At the critical value of β, both compressive and rarefactive m-KdV solitons co-exist. We have also investigated the soliton in the parametric regime where the KdV equation is not valid to study solitonsolution. In this region, it is found that below the critical concentration the system supports rarefactive Gardner solitons and above it compressive Gardner solitons are found. The effects of temperature ratio of two-electron species, cold electron concentration, positron concentration on the characteristics of solitons are also discussed.