Research ArticleModulation Instability of Ion-Acoustic Waves in Plasma withNonthermal Electrons
Basudev Ghosh1 and Sreyasi Banerjee2
1 Department of Physics Jadavpur University Kolkata 700 032 India2Department of Electronics Vidyasagar College Kolkata 700 006 India
Correspondence should be addressed to Basudev Ghosh bsdvghoshgmailcom
Received 2 May 2014 Accepted 3 July 2014 Published 16 July 2014
Academic Editor Milan S Dimitrijevic
Copyright copy 2014 B Ghosh and S Banerjee This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Modulational instability of ion-acoustic waves has been theoretically investigated in an unmagnetized collisionless plasma withnonthermal electrons Boltzmann positrons and warm positive ions To describe the nonlinear evolution of the wave amplitude anonlinear Schrodinger (NLS) equation has been derived by usingmultiple scale perturbation techniqueThenonthermal parameterpositron concentration and ion temperature are shown to play significant role in the modulational instability of ion-acoustic wavesand the formation of envelope solitons
1 Introduction
Electron-positron-ion (119890-119901-119894) plasmas occur in many astro-physical environments such as active galactic nuclei [1]pulsar magnetospheres [2] polar regions of neutron stars[3] centres of our galaxy [4] the early universe [5 6]and the solar atmosphere [7] For this over the last twodecades there has been a great deal of interest in the studyof nonlinear wave phenomena in 119890-119901-119894 plasmas [8ndash12]Positrons are produced by pair production in high energyprocesses occurring in many astrophysical environmentsPopel et al [9] have reported decrease in soliton ampli-tude in the presence of positrons Jehan et al [13] haveshown that solitons become narrower as the concentration ofpositron increasesThe presence of non-Maxwellian electronis common in space and astrophysical plasmas including themagnetosphere [12] and auroral zones [14] The presence ofsuch non-Maxwellian electrons gives rise to many interestingcharacteristics in the nonlinear propagation of waves includ-ing the ion-acoustic solitons [15 16] The solitary structureswith density depression in the magnetosphere observed bythe Freja satellites [17 18] have been explained by Cairnset al [19] by assuming electron distribution to be nonthermalNonlinear ion-acoustic solitary waves in 119890-119901-119894 plasma have
been considered by some authors [9 20 21] assuming ionsto be cold In practice ions have finite temperature and theionic temperature can significantly affect the characteristicsof nonlinear ion-acoustic structures [10 22 23] Chawla et al[24] have considered ion-acoustic waves in 119890-119901-119894 plasma withwarm adiabatic ions and isothermal electrons Baluku andHellberg [25] have considered ion-acoustic solitary waves in119890-119901-119894 plasma with cold ions and nonthermal electrons Henceit is interesting to study the nonlinear ion-acoustic waves in119890-119901-119894 plasma assuming simultaneous presence of nonthermalelectrons warm negative ions and the positrons RecentlyPakzad [11] has shown that the presence of warm ionsand nonthermal electrons can modify parametric regions ofexistence of ion-acoustic solitary waves A nonlinear theoryof ion-acoustic waves in 119890-119901-119894 plasma has been developedby Dubinov and Sazonkin [26] considering polytropic lawsof compression and rarefraction for all plasma componentsSurvey of the past literatures shows that a large number ofworks on KdV type and large amplitude solitary structureformation in 119890-119901-119894 plasmas have been reported Nonlinearpropagation of waves in a dispersive medium is genericallysubject to amplitude modulation due to carrier wave self-interaction or intrinsic nonlinearity of the medium Modula-tional instability is an important phenomenon in connection
Hindawi Publishing CorporationJournal of AstrophysicsVolume 2014 Article ID 785670 8 pageshttpdxdoiorg1011552014785670
2 Journal of Astrophysics
with stablewave propagationHowever only a fewworks havebeen reported in recent years on the modulational instabilityand formation of envelope soliton in 119890-119901-119894 plasmas [20 2124] It has been shown that the presence of positrons shifts thecritical wave number separating the stability and instabilityregions to higher values and for fixed amplitude width ofenvelope solitons decreases with the increase of positronconcentration Mahmood et al [27] have studied modu-lational instability of ion-acoustic waves in 119890-119901-119894 plasmawith warm ions and isothermal electrons and positronsat the same temperature Chawla et al [24] have studiedthe effects of ion temperature positron concentration andpositron temperature on the modulational instability of ion-acoustic waves in 119890-119901-119894 plasma with isothermal electrons andpositrons at different temperatures Bains et al [28] haveconsidered modulational instability of ion-acoustic wavesin 119890-119901-119894 plasma with dust particles Eslami et al [29] haveconsidered modulational instability of ion-acoustic wavesin 119890-119901-119894 plasma with electrons and positrons following q-nonextensive distribution Gill et al [21] have studied mod-ulational instability of ion-acoustic waves in 119890-119901-119894 plasmawith superthermal electrons and isothermal positrons Zhanget al [30] have investigated modulational instability of ion-acoustic waves in 119890-119901-119894 plasmawith nonthermally distributedelectrons and cold ions Modulational instability and excita-tion of ion-acoustic envelope solitons in 119890-119901-119894 plasma withnonthermal electrons have been investigated by Gill et al [31]including ion temperature The purpose of the present paperis to make a detailed study of modulational instability of ion-acoustic waves in 119890-119901-119894 plasma including simultaneously boththe effects of nonthermality of electrons and ion-temperature
2 Basic Formulation
We consider an unmagnetized collisionless plasma consistingof warm positive ions Boltzmann positrons and nonthermalelectrons The normalized basic equations governing iondynamics for one-dimensional propagation in such plasmain dimensionless form are as follows [28]
In aforementioned equations the parameters 119899119894 V119894are
respectively the concentration and velocity of the positiveions 119899
119890and 119899119901are respectively the concentration of electrons
and positrons 120601 denotes the electrostatic potential otherparameters have their usual meaning Different quantities arenormalized as follows the velocities by ion-acoustic speed119862119904
= radic119896119861119879119890119898119894 the densities by equilibrium electron
density 1198991198900 all the length 119909 by the electron Debye length
120582De = radic119896119861119879119890411989021198991198900 time by 120582De119862119904 ion temperature 119879
119894by
119879119890(120590119894= 119879119894119879119890) and the potential 120601 by 119896
119861119879119890119890 where 119896
119861is
the Boltzmannrsquos constantThe nonthermal electron density isgiven by [19]
119899119890= (1 minus 120573120601 + 120573120601
2) exp (120601) (2)
where 120573 = 4120575(1 + 3120575) measures the deviation from thethermalized state and 120575 determines the presence of nonther-mal electrons inside the plasma The density of Boltzmannpositrons is given by
119899119901= 120594 exp (minus120590
119901120601) (3)
where 120594 = 11989911990101198991198900
is the ratio between the unperturbedpositron and electron number densities and 120590
119901= 119879119890119879119901is
the ratio between electron and positron temperatures Theequilibrium charge neutrality condition in normalized formis given by
120594 + 1198991198940= 1 (4)
in which 1198991198940is the equilibrium ion density normalized by the
equilibrium electron densityUsing (2) and (3) Poissonrsquos equation in (1) is rewritten as
1205972120601
1205971199092= (1 minus 120573 + 120573120601
2) exp (120601) minus 120594 exp (minus120590
119901120601) minus 119899
119894 (5)
3 Derivation of the Evolution Equation
Following the usual procedure wemake the following Fourierexpansions for the field quantities [28 32ndash34]
where 119865 stands for the field quantities 119899119894 V119894 and 120601 1198651015840
0and
119865119904are assumed to vary slowly with space and time that is
they are supposed to be functions of 120585 = 120576(119909 minus 119862119892119905) and
120591 = 1205762120591 with 120576 being a small parameter and 119862
119892the group
velocity 120595 = 119896119909 minus 120596119905 (120596 119896 being two constants satisfyinglinear dispersion relation) Substituting the expansion (6) in(1) and (5) and then equating from both sides the coefficientsof exp(119894120595) exp(2119894120595) and terms independent of 120595 we obtainthree sets of equationswhichwe call respectively I II and IIITo solve these equations we make the following perturbationexpansion for the field quantities 1198651015840
0and 119865119904 which we denote
by119883
119883 = 119883(1)
+ 120576119883(2)
+ 120576119883(3)
+ sdot sdot sdot (7)
Solving the lowest order equations obtained from the set ofequations I after substituting the expansion (7) we get thefollowing solutions for the first harmonic quantities in thelowest order
119899(1)
1198941= (1 minus 120573 + 120594120590
119901+ 1198962) sdot 120572
V(1)1198941
=
120596 sdot (1 minus 120573 + 120594120590119901+ 1198962)
(1 minus 120594)
sdot 120572
(8)
Journal of Astrophysics 3
where
120572 = 120601(1)
1 (9)
The linear dispersion relation is obtained as
1205962= 1198962[
(1 minus 120594)
(1 minus 120573 + 120594120590119901+ 1198962)
+
3120590119894
(1 minus 120594)
] (10)
The wave frequency is found to increase with the increasein the nonthermal parameter 120573 and the ion temperature Onthe other hand increase in positron concentration decreasesthe wave frequency In this connection it is pertinent tomention that Pakzad [35] reported an incorrect result and itwas pointed out and corrected by Baluku andHellberg [25] Ifwe put 120573 = 0 120594 = 0 and 120590
119894= 0 we get the linear dispersion
relation for ion-acoustic waves in 119890-119894 plasma as obtained byKakutani and Sugimoto [36] In the limit 119896 rarr 0 (10) leadsto the normalized ion-acoustic speed (119881
119904) modified by the
presence of positrons ion-temperature and non-Maxwellianelectron distribution
1198812
119904=
(1 minus 120594)
(1 minus 120573 + 120594120590119901)
+
3120590119894
(1 minus 120594)
(11)
It agrees with the results obtained by Baluku and Hellberg[25] for the case of cold ions (120590
119894= 0) Equation (11)
shows that for the case of cold ions increase in positronconcentration decreases the phase speed [15] increase inthe nonthermal parameter (120573) leads to an increase in phasespeed and also increase in ion temperature increases thephase speed
First harmonic quantities in the second order areobtained from the solutions (8) by replacing minus119894120596 by minus119894120596 minus
120576119862119892(120597120597120585) + 120576
2(120597120597120591) and 119894119896 by 119894119896 + 120576(120597120597120585) and then picking
out order 120576 terms These are as follows
120601(2)
1= 0
119899(2)
1198941
= minus1198942119896
120597120572
120597120585
V(2)1198941
= [(
120596
1198962minus
119862119892
119896
)(
1 minus 120573 + 120594120590119901
1 minus 120594
)
minus
2120596119896
1 minus 120594
minus
1198962119862119892
1 minus 120594
]
120597120572
120597120585
(12)
The second harmonic quantities in the lowest order obtainedfrom the set of equations II after substituting the expansion(7) are as follows
120601(1)
2= 1198601sdot 1205722
119899(1)
1198942
= [1198601(1 minus 120573 + 120594120590
119901+ 41198962)
2
+
1205941205902
119901
2
minus 120573] sdot 1205722
V(1)1198942
=
120596
119896 (1 minus 120594)
[
[
1198601(1 minus 120573 + 120594120590
119901+ 41198962)
minus
(1 minus 120573 + 120594120590119901+ 1198962)
2
(1 minus 120594)
+(
1205941205902
119901
2
minus 120573)]
]
sdot 1205722
(13)
where
1198601= [(
21205962
119896 (1 minus 120594)
minus
6120590119894
(1 minus 120594)
)(
1205941205902
119901
2
minus 120573)]
minus [(
31205962
119896(1 minus 120594)2minus
3120590119894
(1 minus 120594)2)(1 minus 120573 + 120594120590
119901+ 1198962)
2
]
times ([
6120590119894119896
(1 minus 120594)
(1 minus 120573 + 120594120590119901+ 41198962)] + 2119896
minus
[21205962(1 minus 120573 + 120594120590
119901+ 41198962)]
119896 (1 minus 120594)
)
minus1
(14)
The zeroth harmonic components generated through nonlin-ear self-interaction of the finite amplitude wave are obtainedfrom the set of equations III after substituting the expansion(7)
120601(1)
0= 1198611sdot 120572120572lowast
119899(1)
1198940
= [1198611(1 + 120594120590
119901) minus 2120573] sdot 120572120572
lowast
V(1)1198940
=[
[
[
1198611
119862119892(1 + 120594120590
119901)
1 minus 120594
minus
2120573119862119892
1 minus 120594
minus
2120596(1 minus 120573 + 120594120590119901+ 1198962)
2
119896(1 minus 120594)2
]
]
]
120572120572lowast
(15)
where
1198611=
6120573120590119894
1 minus 120594
minus
21205731198622
119892
1 minus 120594
minus[
[
(1 minus 120573 + 120594120590119901+ 1198962)
2
(1 minus 120594)2
(3120590119894+ 2119862119892
120596
119896
+
1205962
1198962)]
]
times (1 minus
1198622
119892(1 + 120594120590
119901)
1 minus 120594
+ [
3120590119894
1 minus 120594
(1 + 120594120590119901)])
minus1
(16)
4 Journal of Astrophysics
Now in order to derive the NLS equation we need to con-sider first harmonic quantities in the third order Collectingcoefficients of 1205763 from both sides of the set of equations Iafter substituting perturbation expansion (7) we get a setof equations for first harmonic quantities in the third orderfrom which after proper elimination we obtain the followingdesired NLS equation
119894
120597120572
120597120591
+ 119875 sdot
1205972120572
1205971205852= 119876 sdot 120572120572
lowast (17)
119875 =
119896 (1 minus 120594)
2120596 (1 minus 120573 + 120594120590119901+ 1198962)
times [
2120596119862119892
1 minus 120594
minus
1205962
119896 (1 minus 120594)
minus
21205962
(1 minus 120594)2minus
119896119862119892120596
(1 minus 120594)2minus 119862119892
times [(
120596
1198962minus
119862119892
119896
)(
1 minus 120573 + 120594120590119901
1 minus 120594
) minus
2120596119896
1 minus 120594
minus
1198962119862119892
1 minus 120594
]
minus
3120590119894119896
(1 minus 120594)2+
120596
119896
(
120596
1198962minus
119862119892
119896
)(
1 minus 120573 + 120594120590119901
1 minus 120594
)]
(18)
119876 =
119896 (1 minus 120594)
2120596 (1 minus 120573 + 120594120590119901+ 1198962)
[1198652119896 minus
12059621198653
119896 (1 minus 120594)
+
1205961198651
(1 minus 120594)
]
(19)
where
1198651= [(1 + 120594120590
119901) 1198611minus 2120573]
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
+ (1 minus 120573 + 120594120590119901+ 1198962)
times[
[
119862119892(1 + 120594120590
119901)
1 minus 120594
minus
2120573119862119892
1 minus 120594
minus
2120596(1 minus 120573 + 120594120590119901+ 1198962)
2
119896(1 minus 120594)2
]
]
+
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
times[
[
119862119892(1 + 120594120590
119901)
1 minus 120594
minus
2120573119862119892
1 minus 120594
minus
(1 minus 120573 + 120594120590119901+ 1198962)
2
1 minus 120594
]
]
+
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
times [(1 minus 120573 + 120594120590119901+ 41198962)1198601+ (
1205941205902
119901
2
minus 120573)]
1198652=[
[
119862119892(1 + 120594120590
119901)
1 minus 120594
1198611minus
2120573119862119892
1 minus 120594
minus
2120596(1 minus 120573 + 120594120590119901+ 1198962)
2
119896(1 minus 120594)2
]
]
times
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
1205962(1 minus 120573 + 120594120590
119901+ 1198962)
1198962(1 minus 120594)
2
times[
[
(1 minus 120573 + 120594120590119901+ 41198962) + (
1205941205902
119901
2
minus 120573)
minus
(1 minus 120573 + 120594120590119901+ 1198962)
2
(1 minus 120594)
]
]
+
3120590119894
(1 minus 120594)2(1 minus 120573 + 120594120590
119901+ 1198962)
times [ (1 + 120594120590119901) 1198611minus 2120573 + (1 minus 120573 + 120594120590
119901+ 41198962)1198601
+(
1205941205902
119901
2
minus 120573)]
1198653= 1198611(2120573 + 120594120590
2
119901) + 120573 + 120573119860
1
(20)
4 Modulational Instability andEnvelope Solitons
NLS equation (17) describes the nonlinear evolution of theamplitude of IAWs in 119890-119901-119894 plasma with warm ions non-thermal electrons and Boltzmann positrons NLS equation(17) has been studied extensively in connection with thenonlinear propagation of different wave modes It is wellknown that a uniform wave train may be modulationallystable or unstable depending on the sign of the product of thegroup dispersive and the nonlinearity coefficient that is 119875119876As the coefficients depend on the plasma parameters such asnonthermal parameter 120573 ion temperature 120590
119894 and positron
concentration 120594 the product of 119875119876 can have both positiveand negative values over different parametric regions Thewave is modulationally unstable if 119875119876 lt 0 and the growthrate of instability has a maximum value 119892
119898given by
119892119898= |119876| 120572
2
0 (21)
where 1205720is the constant real amplitude of the carrier wave
For 119875119876 gt 0 the IAW ismodulationally stable As the productcan have both positive and negative signs for different valuesof 120573 120590
119894 and 120594 there are accordingly two types of localized
solitary wave solutions of the NLS equation (17) To obtainthe soliton profile we let
120572 = 120588 exp (119894120579) (22)
Journal of Astrophysics 5
where 120588 and 120579 are two real variables Solving the resultingequations for 120588 and 120579with119875119876 lt 0we get the following brightenvelope soliton solution
120588 =
radic2 |119875119876|
119871
sech(
120585 minus 119880120591
119871
) (23)
where 119880 is the envelope speed and 119871 is the spatial width ofthe pulse It encloses high frequency carrier oscillations andvanishes at infinity On the other hand if 119875119876 gt 0 a stablegray or dark soliton (a potential hole or a localized region ofdeceased amplitude) is obtained
120588 =
radic2119875119876
119871119889
radic1 minus 1198892sech2 (120585 minus 119880120591
119871
) (24)
where the parameter 119889 determines the depth of the modula-tion For 119889 = 1 we get a dark soliton
120588 =
radic2119875119876
119871119889
tanh(
120585 minus 119880120591
119871
) (25)
Thus the sign of the product 119875119876 determines the stabil-ityinstability profile of IAWs as well as the type of solitonstructureThe soliton width is determined by the ratio |119875119876|
We have numerically examined different parametricregionswhere someof the above excitationsmay occurAs thecoefficients119875 and119876 depend on nonthermal parameter120573 ion-to-electron temperature ratio 120590
119894 and positron-to-electron
concentration ratio 120594 these parameters would definitelydetermine the modulational instability and the formationof envelope solitons Numerical plots in Figures 1ndash3 show119875119876 as a function of 119896 for different values of 120573 120590
119894 and 120594
It shows that the IAWs remain modulationally stable for 119896
less than certain critical value 119896119888and for 119896 gt 119896
119888the wave is
modulationally unstableIn Figure 1 the variation of 119875119876 with wave number has
been plotted for different values of nonthermal parameter (120573)keeping positron concentration (120594) and ion temperature (120590
119894)
fixed It shows that as 120573 increases the value of critical wavenumber separating stable and unstable regions decreases It isalso noticed that as 120573 increases the width of the dark solitonsincreases but that of the bright solitons decreases
In Figure 2 119875119876 is plotted as function of 119896 for differentvalues of ion temperature (120590
119894) taking other plasmaparameters
such as positron concentration (120594) and nonthermal parame-ter (120573) as constant It is seen that as 120590
119894increases critical wave
number decreases the width of dark solitons increases butthat of bright solitons decreases
Figure 3 is a 119875119876 versus wave number plot for differentvalues of positron concentration (120594) keeping the values ofnonthermal parameter (120573) and ion temperature (120590
119894) constant
It shows that as the value of 120594 increases the critical wavenumber increases The width of dark solitons decreases andthat of bright solitons increases as 120594 increases
Qualitatively these results agree with those obtained byGill et al [31] but quantitatively there are differences Wefind that the critical wave number is more sensitive to thevariation in 120573 120590
119894 and 120594 than that predicted by Gill et al [31]
PQ
minus2
20
2
0
1510
ab
c
a 120573 = 0
b 120573 = 0055
c 120573 = 011
Wave number
Figure 1 Plot of 119875119876 versus wave number 119896 for different values ofnonthermal parameter (120573) Curves labelled a b and c correspondto 120573 = 0 0055 and 011 respectively 120594 = 022 120590
119901= 001 and
120590119894= 002
32
a 120590i = 00145
b 120590i = 00155
c 120590i = 00165
Wave number
abc
PQ
1
0
minus1
Figure 2 Plot of 119875119876 versus wave number 119896 for different values ofion temperature (120590
119894) Curves labelled a b and c correspond to 120590
119894=
00145 00155 and 00165 respectively 120594 = 02 120590119901
= 0015 and120573 = 0022
In addition we have numerically studied the dependence ofgrowth rate of instability on all the plasma parameters 120573 120590
119894
and 120594The results are shown in Figures 4 5 and 6 It is shownthat the growth rate of instability increases with increase inthe nonthermality of electrons and ion temperature but theincrease of positron concentration reduces instability growthrate
6 Journal of AstrophysicsPQ
05
00
minus05
21
a 120594 = 025
b 120594 = 026
c 120594 = 027
Wave number
a b c
Figure 3 Plot of 119875119876 versus wave number 119896 for different values ofpositron concentration (120594) Curves labelled a b and c correspondto 120594 = 025 026 and 027 respectively 120573 = 0022 120590
119901= 001 and
120590119894= 0052
2
1
0
1
a 120573 = 0
b 120573 = 0055
c 120573 = 011
Wave number
a
b
c
Gro
wth
rate
Figure 4 Plot of growth rate versus wave number 119896 for differentvalues of nonthermal parameter (120573) Curves labelled a b and ccorrespond to 120573 = 0 0055 and 011 respectively 120594 = 002 120590
119901=
001 and 120590119894= 0002
5 Conclusions
In the present work we have investigated modulationalinstability and envelope excitations of IAWs in the 119890-119901-119894 plasma in detail including simultaneously the effects ofnonthermality of electrons and temperatures of ions Ourmain findings are summarized below
15
10
05
00
12 16
a 120590i = 00012
b 120590i = 00024
c 120590i = 00036
Wave number
a
b
c
Gro
wth
rate
Figure 5 Plot of growth rate versus wave number 119896 for differentvalues of ion temperature (120590
119894) Curves labelled a b and c correspond
to 120590119894= 00012 00024 and 00036 respectively 120594 = 0001 120590
119901=
001 and 120573 = 0001
a 120594 = 0
b 120594 = 002
c 120594 = 004
Wave number
a
b
c
Gro
wth
rate
2
1
0
201510
Figure 6 Plot of growth rate versus wave number 119896 for differentvalues of positron concentration (120594) Curves labelled a b and ccorrespond to 120594 = 0 002 and 004 respectively 120573 = 001 120590
119901=
001 and 120590119894= 001
(i) The wave frequency increases with increase innonthermality of electrons and the temperature ofions whereas the increase in positron concentrationdecreases the wave frequency
(ii) There exists a critical wave number 119896119888below which
thewave ismodulationally stable and abovewhich thewave is modulationally unstable
Journal of Astrophysics 7
(iii) The value of the critical wave number and the char-acteristics of brightdark envelope solitons dependsignificantly on the nonthermal parameter (120573) iontemperature (120590
119894) and positron concentration (120594)
Finally we would like to mention that the results pre-sented in this paper may be useful to explain modulationalinstability and envelope soliton excitations of IAWs in someastrophysical and space environments where 119890-119901-119894 plasmaswith nonthermal electrons are present
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the reviewers for varioussuggestions and helpful comments in bringing the paper tothe present form
References
[1] H R Miller and P J Witta Active Galactic Nuclei SpringerBerlin Germany 1978
[2] F C Michel ldquoTheory of pulsar magnetospheresrdquo Reviews ofModern Physics vol 54 no 1 pp 1ndash66 1982
[3] F C Michel Theory of Neutron Star Magnetosphere ChicagoUniversity Press Chicago Ill USA 1991
[4] M I Barns Positron Electron Pairs in Astrophysics AmericanInstitute of Physics New York NY USA 1983
[5] W K Misner S Thorne and J A Wheeler GravitationFreeman San Francisco Calif USA 1973
[6] M J Rees G W Gibbons S W Hawking and S SiklasedsTheEarly Universe Cambridge University Press Cambridge UK1983
[7] E Tandberg-Hanssen and A Gordon Emslie The Physics ofSolar Flares CambridgeUniversity Press Cambridge UK 1988
[8] A Cairns R Bingham R O Dendy C M C Nairn P KShukla and A A Mamun ldquoIon sound solitary waves withdensity depressionsrdquo Journal of Physics IV France vol 5 no C6pp 43ndash48 1995
[9] S I Popel S V Vladimirov and P K Shukla ldquoIon-acousticsolitons in electron-positron-ion plasmasrdquo Physics of Plasmasvol 2 no 3 pp 716ndash719 1995
[10] Y N Nejoh ldquoThe effect of the ion temperature on large ampli-tude ion-acoustic waves in an electron-positron-ion plasmardquoPhysics of Plasmas vol 3 no 4 pp 1447ndash1451 1996
[11] H R Pakzad ldquoIon acoustic solitary waves in plasma withnonthermal electron positron and warm ionrdquo Astrophysics andSpace Science vol 323 no 4 pp 345ndash350 2009
[12] S Ghosh and R Bharuthram ldquoIon acoustic solitons and doublelayers in electron-positron-ion plasmas with dust particulatesrdquoAstrophysics and Space Science vol 314 no 1-3 pp 121ndash127 2008
[13] N Jehan W Masood and A M Mirza ldquoPlanar and nonplanardust acoustic solitary waves in electronpositron-ion- dust plas-masrdquo Physica Scripta vol 80 no 3 Article ID 035506 2009
[14] R A Cairns A A Mamun R Bingham and P K ShuklaldquoIon acoustic solitons in a magnetised plasma with nonthermalelectronsrdquo Physica Scripta vol 63 pp 80ndash86 1996
[15] B Ghosh S Banerjee and S N Paul ldquoEffect of non-thermalelectrons andwarmnegative ions on ion-acoustic solitarywavesinmulti-component drifting plasmardquo Indian Journal of Pure andApplied Physics vol 51 no 7 pp 488ndash493 2013
[16] B Ghosh S N Paul C Das I Paul and S Banerjee ldquoElectro-static double layers in amulticomponent drifting plasma havingnonthermal electronsrdquo Brazilian Journal of Physics vol 43 no1-2 pp 28ndash33 2013
[17] P O Dovner A I Eriksson R Bostrom and B Holback ldquoFrejamultiprobe observations of electrostatic solitary structuresrdquoGeophysical Research Letters vol 21 no 17 pp 1827ndash1830 1994
[18] R Bostrom G Gustafsson B Holback G Holmgren HKoskinen and P Kintner ldquoCharacteristics of solitary waves andweak double layers in the magnetospheric plasmardquo PhysicalReview Letters vol 61 no 1 pp 82ndash85 1988
[19] R A Cairns A A Mamun R Bingham et al ldquoElectro-static solitary structures in non-thermal plasmasrdquo GeophysicalResearch Letters vol 22 no 20 pp 2709ndash2712 1995
[20] M Salahuddin H Saleem and M Saddiq ldquoIon-acoustic enve-lope solitons in electron-positron-ion plasmasrdquo Physical ReviewE vol 66 no 3 Article ID 036407 2002
[21] T S Gill C Bedi and A S Bains ldquoEnvelope excitations of ionacoustic solitary waves in a plasma with superthermal electronsand positronsrdquo Physica Scripta vol 81 no 5 Article ID 0555032010
[22] GMurtaza andM Salahuddin ldquoModulational instability of ionacoustic waves in a magnetised plasmardquo Plasma Physics vol 24no 5 pp 451ndash456 1982
[23] Yashvir T N Bhatnagar and S R Sharma ldquoNonlinear ion-acoustic waves and solitons in warm-ion magnetized plasmardquoPlasma Physics and Controlled Fusion vol 26 no 11 article 004pp 1303ndash1310 1984
[24] J K Chawla M K Mishra and R S Tiwari ldquoModulationalinstability of ion-acoustic waves in electron-positron-ion plas-masrdquoAstrophysics and Space Science vol 347 pp 283ndash292 2013
[25] T K Baluku andM A Hellberg ldquoIon acoustic solitary waves inan electron-positron-ion plasma with non-thermal electronsrdquoPlasma Physics and Controlled Fusion vol 53 no 9 Article ID095007 2011
[26] A E Dubinov and M A Sazonkin ldquoNonlinear theory of ion-acoustic waves in an electron-positron-ion plasmardquo PlasmaPhysics Reports vol 35 no 1 pp 14ndash24 2009
[27] S Mahmood S Siddiqui and N Jehan ldquoModulational instabil-ity of ion acousticwavewithwarm ions in electron-positron-ionplasmasrdquo Physics of Plasmas vol 18 no 5 Article ID 0523092011
[28] A S BainsN S Saini andT SGill ldquoModulational instability ofion-acoustic soliton in electron-positron-ion plasma with dustparticulatesrdquo Astrophysics and Space Science vol 343 no 1 pp293ndash299 2013
[29] P Eslami M Mottaghizadeh and H R Pakzad ldquoModulationalinstability of ion acoustic waves in e-p-i plasmas with electronsand positrons following a q-nonextensive distributionrdquo Physicsof Plasmas vol 18 no 10 Article ID 102313 2011
[30] J Zhang Y Wang and L Wu ldquoModulation instability of ionacoustic waves solitons and their interactions in nonthermalelectron-positron-ion plasmasrdquo Physics of Plasmas vol 16 no6 Article ID 062102 2009
[31] T S Gill A S Bains N S Saini and C Bedi ldquoIon-acousticenvelope excitations in electron-positron-ion plasma with non-thermal electronsrdquo Physics Letters A vol 374 no 31-32 pp3210ndash3215 2010
8 Journal of Astrophysics
[32] B Ghosh S N Paul C Das and I Paul ldquoModulationalinstability of high frequency surface waves on warm plasmahalf-spacerdquo Canadian Journal of Physics vol 90 no 3 pp 291ndash297 2012
[33] B Ghosh and K P Das ldquoModulational instability of electronplasma waves in a cylindrical wave guiderdquo Plasma Physics andControlled Fusion vol 27 no 9 pp 969ndash982 1985
[34] B Ghosh S Chandra and S N Paul ldquoAmplitudemodulation ofelectron plasmawaves in a quantumplasmardquoPhysics of Plasmasvol 18 no 1 Article ID 012106 2011
[35] H R Pakzad ldquoIon acoustic solitary waves in plasma withnonthermal electron and positronrdquo Physics Letters A GeneralAtomic and Solid State Physics vol 373 no 8-9 pp 847ndash8502009
[36] T Kakutani and N Sugimoto ldquoKrylov-Bogoliubov-Mitr-opolsky method for nonlinear wave modulationrdquo The Physicsof Fluids vol 17 pp 1617ndash1625 1974
with stablewave propagationHowever only a fewworks havebeen reported in recent years on the modulational instabilityand formation of envelope soliton in 119890-119901-119894 plasmas [20 2124] It has been shown that the presence of positrons shifts thecritical wave number separating the stability and instabilityregions to higher values and for fixed amplitude width ofenvelope solitons decreases with the increase of positronconcentration Mahmood et al [27] have studied modu-lational instability of ion-acoustic waves in 119890-119901-119894 plasmawith warm ions and isothermal electrons and positronsat the same temperature Chawla et al [24] have studiedthe effects of ion temperature positron concentration andpositron temperature on the modulational instability of ion-acoustic waves in 119890-119901-119894 plasma with isothermal electrons andpositrons at different temperatures Bains et al [28] haveconsidered modulational instability of ion-acoustic wavesin 119890-119901-119894 plasma with dust particles Eslami et al [29] haveconsidered modulational instability of ion-acoustic wavesin 119890-119901-119894 plasma with electrons and positrons following q-nonextensive distribution Gill et al [21] have studied mod-ulational instability of ion-acoustic waves in 119890-119901-119894 plasmawith superthermal electrons and isothermal positrons Zhanget al [30] have investigated modulational instability of ion-acoustic waves in 119890-119901-119894 plasmawith nonthermally distributedelectrons and cold ions Modulational instability and excita-tion of ion-acoustic envelope solitons in 119890-119901-119894 plasma withnonthermal electrons have been investigated by Gill et al [31]including ion temperature The purpose of the present paperis to make a detailed study of modulational instability of ion-acoustic waves in 119890-119901-119894 plasma including simultaneously boththe effects of nonthermality of electrons and ion-temperature
2 Basic Formulation
We consider an unmagnetized collisionless plasma consistingof warm positive ions Boltzmann positrons and nonthermalelectrons The normalized basic equations governing iondynamics for one-dimensional propagation in such plasmain dimensionless form are as follows [28]
In aforementioned equations the parameters 119899119894 V119894are
respectively the concentration and velocity of the positiveions 119899
119890and 119899119901are respectively the concentration of electrons
and positrons 120601 denotes the electrostatic potential otherparameters have their usual meaning Different quantities arenormalized as follows the velocities by ion-acoustic speed119862119904
= radic119896119861119879119890119898119894 the densities by equilibrium electron
density 1198991198900 all the length 119909 by the electron Debye length
120582De = radic119896119861119879119890411989021198991198900 time by 120582De119862119904 ion temperature 119879
119894by
119879119890(120590119894= 119879119894119879119890) and the potential 120601 by 119896
119861119879119890119890 where 119896
119861is
the Boltzmannrsquos constantThe nonthermal electron density isgiven by [19]
119899119890= (1 minus 120573120601 + 120573120601
2) exp (120601) (2)
where 120573 = 4120575(1 + 3120575) measures the deviation from thethermalized state and 120575 determines the presence of nonther-mal electrons inside the plasma The density of Boltzmannpositrons is given by
119899119901= 120594 exp (minus120590
119901120601) (3)
where 120594 = 11989911990101198991198900
is the ratio between the unperturbedpositron and electron number densities and 120590
119901= 119879119890119879119901is
the ratio between electron and positron temperatures Theequilibrium charge neutrality condition in normalized formis given by
120594 + 1198991198940= 1 (4)
in which 1198991198940is the equilibrium ion density normalized by the
equilibrium electron densityUsing (2) and (3) Poissonrsquos equation in (1) is rewritten as
1205972120601
1205971199092= (1 minus 120573 + 120573120601
2) exp (120601) minus 120594 exp (minus120590
119901120601) minus 119899
119894 (5)
3 Derivation of the Evolution Equation
Following the usual procedure wemake the following Fourierexpansions for the field quantities [28 32ndash34]
where 119865 stands for the field quantities 119899119894 V119894 and 120601 1198651015840
0and
119865119904are assumed to vary slowly with space and time that is
they are supposed to be functions of 120585 = 120576(119909 minus 119862119892119905) and
120591 = 1205762120591 with 120576 being a small parameter and 119862
119892the group
velocity 120595 = 119896119909 minus 120596119905 (120596 119896 being two constants satisfyinglinear dispersion relation) Substituting the expansion (6) in(1) and (5) and then equating from both sides the coefficientsof exp(119894120595) exp(2119894120595) and terms independent of 120595 we obtainthree sets of equationswhichwe call respectively I II and IIITo solve these equations we make the following perturbationexpansion for the field quantities 1198651015840
0and 119865119904 which we denote
by119883
119883 = 119883(1)
+ 120576119883(2)
+ 120576119883(3)
+ sdot sdot sdot (7)
Solving the lowest order equations obtained from the set ofequations I after substituting the expansion (7) we get thefollowing solutions for the first harmonic quantities in thelowest order
119899(1)
1198941= (1 minus 120573 + 120594120590
119901+ 1198962) sdot 120572
V(1)1198941
=
120596 sdot (1 minus 120573 + 120594120590119901+ 1198962)
(1 minus 120594)
sdot 120572
(8)
Journal of Astrophysics 3
where
120572 = 120601(1)
1 (9)
The linear dispersion relation is obtained as
1205962= 1198962[
(1 minus 120594)
(1 minus 120573 + 120594120590119901+ 1198962)
+
3120590119894
(1 minus 120594)
] (10)
The wave frequency is found to increase with the increasein the nonthermal parameter 120573 and the ion temperature Onthe other hand increase in positron concentration decreasesthe wave frequency In this connection it is pertinent tomention that Pakzad [35] reported an incorrect result and itwas pointed out and corrected by Baluku andHellberg [25] Ifwe put 120573 = 0 120594 = 0 and 120590
119894= 0 we get the linear dispersion
relation for ion-acoustic waves in 119890-119894 plasma as obtained byKakutani and Sugimoto [36] In the limit 119896 rarr 0 (10) leadsto the normalized ion-acoustic speed (119881
119904) modified by the
presence of positrons ion-temperature and non-Maxwellianelectron distribution
1198812
119904=
(1 minus 120594)
(1 minus 120573 + 120594120590119901)
+
3120590119894
(1 minus 120594)
(11)
It agrees with the results obtained by Baluku and Hellberg[25] for the case of cold ions (120590
119894= 0) Equation (11)
shows that for the case of cold ions increase in positronconcentration decreases the phase speed [15] increase inthe nonthermal parameter (120573) leads to an increase in phasespeed and also increase in ion temperature increases thephase speed
First harmonic quantities in the second order areobtained from the solutions (8) by replacing minus119894120596 by minus119894120596 minus
120576119862119892(120597120597120585) + 120576
2(120597120597120591) and 119894119896 by 119894119896 + 120576(120597120597120585) and then picking
out order 120576 terms These are as follows
120601(2)
1= 0
119899(2)
1198941
= minus1198942119896
120597120572
120597120585
V(2)1198941
= [(
120596
1198962minus
119862119892
119896
)(
1 minus 120573 + 120594120590119901
1 minus 120594
)
minus
2120596119896
1 minus 120594
minus
1198962119862119892
1 minus 120594
]
120597120572
120597120585
(12)
The second harmonic quantities in the lowest order obtainedfrom the set of equations II after substituting the expansion(7) are as follows
120601(1)
2= 1198601sdot 1205722
119899(1)
1198942
= [1198601(1 minus 120573 + 120594120590
119901+ 41198962)
2
+
1205941205902
119901
2
minus 120573] sdot 1205722
V(1)1198942
=
120596
119896 (1 minus 120594)
[
[
1198601(1 minus 120573 + 120594120590
119901+ 41198962)
minus
(1 minus 120573 + 120594120590119901+ 1198962)
2
(1 minus 120594)
+(
1205941205902
119901
2
minus 120573)]
]
sdot 1205722
(13)
where
1198601= [(
21205962
119896 (1 minus 120594)
minus
6120590119894
(1 minus 120594)
)(
1205941205902
119901
2
minus 120573)]
minus [(
31205962
119896(1 minus 120594)2minus
3120590119894
(1 minus 120594)2)(1 minus 120573 + 120594120590
119901+ 1198962)
2
]
times ([
6120590119894119896
(1 minus 120594)
(1 minus 120573 + 120594120590119901+ 41198962)] + 2119896
minus
[21205962(1 minus 120573 + 120594120590
119901+ 41198962)]
119896 (1 minus 120594)
)
minus1
(14)
The zeroth harmonic components generated through nonlin-ear self-interaction of the finite amplitude wave are obtainedfrom the set of equations III after substituting the expansion(7)
120601(1)
0= 1198611sdot 120572120572lowast
119899(1)
1198940
= [1198611(1 + 120594120590
119901) minus 2120573] sdot 120572120572
lowast
V(1)1198940
=[
[
[
1198611
119862119892(1 + 120594120590
119901)
1 minus 120594
minus
2120573119862119892
1 minus 120594
minus
2120596(1 minus 120573 + 120594120590119901+ 1198962)
2
119896(1 minus 120594)2
]
]
]
120572120572lowast
(15)
where
1198611=
6120573120590119894
1 minus 120594
minus
21205731198622
119892
1 minus 120594
minus[
[
(1 minus 120573 + 120594120590119901+ 1198962)
2
(1 minus 120594)2
(3120590119894+ 2119862119892
120596
119896
+
1205962
1198962)]
]
times (1 minus
1198622
119892(1 + 120594120590
119901)
1 minus 120594
+ [
3120590119894
1 minus 120594
(1 + 120594120590119901)])
minus1
(16)
4 Journal of Astrophysics
Now in order to derive the NLS equation we need to con-sider first harmonic quantities in the third order Collectingcoefficients of 1205763 from both sides of the set of equations Iafter substituting perturbation expansion (7) we get a setof equations for first harmonic quantities in the third orderfrom which after proper elimination we obtain the followingdesired NLS equation
119894
120597120572
120597120591
+ 119875 sdot
1205972120572
1205971205852= 119876 sdot 120572120572
lowast (17)
119875 =
119896 (1 minus 120594)
2120596 (1 minus 120573 + 120594120590119901+ 1198962)
times [
2120596119862119892
1 minus 120594
minus
1205962
119896 (1 minus 120594)
minus
21205962
(1 minus 120594)2minus
119896119862119892120596
(1 minus 120594)2minus 119862119892
times [(
120596
1198962minus
119862119892
119896
)(
1 minus 120573 + 120594120590119901
1 minus 120594
) minus
2120596119896
1 minus 120594
minus
1198962119862119892
1 minus 120594
]
minus
3120590119894119896
(1 minus 120594)2+
120596
119896
(
120596
1198962minus
119862119892
119896
)(
1 minus 120573 + 120594120590119901
1 minus 120594
)]
(18)
119876 =
119896 (1 minus 120594)
2120596 (1 minus 120573 + 120594120590119901+ 1198962)
[1198652119896 minus
12059621198653
119896 (1 minus 120594)
+
1205961198651
(1 minus 120594)
]
(19)
where
1198651= [(1 + 120594120590
119901) 1198611minus 2120573]
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
+ (1 minus 120573 + 120594120590119901+ 1198962)
times[
[
119862119892(1 + 120594120590
119901)
1 minus 120594
minus
2120573119862119892
1 minus 120594
minus
2120596(1 minus 120573 + 120594120590119901+ 1198962)
2
119896(1 minus 120594)2
]
]
+
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
times[
[
119862119892(1 + 120594120590
119901)
1 minus 120594
minus
2120573119862119892
1 minus 120594
minus
(1 minus 120573 + 120594120590119901+ 1198962)
2
1 minus 120594
]
]
+
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
times [(1 minus 120573 + 120594120590119901+ 41198962)1198601+ (
1205941205902
119901
2
minus 120573)]
1198652=[
[
119862119892(1 + 120594120590
119901)
1 minus 120594
1198611minus
2120573119862119892
1 minus 120594
minus
2120596(1 minus 120573 + 120594120590119901+ 1198962)
2
119896(1 minus 120594)2
]
]
times
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
1205962(1 minus 120573 + 120594120590
119901+ 1198962)
1198962(1 minus 120594)
2
times[
[
(1 minus 120573 + 120594120590119901+ 41198962) + (
1205941205902
119901
2
minus 120573)
minus
(1 minus 120573 + 120594120590119901+ 1198962)
2
(1 minus 120594)
]
]
+
3120590119894
(1 minus 120594)2(1 minus 120573 + 120594120590
119901+ 1198962)
times [ (1 + 120594120590119901) 1198611minus 2120573 + (1 minus 120573 + 120594120590
119901+ 41198962)1198601
+(
1205941205902
119901
2
minus 120573)]
1198653= 1198611(2120573 + 120594120590
2
119901) + 120573 + 120573119860
1
(20)
4 Modulational Instability andEnvelope Solitons
NLS equation (17) describes the nonlinear evolution of theamplitude of IAWs in 119890-119901-119894 plasma with warm ions non-thermal electrons and Boltzmann positrons NLS equation(17) has been studied extensively in connection with thenonlinear propagation of different wave modes It is wellknown that a uniform wave train may be modulationallystable or unstable depending on the sign of the product of thegroup dispersive and the nonlinearity coefficient that is 119875119876As the coefficients depend on the plasma parameters such asnonthermal parameter 120573 ion temperature 120590
119894 and positron
concentration 120594 the product of 119875119876 can have both positiveand negative values over different parametric regions Thewave is modulationally unstable if 119875119876 lt 0 and the growthrate of instability has a maximum value 119892
119898given by
119892119898= |119876| 120572
2
0 (21)
where 1205720is the constant real amplitude of the carrier wave
For 119875119876 gt 0 the IAW ismodulationally stable As the productcan have both positive and negative signs for different valuesof 120573 120590
119894 and 120594 there are accordingly two types of localized
solitary wave solutions of the NLS equation (17) To obtainthe soliton profile we let
120572 = 120588 exp (119894120579) (22)
Journal of Astrophysics 5
where 120588 and 120579 are two real variables Solving the resultingequations for 120588 and 120579with119875119876 lt 0we get the following brightenvelope soliton solution
120588 =
radic2 |119875119876|
119871
sech(
120585 minus 119880120591
119871
) (23)
where 119880 is the envelope speed and 119871 is the spatial width ofthe pulse It encloses high frequency carrier oscillations andvanishes at infinity On the other hand if 119875119876 gt 0 a stablegray or dark soliton (a potential hole or a localized region ofdeceased amplitude) is obtained
120588 =
radic2119875119876
119871119889
radic1 minus 1198892sech2 (120585 minus 119880120591
119871
) (24)
where the parameter 119889 determines the depth of the modula-tion For 119889 = 1 we get a dark soliton
120588 =
radic2119875119876
119871119889
tanh(
120585 minus 119880120591
119871
) (25)
Thus the sign of the product 119875119876 determines the stabil-ityinstability profile of IAWs as well as the type of solitonstructureThe soliton width is determined by the ratio |119875119876|
We have numerically examined different parametricregionswhere someof the above excitationsmay occurAs thecoefficients119875 and119876 depend on nonthermal parameter120573 ion-to-electron temperature ratio 120590
119894 and positron-to-electron
concentration ratio 120594 these parameters would definitelydetermine the modulational instability and the formationof envelope solitons Numerical plots in Figures 1ndash3 show119875119876 as a function of 119896 for different values of 120573 120590
119894 and 120594
It shows that the IAWs remain modulationally stable for 119896
less than certain critical value 119896119888and for 119896 gt 119896
119888the wave is
modulationally unstableIn Figure 1 the variation of 119875119876 with wave number has
been plotted for different values of nonthermal parameter (120573)keeping positron concentration (120594) and ion temperature (120590
119894)
fixed It shows that as 120573 increases the value of critical wavenumber separating stable and unstable regions decreases It isalso noticed that as 120573 increases the width of the dark solitonsincreases but that of the bright solitons decreases
In Figure 2 119875119876 is plotted as function of 119896 for differentvalues of ion temperature (120590
119894) taking other plasmaparameters
such as positron concentration (120594) and nonthermal parame-ter (120573) as constant It is seen that as 120590
119894increases critical wave
number decreases the width of dark solitons increases butthat of bright solitons decreases
Figure 3 is a 119875119876 versus wave number plot for differentvalues of positron concentration (120594) keeping the values ofnonthermal parameter (120573) and ion temperature (120590
119894) constant
It shows that as the value of 120594 increases the critical wavenumber increases The width of dark solitons decreases andthat of bright solitons increases as 120594 increases
Qualitatively these results agree with those obtained byGill et al [31] but quantitatively there are differences Wefind that the critical wave number is more sensitive to thevariation in 120573 120590
119894 and 120594 than that predicted by Gill et al [31]
PQ
minus2
20
2
0
1510
ab
c
a 120573 = 0
b 120573 = 0055
c 120573 = 011
Wave number
Figure 1 Plot of 119875119876 versus wave number 119896 for different values ofnonthermal parameter (120573) Curves labelled a b and c correspondto 120573 = 0 0055 and 011 respectively 120594 = 022 120590
119901= 001 and
120590119894= 002
32
a 120590i = 00145
b 120590i = 00155
c 120590i = 00165
Wave number
abc
PQ
1
0
minus1
Figure 2 Plot of 119875119876 versus wave number 119896 for different values ofion temperature (120590
119894) Curves labelled a b and c correspond to 120590
119894=
00145 00155 and 00165 respectively 120594 = 02 120590119901
= 0015 and120573 = 0022
In addition we have numerically studied the dependence ofgrowth rate of instability on all the plasma parameters 120573 120590
119894
and 120594The results are shown in Figures 4 5 and 6 It is shownthat the growth rate of instability increases with increase inthe nonthermality of electrons and ion temperature but theincrease of positron concentration reduces instability growthrate
6 Journal of AstrophysicsPQ
05
00
minus05
21
a 120594 = 025
b 120594 = 026
c 120594 = 027
Wave number
a b c
Figure 3 Plot of 119875119876 versus wave number 119896 for different values ofpositron concentration (120594) Curves labelled a b and c correspondto 120594 = 025 026 and 027 respectively 120573 = 0022 120590
119901= 001 and
120590119894= 0052
2
1
0
1
a 120573 = 0
b 120573 = 0055
c 120573 = 011
Wave number
a
b
c
Gro
wth
rate
Figure 4 Plot of growth rate versus wave number 119896 for differentvalues of nonthermal parameter (120573) Curves labelled a b and ccorrespond to 120573 = 0 0055 and 011 respectively 120594 = 002 120590
119901=
001 and 120590119894= 0002
5 Conclusions
In the present work we have investigated modulationalinstability and envelope excitations of IAWs in the 119890-119901-119894 plasma in detail including simultaneously the effects ofnonthermality of electrons and temperatures of ions Ourmain findings are summarized below
15
10
05
00
12 16
a 120590i = 00012
b 120590i = 00024
c 120590i = 00036
Wave number
a
b
c
Gro
wth
rate
Figure 5 Plot of growth rate versus wave number 119896 for differentvalues of ion temperature (120590
119894) Curves labelled a b and c correspond
to 120590119894= 00012 00024 and 00036 respectively 120594 = 0001 120590
119901=
001 and 120573 = 0001
a 120594 = 0
b 120594 = 002
c 120594 = 004
Wave number
a
b
c
Gro
wth
rate
2
1
0
201510
Figure 6 Plot of growth rate versus wave number 119896 for differentvalues of positron concentration (120594) Curves labelled a b and ccorrespond to 120594 = 0 002 and 004 respectively 120573 = 001 120590
119901=
001 and 120590119894= 001
(i) The wave frequency increases with increase innonthermality of electrons and the temperature ofions whereas the increase in positron concentrationdecreases the wave frequency
(ii) There exists a critical wave number 119896119888below which
thewave ismodulationally stable and abovewhich thewave is modulationally unstable
Journal of Astrophysics 7
(iii) The value of the critical wave number and the char-acteristics of brightdark envelope solitons dependsignificantly on the nonthermal parameter (120573) iontemperature (120590
119894) and positron concentration (120594)
Finally we would like to mention that the results pre-sented in this paper may be useful to explain modulationalinstability and envelope soliton excitations of IAWs in someastrophysical and space environments where 119890-119901-119894 plasmaswith nonthermal electrons are present
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the reviewers for varioussuggestions and helpful comments in bringing the paper tothe present form
References
[1] H R Miller and P J Witta Active Galactic Nuclei SpringerBerlin Germany 1978
[2] F C Michel ldquoTheory of pulsar magnetospheresrdquo Reviews ofModern Physics vol 54 no 1 pp 1ndash66 1982
[3] F C Michel Theory of Neutron Star Magnetosphere ChicagoUniversity Press Chicago Ill USA 1991
[4] M I Barns Positron Electron Pairs in Astrophysics AmericanInstitute of Physics New York NY USA 1983
[5] W K Misner S Thorne and J A Wheeler GravitationFreeman San Francisco Calif USA 1973
[6] M J Rees G W Gibbons S W Hawking and S SiklasedsTheEarly Universe Cambridge University Press Cambridge UK1983
[7] E Tandberg-Hanssen and A Gordon Emslie The Physics ofSolar Flares CambridgeUniversity Press Cambridge UK 1988
[8] A Cairns R Bingham R O Dendy C M C Nairn P KShukla and A A Mamun ldquoIon sound solitary waves withdensity depressionsrdquo Journal of Physics IV France vol 5 no C6pp 43ndash48 1995
[9] S I Popel S V Vladimirov and P K Shukla ldquoIon-acousticsolitons in electron-positron-ion plasmasrdquo Physics of Plasmasvol 2 no 3 pp 716ndash719 1995
[10] Y N Nejoh ldquoThe effect of the ion temperature on large ampli-tude ion-acoustic waves in an electron-positron-ion plasmardquoPhysics of Plasmas vol 3 no 4 pp 1447ndash1451 1996
[11] H R Pakzad ldquoIon acoustic solitary waves in plasma withnonthermal electron positron and warm ionrdquo Astrophysics andSpace Science vol 323 no 4 pp 345ndash350 2009
[12] S Ghosh and R Bharuthram ldquoIon acoustic solitons and doublelayers in electron-positron-ion plasmas with dust particulatesrdquoAstrophysics and Space Science vol 314 no 1-3 pp 121ndash127 2008
[13] N Jehan W Masood and A M Mirza ldquoPlanar and nonplanardust acoustic solitary waves in electronpositron-ion- dust plas-masrdquo Physica Scripta vol 80 no 3 Article ID 035506 2009
[14] R A Cairns A A Mamun R Bingham and P K ShuklaldquoIon acoustic solitons in a magnetised plasma with nonthermalelectronsrdquo Physica Scripta vol 63 pp 80ndash86 1996
[15] B Ghosh S Banerjee and S N Paul ldquoEffect of non-thermalelectrons andwarmnegative ions on ion-acoustic solitarywavesinmulti-component drifting plasmardquo Indian Journal of Pure andApplied Physics vol 51 no 7 pp 488ndash493 2013
[16] B Ghosh S N Paul C Das I Paul and S Banerjee ldquoElectro-static double layers in amulticomponent drifting plasma havingnonthermal electronsrdquo Brazilian Journal of Physics vol 43 no1-2 pp 28ndash33 2013
[17] P O Dovner A I Eriksson R Bostrom and B Holback ldquoFrejamultiprobe observations of electrostatic solitary structuresrdquoGeophysical Research Letters vol 21 no 17 pp 1827ndash1830 1994
[18] R Bostrom G Gustafsson B Holback G Holmgren HKoskinen and P Kintner ldquoCharacteristics of solitary waves andweak double layers in the magnetospheric plasmardquo PhysicalReview Letters vol 61 no 1 pp 82ndash85 1988
[19] R A Cairns A A Mamun R Bingham et al ldquoElectro-static solitary structures in non-thermal plasmasrdquo GeophysicalResearch Letters vol 22 no 20 pp 2709ndash2712 1995
[20] M Salahuddin H Saleem and M Saddiq ldquoIon-acoustic enve-lope solitons in electron-positron-ion plasmasrdquo Physical ReviewE vol 66 no 3 Article ID 036407 2002
[21] T S Gill C Bedi and A S Bains ldquoEnvelope excitations of ionacoustic solitary waves in a plasma with superthermal electronsand positronsrdquo Physica Scripta vol 81 no 5 Article ID 0555032010
[22] GMurtaza andM Salahuddin ldquoModulational instability of ionacoustic waves in a magnetised plasmardquo Plasma Physics vol 24no 5 pp 451ndash456 1982
[23] Yashvir T N Bhatnagar and S R Sharma ldquoNonlinear ion-acoustic waves and solitons in warm-ion magnetized plasmardquoPlasma Physics and Controlled Fusion vol 26 no 11 article 004pp 1303ndash1310 1984
[24] J K Chawla M K Mishra and R S Tiwari ldquoModulationalinstability of ion-acoustic waves in electron-positron-ion plas-masrdquoAstrophysics and Space Science vol 347 pp 283ndash292 2013
[25] T K Baluku andM A Hellberg ldquoIon acoustic solitary waves inan electron-positron-ion plasma with non-thermal electronsrdquoPlasma Physics and Controlled Fusion vol 53 no 9 Article ID095007 2011
[26] A E Dubinov and M A Sazonkin ldquoNonlinear theory of ion-acoustic waves in an electron-positron-ion plasmardquo PlasmaPhysics Reports vol 35 no 1 pp 14ndash24 2009
[27] S Mahmood S Siddiqui and N Jehan ldquoModulational instabil-ity of ion acousticwavewithwarm ions in electron-positron-ionplasmasrdquo Physics of Plasmas vol 18 no 5 Article ID 0523092011
[28] A S BainsN S Saini andT SGill ldquoModulational instability ofion-acoustic soliton in electron-positron-ion plasma with dustparticulatesrdquo Astrophysics and Space Science vol 343 no 1 pp293ndash299 2013
[29] P Eslami M Mottaghizadeh and H R Pakzad ldquoModulationalinstability of ion acoustic waves in e-p-i plasmas with electronsand positrons following a q-nonextensive distributionrdquo Physicsof Plasmas vol 18 no 10 Article ID 102313 2011
[30] J Zhang Y Wang and L Wu ldquoModulation instability of ionacoustic waves solitons and their interactions in nonthermalelectron-positron-ion plasmasrdquo Physics of Plasmas vol 16 no6 Article ID 062102 2009
[31] T S Gill A S Bains N S Saini and C Bedi ldquoIon-acousticenvelope excitations in electron-positron-ion plasma with non-thermal electronsrdquo Physics Letters A vol 374 no 31-32 pp3210ndash3215 2010
8 Journal of Astrophysics
[32] B Ghosh S N Paul C Das and I Paul ldquoModulationalinstability of high frequency surface waves on warm plasmahalf-spacerdquo Canadian Journal of Physics vol 90 no 3 pp 291ndash297 2012
[33] B Ghosh and K P Das ldquoModulational instability of electronplasma waves in a cylindrical wave guiderdquo Plasma Physics andControlled Fusion vol 27 no 9 pp 969ndash982 1985
[34] B Ghosh S Chandra and S N Paul ldquoAmplitudemodulation ofelectron plasmawaves in a quantumplasmardquoPhysics of Plasmasvol 18 no 1 Article ID 012106 2011
[35] H R Pakzad ldquoIon acoustic solitary waves in plasma withnonthermal electron and positronrdquo Physics Letters A GeneralAtomic and Solid State Physics vol 373 no 8-9 pp 847ndash8502009
[36] T Kakutani and N Sugimoto ldquoKrylov-Bogoliubov-Mitr-opolsky method for nonlinear wave modulationrdquo The Physicsof Fluids vol 17 pp 1617ndash1625 1974
The wave frequency is found to increase with the increasein the nonthermal parameter 120573 and the ion temperature Onthe other hand increase in positron concentration decreasesthe wave frequency In this connection it is pertinent tomention that Pakzad [35] reported an incorrect result and itwas pointed out and corrected by Baluku andHellberg [25] Ifwe put 120573 = 0 120594 = 0 and 120590
119894= 0 we get the linear dispersion
relation for ion-acoustic waves in 119890-119894 plasma as obtained byKakutani and Sugimoto [36] In the limit 119896 rarr 0 (10) leadsto the normalized ion-acoustic speed (119881
119904) modified by the
presence of positrons ion-temperature and non-Maxwellianelectron distribution
1198812
119904=
(1 minus 120594)
(1 minus 120573 + 120594120590119901)
+
3120590119894
(1 minus 120594)
(11)
It agrees with the results obtained by Baluku and Hellberg[25] for the case of cold ions (120590
119894= 0) Equation (11)
shows that for the case of cold ions increase in positronconcentration decreases the phase speed [15] increase inthe nonthermal parameter (120573) leads to an increase in phasespeed and also increase in ion temperature increases thephase speed
First harmonic quantities in the second order areobtained from the solutions (8) by replacing minus119894120596 by minus119894120596 minus
120576119862119892(120597120597120585) + 120576
2(120597120597120591) and 119894119896 by 119894119896 + 120576(120597120597120585) and then picking
out order 120576 terms These are as follows
120601(2)
1= 0
119899(2)
1198941
= minus1198942119896
120597120572
120597120585
V(2)1198941
= [(
120596
1198962minus
119862119892
119896
)(
1 minus 120573 + 120594120590119901
1 minus 120594
)
minus
2120596119896
1 minus 120594
minus
1198962119862119892
1 minus 120594
]
120597120572
120597120585
(12)
The second harmonic quantities in the lowest order obtainedfrom the set of equations II after substituting the expansion(7) are as follows
120601(1)
2= 1198601sdot 1205722
119899(1)
1198942
= [1198601(1 minus 120573 + 120594120590
119901+ 41198962)
2
+
1205941205902
119901
2
minus 120573] sdot 1205722
V(1)1198942
=
120596
119896 (1 minus 120594)
[
[
1198601(1 minus 120573 + 120594120590
119901+ 41198962)
minus
(1 minus 120573 + 120594120590119901+ 1198962)
2
(1 minus 120594)
+(
1205941205902
119901
2
minus 120573)]
]
sdot 1205722
(13)
where
1198601= [(
21205962
119896 (1 minus 120594)
minus
6120590119894
(1 minus 120594)
)(
1205941205902
119901
2
minus 120573)]
minus [(
31205962
119896(1 minus 120594)2minus
3120590119894
(1 minus 120594)2)(1 minus 120573 + 120594120590
119901+ 1198962)
2
]
times ([
6120590119894119896
(1 minus 120594)
(1 minus 120573 + 120594120590119901+ 41198962)] + 2119896
minus
[21205962(1 minus 120573 + 120594120590
119901+ 41198962)]
119896 (1 minus 120594)
)
minus1
(14)
The zeroth harmonic components generated through nonlin-ear self-interaction of the finite amplitude wave are obtainedfrom the set of equations III after substituting the expansion(7)
120601(1)
0= 1198611sdot 120572120572lowast
119899(1)
1198940
= [1198611(1 + 120594120590
119901) minus 2120573] sdot 120572120572
lowast
V(1)1198940
=[
[
[
1198611
119862119892(1 + 120594120590
119901)
1 minus 120594
minus
2120573119862119892
1 minus 120594
minus
2120596(1 minus 120573 + 120594120590119901+ 1198962)
2
119896(1 minus 120594)2
]
]
]
120572120572lowast
(15)
where
1198611=
6120573120590119894
1 minus 120594
minus
21205731198622
119892
1 minus 120594
minus[
[
(1 minus 120573 + 120594120590119901+ 1198962)
2
(1 minus 120594)2
(3120590119894+ 2119862119892
120596
119896
+
1205962
1198962)]
]
times (1 minus
1198622
119892(1 + 120594120590
119901)
1 minus 120594
+ [
3120590119894
1 minus 120594
(1 + 120594120590119901)])
minus1
(16)
4 Journal of Astrophysics
Now in order to derive the NLS equation we need to con-sider first harmonic quantities in the third order Collectingcoefficients of 1205763 from both sides of the set of equations Iafter substituting perturbation expansion (7) we get a setof equations for first harmonic quantities in the third orderfrom which after proper elimination we obtain the followingdesired NLS equation
119894
120597120572
120597120591
+ 119875 sdot
1205972120572
1205971205852= 119876 sdot 120572120572
lowast (17)
119875 =
119896 (1 minus 120594)
2120596 (1 minus 120573 + 120594120590119901+ 1198962)
times [
2120596119862119892
1 minus 120594
minus
1205962
119896 (1 minus 120594)
minus
21205962
(1 minus 120594)2minus
119896119862119892120596
(1 minus 120594)2minus 119862119892
times [(
120596
1198962minus
119862119892
119896
)(
1 minus 120573 + 120594120590119901
1 minus 120594
) minus
2120596119896
1 minus 120594
minus
1198962119862119892
1 minus 120594
]
minus
3120590119894119896
(1 minus 120594)2+
120596
119896
(
120596
1198962minus
119862119892
119896
)(
1 minus 120573 + 120594120590119901
1 minus 120594
)]
(18)
119876 =
119896 (1 minus 120594)
2120596 (1 minus 120573 + 120594120590119901+ 1198962)
[1198652119896 minus
12059621198653
119896 (1 minus 120594)
+
1205961198651
(1 minus 120594)
]
(19)
where
1198651= [(1 + 120594120590
119901) 1198611minus 2120573]
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
+ (1 minus 120573 + 120594120590119901+ 1198962)
times[
[
119862119892(1 + 120594120590
119901)
1 minus 120594
minus
2120573119862119892
1 minus 120594
minus
2120596(1 minus 120573 + 120594120590119901+ 1198962)
2
119896(1 minus 120594)2
]
]
+
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
times[
[
119862119892(1 + 120594120590
119901)
1 minus 120594
minus
2120573119862119892
1 minus 120594
minus
(1 minus 120573 + 120594120590119901+ 1198962)
2
1 minus 120594
]
]
+
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
times [(1 minus 120573 + 120594120590119901+ 41198962)1198601+ (
1205941205902
119901
2
minus 120573)]
1198652=[
[
119862119892(1 + 120594120590
119901)
1 minus 120594
1198611minus
2120573119862119892
1 minus 120594
minus
2120596(1 minus 120573 + 120594120590119901+ 1198962)
2
119896(1 minus 120594)2
]
]
times
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
1205962(1 minus 120573 + 120594120590
119901+ 1198962)
1198962(1 minus 120594)
2
times[
[
(1 minus 120573 + 120594120590119901+ 41198962) + (
1205941205902
119901
2
minus 120573)
minus
(1 minus 120573 + 120594120590119901+ 1198962)
2
(1 minus 120594)
]
]
+
3120590119894
(1 minus 120594)2(1 minus 120573 + 120594120590
119901+ 1198962)
times [ (1 + 120594120590119901) 1198611minus 2120573 + (1 minus 120573 + 120594120590
119901+ 41198962)1198601
+(
1205941205902
119901
2
minus 120573)]
1198653= 1198611(2120573 + 120594120590
2
119901) + 120573 + 120573119860
1
(20)
4 Modulational Instability andEnvelope Solitons
NLS equation (17) describes the nonlinear evolution of theamplitude of IAWs in 119890-119901-119894 plasma with warm ions non-thermal electrons and Boltzmann positrons NLS equation(17) has been studied extensively in connection with thenonlinear propagation of different wave modes It is wellknown that a uniform wave train may be modulationallystable or unstable depending on the sign of the product of thegroup dispersive and the nonlinearity coefficient that is 119875119876As the coefficients depend on the plasma parameters such asnonthermal parameter 120573 ion temperature 120590
119894 and positron
concentration 120594 the product of 119875119876 can have both positiveand negative values over different parametric regions Thewave is modulationally unstable if 119875119876 lt 0 and the growthrate of instability has a maximum value 119892
119898given by
119892119898= |119876| 120572
2
0 (21)
where 1205720is the constant real amplitude of the carrier wave
For 119875119876 gt 0 the IAW ismodulationally stable As the productcan have both positive and negative signs for different valuesof 120573 120590
119894 and 120594 there are accordingly two types of localized
solitary wave solutions of the NLS equation (17) To obtainthe soliton profile we let
120572 = 120588 exp (119894120579) (22)
Journal of Astrophysics 5
where 120588 and 120579 are two real variables Solving the resultingequations for 120588 and 120579with119875119876 lt 0we get the following brightenvelope soliton solution
120588 =
radic2 |119875119876|
119871
sech(
120585 minus 119880120591
119871
) (23)
where 119880 is the envelope speed and 119871 is the spatial width ofthe pulse It encloses high frequency carrier oscillations andvanishes at infinity On the other hand if 119875119876 gt 0 a stablegray or dark soliton (a potential hole or a localized region ofdeceased amplitude) is obtained
120588 =
radic2119875119876
119871119889
radic1 minus 1198892sech2 (120585 minus 119880120591
119871
) (24)
where the parameter 119889 determines the depth of the modula-tion For 119889 = 1 we get a dark soliton
120588 =
radic2119875119876
119871119889
tanh(
120585 minus 119880120591
119871
) (25)
Thus the sign of the product 119875119876 determines the stabil-ityinstability profile of IAWs as well as the type of solitonstructureThe soliton width is determined by the ratio |119875119876|
We have numerically examined different parametricregionswhere someof the above excitationsmay occurAs thecoefficients119875 and119876 depend on nonthermal parameter120573 ion-to-electron temperature ratio 120590
119894 and positron-to-electron
concentration ratio 120594 these parameters would definitelydetermine the modulational instability and the formationof envelope solitons Numerical plots in Figures 1ndash3 show119875119876 as a function of 119896 for different values of 120573 120590
119894 and 120594
It shows that the IAWs remain modulationally stable for 119896
less than certain critical value 119896119888and for 119896 gt 119896
119888the wave is
modulationally unstableIn Figure 1 the variation of 119875119876 with wave number has
been plotted for different values of nonthermal parameter (120573)keeping positron concentration (120594) and ion temperature (120590
119894)
fixed It shows that as 120573 increases the value of critical wavenumber separating stable and unstable regions decreases It isalso noticed that as 120573 increases the width of the dark solitonsincreases but that of the bright solitons decreases
In Figure 2 119875119876 is plotted as function of 119896 for differentvalues of ion temperature (120590
119894) taking other plasmaparameters
such as positron concentration (120594) and nonthermal parame-ter (120573) as constant It is seen that as 120590
119894increases critical wave
number decreases the width of dark solitons increases butthat of bright solitons decreases
Figure 3 is a 119875119876 versus wave number plot for differentvalues of positron concentration (120594) keeping the values ofnonthermal parameter (120573) and ion temperature (120590
119894) constant
It shows that as the value of 120594 increases the critical wavenumber increases The width of dark solitons decreases andthat of bright solitons increases as 120594 increases
Qualitatively these results agree with those obtained byGill et al [31] but quantitatively there are differences Wefind that the critical wave number is more sensitive to thevariation in 120573 120590
119894 and 120594 than that predicted by Gill et al [31]
PQ
minus2
20
2
0
1510
ab
c
a 120573 = 0
b 120573 = 0055
c 120573 = 011
Wave number
Figure 1 Plot of 119875119876 versus wave number 119896 for different values ofnonthermal parameter (120573) Curves labelled a b and c correspondto 120573 = 0 0055 and 011 respectively 120594 = 022 120590
119901= 001 and
120590119894= 002
32
a 120590i = 00145
b 120590i = 00155
c 120590i = 00165
Wave number
abc
PQ
1
0
minus1
Figure 2 Plot of 119875119876 versus wave number 119896 for different values ofion temperature (120590
119894) Curves labelled a b and c correspond to 120590
119894=
00145 00155 and 00165 respectively 120594 = 02 120590119901
= 0015 and120573 = 0022
In addition we have numerically studied the dependence ofgrowth rate of instability on all the plasma parameters 120573 120590
119894
and 120594The results are shown in Figures 4 5 and 6 It is shownthat the growth rate of instability increases with increase inthe nonthermality of electrons and ion temperature but theincrease of positron concentration reduces instability growthrate
6 Journal of AstrophysicsPQ
05
00
minus05
21
a 120594 = 025
b 120594 = 026
c 120594 = 027
Wave number
a b c
Figure 3 Plot of 119875119876 versus wave number 119896 for different values ofpositron concentration (120594) Curves labelled a b and c correspondto 120594 = 025 026 and 027 respectively 120573 = 0022 120590
119901= 001 and
120590119894= 0052
2
1
0
1
a 120573 = 0
b 120573 = 0055
c 120573 = 011
Wave number
a
b
c
Gro
wth
rate
Figure 4 Plot of growth rate versus wave number 119896 for differentvalues of nonthermal parameter (120573) Curves labelled a b and ccorrespond to 120573 = 0 0055 and 011 respectively 120594 = 002 120590
119901=
001 and 120590119894= 0002
5 Conclusions
In the present work we have investigated modulationalinstability and envelope excitations of IAWs in the 119890-119901-119894 plasma in detail including simultaneously the effects ofnonthermality of electrons and temperatures of ions Ourmain findings are summarized below
15
10
05
00
12 16
a 120590i = 00012
b 120590i = 00024
c 120590i = 00036
Wave number
a
b
c
Gro
wth
rate
Figure 5 Plot of growth rate versus wave number 119896 for differentvalues of ion temperature (120590
119894) Curves labelled a b and c correspond
to 120590119894= 00012 00024 and 00036 respectively 120594 = 0001 120590
119901=
001 and 120573 = 0001
a 120594 = 0
b 120594 = 002
c 120594 = 004
Wave number
a
b
c
Gro
wth
rate
2
1
0
201510
Figure 6 Plot of growth rate versus wave number 119896 for differentvalues of positron concentration (120594) Curves labelled a b and ccorrespond to 120594 = 0 002 and 004 respectively 120573 = 001 120590
119901=
001 and 120590119894= 001
(i) The wave frequency increases with increase innonthermality of electrons and the temperature ofions whereas the increase in positron concentrationdecreases the wave frequency
(ii) There exists a critical wave number 119896119888below which
thewave ismodulationally stable and abovewhich thewave is modulationally unstable
Journal of Astrophysics 7
(iii) The value of the critical wave number and the char-acteristics of brightdark envelope solitons dependsignificantly on the nonthermal parameter (120573) iontemperature (120590
119894) and positron concentration (120594)
Finally we would like to mention that the results pre-sented in this paper may be useful to explain modulationalinstability and envelope soliton excitations of IAWs in someastrophysical and space environments where 119890-119901-119894 plasmaswith nonthermal electrons are present
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the reviewers for varioussuggestions and helpful comments in bringing the paper tothe present form
References
[1] H R Miller and P J Witta Active Galactic Nuclei SpringerBerlin Germany 1978
[2] F C Michel ldquoTheory of pulsar magnetospheresrdquo Reviews ofModern Physics vol 54 no 1 pp 1ndash66 1982
[3] F C Michel Theory of Neutron Star Magnetosphere ChicagoUniversity Press Chicago Ill USA 1991
[4] M I Barns Positron Electron Pairs in Astrophysics AmericanInstitute of Physics New York NY USA 1983
[5] W K Misner S Thorne and J A Wheeler GravitationFreeman San Francisco Calif USA 1973
[6] M J Rees G W Gibbons S W Hawking and S SiklasedsTheEarly Universe Cambridge University Press Cambridge UK1983
[7] E Tandberg-Hanssen and A Gordon Emslie The Physics ofSolar Flares CambridgeUniversity Press Cambridge UK 1988
[8] A Cairns R Bingham R O Dendy C M C Nairn P KShukla and A A Mamun ldquoIon sound solitary waves withdensity depressionsrdquo Journal of Physics IV France vol 5 no C6pp 43ndash48 1995
[9] S I Popel S V Vladimirov and P K Shukla ldquoIon-acousticsolitons in electron-positron-ion plasmasrdquo Physics of Plasmasvol 2 no 3 pp 716ndash719 1995
[10] Y N Nejoh ldquoThe effect of the ion temperature on large ampli-tude ion-acoustic waves in an electron-positron-ion plasmardquoPhysics of Plasmas vol 3 no 4 pp 1447ndash1451 1996
[11] H R Pakzad ldquoIon acoustic solitary waves in plasma withnonthermal electron positron and warm ionrdquo Astrophysics andSpace Science vol 323 no 4 pp 345ndash350 2009
[12] S Ghosh and R Bharuthram ldquoIon acoustic solitons and doublelayers in electron-positron-ion plasmas with dust particulatesrdquoAstrophysics and Space Science vol 314 no 1-3 pp 121ndash127 2008
[13] N Jehan W Masood and A M Mirza ldquoPlanar and nonplanardust acoustic solitary waves in electronpositron-ion- dust plas-masrdquo Physica Scripta vol 80 no 3 Article ID 035506 2009
[14] R A Cairns A A Mamun R Bingham and P K ShuklaldquoIon acoustic solitons in a magnetised plasma with nonthermalelectronsrdquo Physica Scripta vol 63 pp 80ndash86 1996
[15] B Ghosh S Banerjee and S N Paul ldquoEffect of non-thermalelectrons andwarmnegative ions on ion-acoustic solitarywavesinmulti-component drifting plasmardquo Indian Journal of Pure andApplied Physics vol 51 no 7 pp 488ndash493 2013
[16] B Ghosh S N Paul C Das I Paul and S Banerjee ldquoElectro-static double layers in amulticomponent drifting plasma havingnonthermal electronsrdquo Brazilian Journal of Physics vol 43 no1-2 pp 28ndash33 2013
[17] P O Dovner A I Eriksson R Bostrom and B Holback ldquoFrejamultiprobe observations of electrostatic solitary structuresrdquoGeophysical Research Letters vol 21 no 17 pp 1827ndash1830 1994
[18] R Bostrom G Gustafsson B Holback G Holmgren HKoskinen and P Kintner ldquoCharacteristics of solitary waves andweak double layers in the magnetospheric plasmardquo PhysicalReview Letters vol 61 no 1 pp 82ndash85 1988
[19] R A Cairns A A Mamun R Bingham et al ldquoElectro-static solitary structures in non-thermal plasmasrdquo GeophysicalResearch Letters vol 22 no 20 pp 2709ndash2712 1995
[20] M Salahuddin H Saleem and M Saddiq ldquoIon-acoustic enve-lope solitons in electron-positron-ion plasmasrdquo Physical ReviewE vol 66 no 3 Article ID 036407 2002
[21] T S Gill C Bedi and A S Bains ldquoEnvelope excitations of ionacoustic solitary waves in a plasma with superthermal electronsand positronsrdquo Physica Scripta vol 81 no 5 Article ID 0555032010
[22] GMurtaza andM Salahuddin ldquoModulational instability of ionacoustic waves in a magnetised plasmardquo Plasma Physics vol 24no 5 pp 451ndash456 1982
[23] Yashvir T N Bhatnagar and S R Sharma ldquoNonlinear ion-acoustic waves and solitons in warm-ion magnetized plasmardquoPlasma Physics and Controlled Fusion vol 26 no 11 article 004pp 1303ndash1310 1984
[24] J K Chawla M K Mishra and R S Tiwari ldquoModulationalinstability of ion-acoustic waves in electron-positron-ion plas-masrdquoAstrophysics and Space Science vol 347 pp 283ndash292 2013
[25] T K Baluku andM A Hellberg ldquoIon acoustic solitary waves inan electron-positron-ion plasma with non-thermal electronsrdquoPlasma Physics and Controlled Fusion vol 53 no 9 Article ID095007 2011
[26] A E Dubinov and M A Sazonkin ldquoNonlinear theory of ion-acoustic waves in an electron-positron-ion plasmardquo PlasmaPhysics Reports vol 35 no 1 pp 14ndash24 2009
[27] S Mahmood S Siddiqui and N Jehan ldquoModulational instabil-ity of ion acousticwavewithwarm ions in electron-positron-ionplasmasrdquo Physics of Plasmas vol 18 no 5 Article ID 0523092011
[28] A S BainsN S Saini andT SGill ldquoModulational instability ofion-acoustic soliton in electron-positron-ion plasma with dustparticulatesrdquo Astrophysics and Space Science vol 343 no 1 pp293ndash299 2013
[29] P Eslami M Mottaghizadeh and H R Pakzad ldquoModulationalinstability of ion acoustic waves in e-p-i plasmas with electronsand positrons following a q-nonextensive distributionrdquo Physicsof Plasmas vol 18 no 10 Article ID 102313 2011
[30] J Zhang Y Wang and L Wu ldquoModulation instability of ionacoustic waves solitons and their interactions in nonthermalelectron-positron-ion plasmasrdquo Physics of Plasmas vol 16 no6 Article ID 062102 2009
[31] T S Gill A S Bains N S Saini and C Bedi ldquoIon-acousticenvelope excitations in electron-positron-ion plasma with non-thermal electronsrdquo Physics Letters A vol 374 no 31-32 pp3210ndash3215 2010
8 Journal of Astrophysics
[32] B Ghosh S N Paul C Das and I Paul ldquoModulationalinstability of high frequency surface waves on warm plasmahalf-spacerdquo Canadian Journal of Physics vol 90 no 3 pp 291ndash297 2012
[33] B Ghosh and K P Das ldquoModulational instability of electronplasma waves in a cylindrical wave guiderdquo Plasma Physics andControlled Fusion vol 27 no 9 pp 969ndash982 1985
[34] B Ghosh S Chandra and S N Paul ldquoAmplitudemodulation ofelectron plasmawaves in a quantumplasmardquoPhysics of Plasmasvol 18 no 1 Article ID 012106 2011
[35] H R Pakzad ldquoIon acoustic solitary waves in plasma withnonthermal electron and positronrdquo Physics Letters A GeneralAtomic and Solid State Physics vol 373 no 8-9 pp 847ndash8502009
[36] T Kakutani and N Sugimoto ldquoKrylov-Bogoliubov-Mitr-opolsky method for nonlinear wave modulationrdquo The Physicsof Fluids vol 17 pp 1617ndash1625 1974
Now in order to derive the NLS equation we need to con-sider first harmonic quantities in the third order Collectingcoefficients of 1205763 from both sides of the set of equations Iafter substituting perturbation expansion (7) we get a setof equations for first harmonic quantities in the third orderfrom which after proper elimination we obtain the followingdesired NLS equation
119894
120597120572
120597120591
+ 119875 sdot
1205972120572
1205971205852= 119876 sdot 120572120572
lowast (17)
119875 =
119896 (1 minus 120594)
2120596 (1 minus 120573 + 120594120590119901+ 1198962)
times [
2120596119862119892
1 minus 120594
minus
1205962
119896 (1 minus 120594)
minus
21205962
(1 minus 120594)2minus
119896119862119892120596
(1 minus 120594)2minus 119862119892
times [(
120596
1198962minus
119862119892
119896
)(
1 minus 120573 + 120594120590119901
1 minus 120594
) minus
2120596119896
1 minus 120594
minus
1198962119862119892
1 minus 120594
]
minus
3120590119894119896
(1 minus 120594)2+
120596
119896
(
120596
1198962minus
119862119892
119896
)(
1 minus 120573 + 120594120590119901
1 minus 120594
)]
(18)
119876 =
119896 (1 minus 120594)
2120596 (1 minus 120573 + 120594120590119901+ 1198962)
[1198652119896 minus
12059621198653
119896 (1 minus 120594)
+
1205961198651
(1 minus 120594)
]
(19)
where
1198651= [(1 + 120594120590
119901) 1198611minus 2120573]
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
+ (1 minus 120573 + 120594120590119901+ 1198962)
times[
[
119862119892(1 + 120594120590
119901)
1 minus 120594
minus
2120573119862119892
1 minus 120594
minus
2120596(1 minus 120573 + 120594120590119901+ 1198962)
2
119896(1 minus 120594)2
]
]
+
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
times[
[
119862119892(1 + 120594120590
119901)
1 minus 120594
minus
2120573119862119892
1 minus 120594
minus
(1 minus 120573 + 120594120590119901+ 1198962)
2
1 minus 120594
]
]
+
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
times [(1 minus 120573 + 120594120590119901+ 41198962)1198601+ (
1205941205902
119901
2
minus 120573)]
1198652=[
[
119862119892(1 + 120594120590
119901)
1 minus 120594
1198611minus
2120573119862119892
1 minus 120594
minus
2120596(1 minus 120573 + 120594120590119901+ 1198962)
2
119896(1 minus 120594)2
]
]
times
120596 (1 minus 120573 + 120594120590119901+ 1198962)
119896 (1 minus 120594)
1205962(1 minus 120573 + 120594120590
119901+ 1198962)
1198962(1 minus 120594)
2
times[
[
(1 minus 120573 + 120594120590119901+ 41198962) + (
1205941205902
119901
2
minus 120573)
minus
(1 minus 120573 + 120594120590119901+ 1198962)
2
(1 minus 120594)
]
]
+
3120590119894
(1 minus 120594)2(1 minus 120573 + 120594120590
119901+ 1198962)
times [ (1 + 120594120590119901) 1198611minus 2120573 + (1 minus 120573 + 120594120590
119901+ 41198962)1198601
+(
1205941205902
119901
2
minus 120573)]
1198653= 1198611(2120573 + 120594120590
2
119901) + 120573 + 120573119860
1
(20)
4 Modulational Instability andEnvelope Solitons
NLS equation (17) describes the nonlinear evolution of theamplitude of IAWs in 119890-119901-119894 plasma with warm ions non-thermal electrons and Boltzmann positrons NLS equation(17) has been studied extensively in connection with thenonlinear propagation of different wave modes It is wellknown that a uniform wave train may be modulationallystable or unstable depending on the sign of the product of thegroup dispersive and the nonlinearity coefficient that is 119875119876As the coefficients depend on the plasma parameters such asnonthermal parameter 120573 ion temperature 120590
119894 and positron
concentration 120594 the product of 119875119876 can have both positiveand negative values over different parametric regions Thewave is modulationally unstable if 119875119876 lt 0 and the growthrate of instability has a maximum value 119892
119898given by
119892119898= |119876| 120572
2
0 (21)
where 1205720is the constant real amplitude of the carrier wave
For 119875119876 gt 0 the IAW ismodulationally stable As the productcan have both positive and negative signs for different valuesof 120573 120590
119894 and 120594 there are accordingly two types of localized
solitary wave solutions of the NLS equation (17) To obtainthe soliton profile we let
120572 = 120588 exp (119894120579) (22)
Journal of Astrophysics 5
where 120588 and 120579 are two real variables Solving the resultingequations for 120588 and 120579with119875119876 lt 0we get the following brightenvelope soliton solution
120588 =
radic2 |119875119876|
119871
sech(
120585 minus 119880120591
119871
) (23)
where 119880 is the envelope speed and 119871 is the spatial width ofthe pulse It encloses high frequency carrier oscillations andvanishes at infinity On the other hand if 119875119876 gt 0 a stablegray or dark soliton (a potential hole or a localized region ofdeceased amplitude) is obtained
120588 =
radic2119875119876
119871119889
radic1 minus 1198892sech2 (120585 minus 119880120591
119871
) (24)
where the parameter 119889 determines the depth of the modula-tion For 119889 = 1 we get a dark soliton
120588 =
radic2119875119876
119871119889
tanh(
120585 minus 119880120591
119871
) (25)
Thus the sign of the product 119875119876 determines the stabil-ityinstability profile of IAWs as well as the type of solitonstructureThe soliton width is determined by the ratio |119875119876|
We have numerically examined different parametricregionswhere someof the above excitationsmay occurAs thecoefficients119875 and119876 depend on nonthermal parameter120573 ion-to-electron temperature ratio 120590
119894 and positron-to-electron
concentration ratio 120594 these parameters would definitelydetermine the modulational instability and the formationof envelope solitons Numerical plots in Figures 1ndash3 show119875119876 as a function of 119896 for different values of 120573 120590
119894 and 120594
It shows that the IAWs remain modulationally stable for 119896
less than certain critical value 119896119888and for 119896 gt 119896
119888the wave is
modulationally unstableIn Figure 1 the variation of 119875119876 with wave number has
been plotted for different values of nonthermal parameter (120573)keeping positron concentration (120594) and ion temperature (120590
119894)
fixed It shows that as 120573 increases the value of critical wavenumber separating stable and unstable regions decreases It isalso noticed that as 120573 increases the width of the dark solitonsincreases but that of the bright solitons decreases
In Figure 2 119875119876 is plotted as function of 119896 for differentvalues of ion temperature (120590
119894) taking other plasmaparameters
such as positron concentration (120594) and nonthermal parame-ter (120573) as constant It is seen that as 120590
119894increases critical wave
number decreases the width of dark solitons increases butthat of bright solitons decreases
Figure 3 is a 119875119876 versus wave number plot for differentvalues of positron concentration (120594) keeping the values ofnonthermal parameter (120573) and ion temperature (120590
119894) constant
It shows that as the value of 120594 increases the critical wavenumber increases The width of dark solitons decreases andthat of bright solitons increases as 120594 increases
Qualitatively these results agree with those obtained byGill et al [31] but quantitatively there are differences Wefind that the critical wave number is more sensitive to thevariation in 120573 120590
119894 and 120594 than that predicted by Gill et al [31]
PQ
minus2
20
2
0
1510
ab
c
a 120573 = 0
b 120573 = 0055
c 120573 = 011
Wave number
Figure 1 Plot of 119875119876 versus wave number 119896 for different values ofnonthermal parameter (120573) Curves labelled a b and c correspondto 120573 = 0 0055 and 011 respectively 120594 = 022 120590
119901= 001 and
120590119894= 002
32
a 120590i = 00145
b 120590i = 00155
c 120590i = 00165
Wave number
abc
PQ
1
0
minus1
Figure 2 Plot of 119875119876 versus wave number 119896 for different values ofion temperature (120590
119894) Curves labelled a b and c correspond to 120590
119894=
00145 00155 and 00165 respectively 120594 = 02 120590119901
= 0015 and120573 = 0022
In addition we have numerically studied the dependence ofgrowth rate of instability on all the plasma parameters 120573 120590
119894
and 120594The results are shown in Figures 4 5 and 6 It is shownthat the growth rate of instability increases with increase inthe nonthermality of electrons and ion temperature but theincrease of positron concentration reduces instability growthrate
6 Journal of AstrophysicsPQ
05
00
minus05
21
a 120594 = 025
b 120594 = 026
c 120594 = 027
Wave number
a b c
Figure 3 Plot of 119875119876 versus wave number 119896 for different values ofpositron concentration (120594) Curves labelled a b and c correspondto 120594 = 025 026 and 027 respectively 120573 = 0022 120590
119901= 001 and
120590119894= 0052
2
1
0
1
a 120573 = 0
b 120573 = 0055
c 120573 = 011
Wave number
a
b
c
Gro
wth
rate
Figure 4 Plot of growth rate versus wave number 119896 for differentvalues of nonthermal parameter (120573) Curves labelled a b and ccorrespond to 120573 = 0 0055 and 011 respectively 120594 = 002 120590
119901=
001 and 120590119894= 0002
5 Conclusions
In the present work we have investigated modulationalinstability and envelope excitations of IAWs in the 119890-119901-119894 plasma in detail including simultaneously the effects ofnonthermality of electrons and temperatures of ions Ourmain findings are summarized below
15
10
05
00
12 16
a 120590i = 00012
b 120590i = 00024
c 120590i = 00036
Wave number
a
b
c
Gro
wth
rate
Figure 5 Plot of growth rate versus wave number 119896 for differentvalues of ion temperature (120590
119894) Curves labelled a b and c correspond
to 120590119894= 00012 00024 and 00036 respectively 120594 = 0001 120590
119901=
001 and 120573 = 0001
a 120594 = 0
b 120594 = 002
c 120594 = 004
Wave number
a
b
c
Gro
wth
rate
2
1
0
201510
Figure 6 Plot of growth rate versus wave number 119896 for differentvalues of positron concentration (120594) Curves labelled a b and ccorrespond to 120594 = 0 002 and 004 respectively 120573 = 001 120590
119901=
001 and 120590119894= 001
(i) The wave frequency increases with increase innonthermality of electrons and the temperature ofions whereas the increase in positron concentrationdecreases the wave frequency
(ii) There exists a critical wave number 119896119888below which
thewave ismodulationally stable and abovewhich thewave is modulationally unstable
Journal of Astrophysics 7
(iii) The value of the critical wave number and the char-acteristics of brightdark envelope solitons dependsignificantly on the nonthermal parameter (120573) iontemperature (120590
119894) and positron concentration (120594)
Finally we would like to mention that the results pre-sented in this paper may be useful to explain modulationalinstability and envelope soliton excitations of IAWs in someastrophysical and space environments where 119890-119901-119894 plasmaswith nonthermal electrons are present
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the reviewers for varioussuggestions and helpful comments in bringing the paper tothe present form
References
[1] H R Miller and P J Witta Active Galactic Nuclei SpringerBerlin Germany 1978
[2] F C Michel ldquoTheory of pulsar magnetospheresrdquo Reviews ofModern Physics vol 54 no 1 pp 1ndash66 1982
[3] F C Michel Theory of Neutron Star Magnetosphere ChicagoUniversity Press Chicago Ill USA 1991
[4] M I Barns Positron Electron Pairs in Astrophysics AmericanInstitute of Physics New York NY USA 1983
[5] W K Misner S Thorne and J A Wheeler GravitationFreeman San Francisco Calif USA 1973
[6] M J Rees G W Gibbons S W Hawking and S SiklasedsTheEarly Universe Cambridge University Press Cambridge UK1983
[7] E Tandberg-Hanssen and A Gordon Emslie The Physics ofSolar Flares CambridgeUniversity Press Cambridge UK 1988
[8] A Cairns R Bingham R O Dendy C M C Nairn P KShukla and A A Mamun ldquoIon sound solitary waves withdensity depressionsrdquo Journal of Physics IV France vol 5 no C6pp 43ndash48 1995
[9] S I Popel S V Vladimirov and P K Shukla ldquoIon-acousticsolitons in electron-positron-ion plasmasrdquo Physics of Plasmasvol 2 no 3 pp 716ndash719 1995
[10] Y N Nejoh ldquoThe effect of the ion temperature on large ampli-tude ion-acoustic waves in an electron-positron-ion plasmardquoPhysics of Plasmas vol 3 no 4 pp 1447ndash1451 1996
[11] H R Pakzad ldquoIon acoustic solitary waves in plasma withnonthermal electron positron and warm ionrdquo Astrophysics andSpace Science vol 323 no 4 pp 345ndash350 2009
[12] S Ghosh and R Bharuthram ldquoIon acoustic solitons and doublelayers in electron-positron-ion plasmas with dust particulatesrdquoAstrophysics and Space Science vol 314 no 1-3 pp 121ndash127 2008
[13] N Jehan W Masood and A M Mirza ldquoPlanar and nonplanardust acoustic solitary waves in electronpositron-ion- dust plas-masrdquo Physica Scripta vol 80 no 3 Article ID 035506 2009
[14] R A Cairns A A Mamun R Bingham and P K ShuklaldquoIon acoustic solitons in a magnetised plasma with nonthermalelectronsrdquo Physica Scripta vol 63 pp 80ndash86 1996
[15] B Ghosh S Banerjee and S N Paul ldquoEffect of non-thermalelectrons andwarmnegative ions on ion-acoustic solitarywavesinmulti-component drifting plasmardquo Indian Journal of Pure andApplied Physics vol 51 no 7 pp 488ndash493 2013
[16] B Ghosh S N Paul C Das I Paul and S Banerjee ldquoElectro-static double layers in amulticomponent drifting plasma havingnonthermal electronsrdquo Brazilian Journal of Physics vol 43 no1-2 pp 28ndash33 2013
[17] P O Dovner A I Eriksson R Bostrom and B Holback ldquoFrejamultiprobe observations of electrostatic solitary structuresrdquoGeophysical Research Letters vol 21 no 17 pp 1827ndash1830 1994
[18] R Bostrom G Gustafsson B Holback G Holmgren HKoskinen and P Kintner ldquoCharacteristics of solitary waves andweak double layers in the magnetospheric plasmardquo PhysicalReview Letters vol 61 no 1 pp 82ndash85 1988
[19] R A Cairns A A Mamun R Bingham et al ldquoElectro-static solitary structures in non-thermal plasmasrdquo GeophysicalResearch Letters vol 22 no 20 pp 2709ndash2712 1995
[20] M Salahuddin H Saleem and M Saddiq ldquoIon-acoustic enve-lope solitons in electron-positron-ion plasmasrdquo Physical ReviewE vol 66 no 3 Article ID 036407 2002
[21] T S Gill C Bedi and A S Bains ldquoEnvelope excitations of ionacoustic solitary waves in a plasma with superthermal electronsand positronsrdquo Physica Scripta vol 81 no 5 Article ID 0555032010
[22] GMurtaza andM Salahuddin ldquoModulational instability of ionacoustic waves in a magnetised plasmardquo Plasma Physics vol 24no 5 pp 451ndash456 1982
[23] Yashvir T N Bhatnagar and S R Sharma ldquoNonlinear ion-acoustic waves and solitons in warm-ion magnetized plasmardquoPlasma Physics and Controlled Fusion vol 26 no 11 article 004pp 1303ndash1310 1984
[24] J K Chawla M K Mishra and R S Tiwari ldquoModulationalinstability of ion-acoustic waves in electron-positron-ion plas-masrdquoAstrophysics and Space Science vol 347 pp 283ndash292 2013
[25] T K Baluku andM A Hellberg ldquoIon acoustic solitary waves inan electron-positron-ion plasma with non-thermal electronsrdquoPlasma Physics and Controlled Fusion vol 53 no 9 Article ID095007 2011
[26] A E Dubinov and M A Sazonkin ldquoNonlinear theory of ion-acoustic waves in an electron-positron-ion plasmardquo PlasmaPhysics Reports vol 35 no 1 pp 14ndash24 2009
[27] S Mahmood S Siddiqui and N Jehan ldquoModulational instabil-ity of ion acousticwavewithwarm ions in electron-positron-ionplasmasrdquo Physics of Plasmas vol 18 no 5 Article ID 0523092011
[28] A S BainsN S Saini andT SGill ldquoModulational instability ofion-acoustic soliton in electron-positron-ion plasma with dustparticulatesrdquo Astrophysics and Space Science vol 343 no 1 pp293ndash299 2013
[29] P Eslami M Mottaghizadeh and H R Pakzad ldquoModulationalinstability of ion acoustic waves in e-p-i plasmas with electronsand positrons following a q-nonextensive distributionrdquo Physicsof Plasmas vol 18 no 10 Article ID 102313 2011
[30] J Zhang Y Wang and L Wu ldquoModulation instability of ionacoustic waves solitons and their interactions in nonthermalelectron-positron-ion plasmasrdquo Physics of Plasmas vol 16 no6 Article ID 062102 2009
[31] T S Gill A S Bains N S Saini and C Bedi ldquoIon-acousticenvelope excitations in electron-positron-ion plasma with non-thermal electronsrdquo Physics Letters A vol 374 no 31-32 pp3210ndash3215 2010
8 Journal of Astrophysics
[32] B Ghosh S N Paul C Das and I Paul ldquoModulationalinstability of high frequency surface waves on warm plasmahalf-spacerdquo Canadian Journal of Physics vol 90 no 3 pp 291ndash297 2012
[33] B Ghosh and K P Das ldquoModulational instability of electronplasma waves in a cylindrical wave guiderdquo Plasma Physics andControlled Fusion vol 27 no 9 pp 969ndash982 1985
[34] B Ghosh S Chandra and S N Paul ldquoAmplitudemodulation ofelectron plasmawaves in a quantumplasmardquoPhysics of Plasmasvol 18 no 1 Article ID 012106 2011
[35] H R Pakzad ldquoIon acoustic solitary waves in plasma withnonthermal electron and positronrdquo Physics Letters A GeneralAtomic and Solid State Physics vol 373 no 8-9 pp 847ndash8502009
[36] T Kakutani and N Sugimoto ldquoKrylov-Bogoliubov-Mitr-opolsky method for nonlinear wave modulationrdquo The Physicsof Fluids vol 17 pp 1617ndash1625 1974
where 120588 and 120579 are two real variables Solving the resultingequations for 120588 and 120579with119875119876 lt 0we get the following brightenvelope soliton solution
120588 =
radic2 |119875119876|
119871
sech(
120585 minus 119880120591
119871
) (23)
where 119880 is the envelope speed and 119871 is the spatial width ofthe pulse It encloses high frequency carrier oscillations andvanishes at infinity On the other hand if 119875119876 gt 0 a stablegray or dark soliton (a potential hole or a localized region ofdeceased amplitude) is obtained
120588 =
radic2119875119876
119871119889
radic1 minus 1198892sech2 (120585 minus 119880120591
119871
) (24)
where the parameter 119889 determines the depth of the modula-tion For 119889 = 1 we get a dark soliton
120588 =
radic2119875119876
119871119889
tanh(
120585 minus 119880120591
119871
) (25)
Thus the sign of the product 119875119876 determines the stabil-ityinstability profile of IAWs as well as the type of solitonstructureThe soliton width is determined by the ratio |119875119876|
We have numerically examined different parametricregionswhere someof the above excitationsmay occurAs thecoefficients119875 and119876 depend on nonthermal parameter120573 ion-to-electron temperature ratio 120590
119894 and positron-to-electron
concentration ratio 120594 these parameters would definitelydetermine the modulational instability and the formationof envelope solitons Numerical plots in Figures 1ndash3 show119875119876 as a function of 119896 for different values of 120573 120590
119894 and 120594
It shows that the IAWs remain modulationally stable for 119896
less than certain critical value 119896119888and for 119896 gt 119896
119888the wave is
modulationally unstableIn Figure 1 the variation of 119875119876 with wave number has
been plotted for different values of nonthermal parameter (120573)keeping positron concentration (120594) and ion temperature (120590
119894)
fixed It shows that as 120573 increases the value of critical wavenumber separating stable and unstable regions decreases It isalso noticed that as 120573 increases the width of the dark solitonsincreases but that of the bright solitons decreases
In Figure 2 119875119876 is plotted as function of 119896 for differentvalues of ion temperature (120590
119894) taking other plasmaparameters
such as positron concentration (120594) and nonthermal parame-ter (120573) as constant It is seen that as 120590
119894increases critical wave
number decreases the width of dark solitons increases butthat of bright solitons decreases
Figure 3 is a 119875119876 versus wave number plot for differentvalues of positron concentration (120594) keeping the values ofnonthermal parameter (120573) and ion temperature (120590
119894) constant
It shows that as the value of 120594 increases the critical wavenumber increases The width of dark solitons decreases andthat of bright solitons increases as 120594 increases
Qualitatively these results agree with those obtained byGill et al [31] but quantitatively there are differences Wefind that the critical wave number is more sensitive to thevariation in 120573 120590
119894 and 120594 than that predicted by Gill et al [31]
PQ
minus2
20
2
0
1510
ab
c
a 120573 = 0
b 120573 = 0055
c 120573 = 011
Wave number
Figure 1 Plot of 119875119876 versus wave number 119896 for different values ofnonthermal parameter (120573) Curves labelled a b and c correspondto 120573 = 0 0055 and 011 respectively 120594 = 022 120590
119901= 001 and
120590119894= 002
32
a 120590i = 00145
b 120590i = 00155
c 120590i = 00165
Wave number
abc
PQ
1
0
minus1
Figure 2 Plot of 119875119876 versus wave number 119896 for different values ofion temperature (120590
119894) Curves labelled a b and c correspond to 120590
119894=
00145 00155 and 00165 respectively 120594 = 02 120590119901
= 0015 and120573 = 0022
In addition we have numerically studied the dependence ofgrowth rate of instability on all the plasma parameters 120573 120590
119894
and 120594The results are shown in Figures 4 5 and 6 It is shownthat the growth rate of instability increases with increase inthe nonthermality of electrons and ion temperature but theincrease of positron concentration reduces instability growthrate
6 Journal of AstrophysicsPQ
05
00
minus05
21
a 120594 = 025
b 120594 = 026
c 120594 = 027
Wave number
a b c
Figure 3 Plot of 119875119876 versus wave number 119896 for different values ofpositron concentration (120594) Curves labelled a b and c correspondto 120594 = 025 026 and 027 respectively 120573 = 0022 120590
119901= 001 and
120590119894= 0052
2
1
0
1
a 120573 = 0
b 120573 = 0055
c 120573 = 011
Wave number
a
b
c
Gro
wth
rate
Figure 4 Plot of growth rate versus wave number 119896 for differentvalues of nonthermal parameter (120573) Curves labelled a b and ccorrespond to 120573 = 0 0055 and 011 respectively 120594 = 002 120590
119901=
001 and 120590119894= 0002
5 Conclusions
In the present work we have investigated modulationalinstability and envelope excitations of IAWs in the 119890-119901-119894 plasma in detail including simultaneously the effects ofnonthermality of electrons and temperatures of ions Ourmain findings are summarized below
15
10
05
00
12 16
a 120590i = 00012
b 120590i = 00024
c 120590i = 00036
Wave number
a
b
c
Gro
wth
rate
Figure 5 Plot of growth rate versus wave number 119896 for differentvalues of ion temperature (120590
119894) Curves labelled a b and c correspond
to 120590119894= 00012 00024 and 00036 respectively 120594 = 0001 120590
119901=
001 and 120573 = 0001
a 120594 = 0
b 120594 = 002
c 120594 = 004
Wave number
a
b
c
Gro
wth
rate
2
1
0
201510
Figure 6 Plot of growth rate versus wave number 119896 for differentvalues of positron concentration (120594) Curves labelled a b and ccorrespond to 120594 = 0 002 and 004 respectively 120573 = 001 120590
119901=
001 and 120590119894= 001
(i) The wave frequency increases with increase innonthermality of electrons and the temperature ofions whereas the increase in positron concentrationdecreases the wave frequency
(ii) There exists a critical wave number 119896119888below which
thewave ismodulationally stable and abovewhich thewave is modulationally unstable
Journal of Astrophysics 7
(iii) The value of the critical wave number and the char-acteristics of brightdark envelope solitons dependsignificantly on the nonthermal parameter (120573) iontemperature (120590
119894) and positron concentration (120594)
Finally we would like to mention that the results pre-sented in this paper may be useful to explain modulationalinstability and envelope soliton excitations of IAWs in someastrophysical and space environments where 119890-119901-119894 plasmaswith nonthermal electrons are present
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the reviewers for varioussuggestions and helpful comments in bringing the paper tothe present form
References
[1] H R Miller and P J Witta Active Galactic Nuclei SpringerBerlin Germany 1978
[2] F C Michel ldquoTheory of pulsar magnetospheresrdquo Reviews ofModern Physics vol 54 no 1 pp 1ndash66 1982
[3] F C Michel Theory of Neutron Star Magnetosphere ChicagoUniversity Press Chicago Ill USA 1991
[4] M I Barns Positron Electron Pairs in Astrophysics AmericanInstitute of Physics New York NY USA 1983
[5] W K Misner S Thorne and J A Wheeler GravitationFreeman San Francisco Calif USA 1973
[6] M J Rees G W Gibbons S W Hawking and S SiklasedsTheEarly Universe Cambridge University Press Cambridge UK1983
[7] E Tandberg-Hanssen and A Gordon Emslie The Physics ofSolar Flares CambridgeUniversity Press Cambridge UK 1988
[8] A Cairns R Bingham R O Dendy C M C Nairn P KShukla and A A Mamun ldquoIon sound solitary waves withdensity depressionsrdquo Journal of Physics IV France vol 5 no C6pp 43ndash48 1995
[9] S I Popel S V Vladimirov and P K Shukla ldquoIon-acousticsolitons in electron-positron-ion plasmasrdquo Physics of Plasmasvol 2 no 3 pp 716ndash719 1995
[10] Y N Nejoh ldquoThe effect of the ion temperature on large ampli-tude ion-acoustic waves in an electron-positron-ion plasmardquoPhysics of Plasmas vol 3 no 4 pp 1447ndash1451 1996
[11] H R Pakzad ldquoIon acoustic solitary waves in plasma withnonthermal electron positron and warm ionrdquo Astrophysics andSpace Science vol 323 no 4 pp 345ndash350 2009
[12] S Ghosh and R Bharuthram ldquoIon acoustic solitons and doublelayers in electron-positron-ion plasmas with dust particulatesrdquoAstrophysics and Space Science vol 314 no 1-3 pp 121ndash127 2008
[13] N Jehan W Masood and A M Mirza ldquoPlanar and nonplanardust acoustic solitary waves in electronpositron-ion- dust plas-masrdquo Physica Scripta vol 80 no 3 Article ID 035506 2009
[14] R A Cairns A A Mamun R Bingham and P K ShuklaldquoIon acoustic solitons in a magnetised plasma with nonthermalelectronsrdquo Physica Scripta vol 63 pp 80ndash86 1996
[15] B Ghosh S Banerjee and S N Paul ldquoEffect of non-thermalelectrons andwarmnegative ions on ion-acoustic solitarywavesinmulti-component drifting plasmardquo Indian Journal of Pure andApplied Physics vol 51 no 7 pp 488ndash493 2013
[16] B Ghosh S N Paul C Das I Paul and S Banerjee ldquoElectro-static double layers in amulticomponent drifting plasma havingnonthermal electronsrdquo Brazilian Journal of Physics vol 43 no1-2 pp 28ndash33 2013
[17] P O Dovner A I Eriksson R Bostrom and B Holback ldquoFrejamultiprobe observations of electrostatic solitary structuresrdquoGeophysical Research Letters vol 21 no 17 pp 1827ndash1830 1994
[18] R Bostrom G Gustafsson B Holback G Holmgren HKoskinen and P Kintner ldquoCharacteristics of solitary waves andweak double layers in the magnetospheric plasmardquo PhysicalReview Letters vol 61 no 1 pp 82ndash85 1988
[19] R A Cairns A A Mamun R Bingham et al ldquoElectro-static solitary structures in non-thermal plasmasrdquo GeophysicalResearch Letters vol 22 no 20 pp 2709ndash2712 1995
[20] M Salahuddin H Saleem and M Saddiq ldquoIon-acoustic enve-lope solitons in electron-positron-ion plasmasrdquo Physical ReviewE vol 66 no 3 Article ID 036407 2002
[21] T S Gill C Bedi and A S Bains ldquoEnvelope excitations of ionacoustic solitary waves in a plasma with superthermal electronsand positronsrdquo Physica Scripta vol 81 no 5 Article ID 0555032010
[22] GMurtaza andM Salahuddin ldquoModulational instability of ionacoustic waves in a magnetised plasmardquo Plasma Physics vol 24no 5 pp 451ndash456 1982
[23] Yashvir T N Bhatnagar and S R Sharma ldquoNonlinear ion-acoustic waves and solitons in warm-ion magnetized plasmardquoPlasma Physics and Controlled Fusion vol 26 no 11 article 004pp 1303ndash1310 1984
[24] J K Chawla M K Mishra and R S Tiwari ldquoModulationalinstability of ion-acoustic waves in electron-positron-ion plas-masrdquoAstrophysics and Space Science vol 347 pp 283ndash292 2013
[25] T K Baluku andM A Hellberg ldquoIon acoustic solitary waves inan electron-positron-ion plasma with non-thermal electronsrdquoPlasma Physics and Controlled Fusion vol 53 no 9 Article ID095007 2011
[26] A E Dubinov and M A Sazonkin ldquoNonlinear theory of ion-acoustic waves in an electron-positron-ion plasmardquo PlasmaPhysics Reports vol 35 no 1 pp 14ndash24 2009
[27] S Mahmood S Siddiqui and N Jehan ldquoModulational instabil-ity of ion acousticwavewithwarm ions in electron-positron-ionplasmasrdquo Physics of Plasmas vol 18 no 5 Article ID 0523092011
[28] A S BainsN S Saini andT SGill ldquoModulational instability ofion-acoustic soliton in electron-positron-ion plasma with dustparticulatesrdquo Astrophysics and Space Science vol 343 no 1 pp293ndash299 2013
[29] P Eslami M Mottaghizadeh and H R Pakzad ldquoModulationalinstability of ion acoustic waves in e-p-i plasmas with electronsand positrons following a q-nonextensive distributionrdquo Physicsof Plasmas vol 18 no 10 Article ID 102313 2011
[30] J Zhang Y Wang and L Wu ldquoModulation instability of ionacoustic waves solitons and their interactions in nonthermalelectron-positron-ion plasmasrdquo Physics of Plasmas vol 16 no6 Article ID 062102 2009
[31] T S Gill A S Bains N S Saini and C Bedi ldquoIon-acousticenvelope excitations in electron-positron-ion plasma with non-thermal electronsrdquo Physics Letters A vol 374 no 31-32 pp3210ndash3215 2010
8 Journal of Astrophysics
[32] B Ghosh S N Paul C Das and I Paul ldquoModulationalinstability of high frequency surface waves on warm plasmahalf-spacerdquo Canadian Journal of Physics vol 90 no 3 pp 291ndash297 2012
[33] B Ghosh and K P Das ldquoModulational instability of electronplasma waves in a cylindrical wave guiderdquo Plasma Physics andControlled Fusion vol 27 no 9 pp 969ndash982 1985
[34] B Ghosh S Chandra and S N Paul ldquoAmplitudemodulation ofelectron plasmawaves in a quantumplasmardquoPhysics of Plasmasvol 18 no 1 Article ID 012106 2011
[35] H R Pakzad ldquoIon acoustic solitary waves in plasma withnonthermal electron and positronrdquo Physics Letters A GeneralAtomic and Solid State Physics vol 373 no 8-9 pp 847ndash8502009
[36] T Kakutani and N Sugimoto ldquoKrylov-Bogoliubov-Mitr-opolsky method for nonlinear wave modulationrdquo The Physicsof Fluids vol 17 pp 1617ndash1625 1974
Figure 3 Plot of 119875119876 versus wave number 119896 for different values ofpositron concentration (120594) Curves labelled a b and c correspondto 120594 = 025 026 and 027 respectively 120573 = 0022 120590
119901= 001 and
120590119894= 0052
2
1
0
1
a 120573 = 0
b 120573 = 0055
c 120573 = 011
Wave number
a
b
c
Gro
wth
rate
Figure 4 Plot of growth rate versus wave number 119896 for differentvalues of nonthermal parameter (120573) Curves labelled a b and ccorrespond to 120573 = 0 0055 and 011 respectively 120594 = 002 120590
119901=
001 and 120590119894= 0002
5 Conclusions
In the present work we have investigated modulationalinstability and envelope excitations of IAWs in the 119890-119901-119894 plasma in detail including simultaneously the effects ofnonthermality of electrons and temperatures of ions Ourmain findings are summarized below
15
10
05
00
12 16
a 120590i = 00012
b 120590i = 00024
c 120590i = 00036
Wave number
a
b
c
Gro
wth
rate
Figure 5 Plot of growth rate versus wave number 119896 for differentvalues of ion temperature (120590
119894) Curves labelled a b and c correspond
to 120590119894= 00012 00024 and 00036 respectively 120594 = 0001 120590
119901=
001 and 120573 = 0001
a 120594 = 0
b 120594 = 002
c 120594 = 004
Wave number
a
b
c
Gro
wth
rate
2
1
0
201510
Figure 6 Plot of growth rate versus wave number 119896 for differentvalues of positron concentration (120594) Curves labelled a b and ccorrespond to 120594 = 0 002 and 004 respectively 120573 = 001 120590
119901=
001 and 120590119894= 001
(i) The wave frequency increases with increase innonthermality of electrons and the temperature ofions whereas the increase in positron concentrationdecreases the wave frequency
(ii) There exists a critical wave number 119896119888below which
thewave ismodulationally stable and abovewhich thewave is modulationally unstable
Journal of Astrophysics 7
(iii) The value of the critical wave number and the char-acteristics of brightdark envelope solitons dependsignificantly on the nonthermal parameter (120573) iontemperature (120590
119894) and positron concentration (120594)
Finally we would like to mention that the results pre-sented in this paper may be useful to explain modulationalinstability and envelope soliton excitations of IAWs in someastrophysical and space environments where 119890-119901-119894 plasmaswith nonthermal electrons are present
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the reviewers for varioussuggestions and helpful comments in bringing the paper tothe present form
References
[1] H R Miller and P J Witta Active Galactic Nuclei SpringerBerlin Germany 1978
[2] F C Michel ldquoTheory of pulsar magnetospheresrdquo Reviews ofModern Physics vol 54 no 1 pp 1ndash66 1982
[3] F C Michel Theory of Neutron Star Magnetosphere ChicagoUniversity Press Chicago Ill USA 1991
[4] M I Barns Positron Electron Pairs in Astrophysics AmericanInstitute of Physics New York NY USA 1983
[5] W K Misner S Thorne and J A Wheeler GravitationFreeman San Francisco Calif USA 1973
[6] M J Rees G W Gibbons S W Hawking and S SiklasedsTheEarly Universe Cambridge University Press Cambridge UK1983
[7] E Tandberg-Hanssen and A Gordon Emslie The Physics ofSolar Flares CambridgeUniversity Press Cambridge UK 1988
[8] A Cairns R Bingham R O Dendy C M C Nairn P KShukla and A A Mamun ldquoIon sound solitary waves withdensity depressionsrdquo Journal of Physics IV France vol 5 no C6pp 43ndash48 1995
[9] S I Popel S V Vladimirov and P K Shukla ldquoIon-acousticsolitons in electron-positron-ion plasmasrdquo Physics of Plasmasvol 2 no 3 pp 716ndash719 1995
[10] Y N Nejoh ldquoThe effect of the ion temperature on large ampli-tude ion-acoustic waves in an electron-positron-ion plasmardquoPhysics of Plasmas vol 3 no 4 pp 1447ndash1451 1996
[11] H R Pakzad ldquoIon acoustic solitary waves in plasma withnonthermal electron positron and warm ionrdquo Astrophysics andSpace Science vol 323 no 4 pp 345ndash350 2009
[12] S Ghosh and R Bharuthram ldquoIon acoustic solitons and doublelayers in electron-positron-ion plasmas with dust particulatesrdquoAstrophysics and Space Science vol 314 no 1-3 pp 121ndash127 2008
[13] N Jehan W Masood and A M Mirza ldquoPlanar and nonplanardust acoustic solitary waves in electronpositron-ion- dust plas-masrdquo Physica Scripta vol 80 no 3 Article ID 035506 2009
[14] R A Cairns A A Mamun R Bingham and P K ShuklaldquoIon acoustic solitons in a magnetised plasma with nonthermalelectronsrdquo Physica Scripta vol 63 pp 80ndash86 1996
[15] B Ghosh S Banerjee and S N Paul ldquoEffect of non-thermalelectrons andwarmnegative ions on ion-acoustic solitarywavesinmulti-component drifting plasmardquo Indian Journal of Pure andApplied Physics vol 51 no 7 pp 488ndash493 2013
[16] B Ghosh S N Paul C Das I Paul and S Banerjee ldquoElectro-static double layers in amulticomponent drifting plasma havingnonthermal electronsrdquo Brazilian Journal of Physics vol 43 no1-2 pp 28ndash33 2013
[17] P O Dovner A I Eriksson R Bostrom and B Holback ldquoFrejamultiprobe observations of electrostatic solitary structuresrdquoGeophysical Research Letters vol 21 no 17 pp 1827ndash1830 1994
[18] R Bostrom G Gustafsson B Holback G Holmgren HKoskinen and P Kintner ldquoCharacteristics of solitary waves andweak double layers in the magnetospheric plasmardquo PhysicalReview Letters vol 61 no 1 pp 82ndash85 1988
[19] R A Cairns A A Mamun R Bingham et al ldquoElectro-static solitary structures in non-thermal plasmasrdquo GeophysicalResearch Letters vol 22 no 20 pp 2709ndash2712 1995
[20] M Salahuddin H Saleem and M Saddiq ldquoIon-acoustic enve-lope solitons in electron-positron-ion plasmasrdquo Physical ReviewE vol 66 no 3 Article ID 036407 2002
[21] T S Gill C Bedi and A S Bains ldquoEnvelope excitations of ionacoustic solitary waves in a plasma with superthermal electronsand positronsrdquo Physica Scripta vol 81 no 5 Article ID 0555032010
[22] GMurtaza andM Salahuddin ldquoModulational instability of ionacoustic waves in a magnetised plasmardquo Plasma Physics vol 24no 5 pp 451ndash456 1982
[23] Yashvir T N Bhatnagar and S R Sharma ldquoNonlinear ion-acoustic waves and solitons in warm-ion magnetized plasmardquoPlasma Physics and Controlled Fusion vol 26 no 11 article 004pp 1303ndash1310 1984
[24] J K Chawla M K Mishra and R S Tiwari ldquoModulationalinstability of ion-acoustic waves in electron-positron-ion plas-masrdquoAstrophysics and Space Science vol 347 pp 283ndash292 2013
[25] T K Baluku andM A Hellberg ldquoIon acoustic solitary waves inan electron-positron-ion plasma with non-thermal electronsrdquoPlasma Physics and Controlled Fusion vol 53 no 9 Article ID095007 2011
[26] A E Dubinov and M A Sazonkin ldquoNonlinear theory of ion-acoustic waves in an electron-positron-ion plasmardquo PlasmaPhysics Reports vol 35 no 1 pp 14ndash24 2009
[27] S Mahmood S Siddiqui and N Jehan ldquoModulational instabil-ity of ion acousticwavewithwarm ions in electron-positron-ionplasmasrdquo Physics of Plasmas vol 18 no 5 Article ID 0523092011
[28] A S BainsN S Saini andT SGill ldquoModulational instability ofion-acoustic soliton in electron-positron-ion plasma with dustparticulatesrdquo Astrophysics and Space Science vol 343 no 1 pp293ndash299 2013
[29] P Eslami M Mottaghizadeh and H R Pakzad ldquoModulationalinstability of ion acoustic waves in e-p-i plasmas with electronsand positrons following a q-nonextensive distributionrdquo Physicsof Plasmas vol 18 no 10 Article ID 102313 2011
[30] J Zhang Y Wang and L Wu ldquoModulation instability of ionacoustic waves solitons and their interactions in nonthermalelectron-positron-ion plasmasrdquo Physics of Plasmas vol 16 no6 Article ID 062102 2009
[31] T S Gill A S Bains N S Saini and C Bedi ldquoIon-acousticenvelope excitations in electron-positron-ion plasma with non-thermal electronsrdquo Physics Letters A vol 374 no 31-32 pp3210ndash3215 2010
8 Journal of Astrophysics
[32] B Ghosh S N Paul C Das and I Paul ldquoModulationalinstability of high frequency surface waves on warm plasmahalf-spacerdquo Canadian Journal of Physics vol 90 no 3 pp 291ndash297 2012
[33] B Ghosh and K P Das ldquoModulational instability of electronplasma waves in a cylindrical wave guiderdquo Plasma Physics andControlled Fusion vol 27 no 9 pp 969ndash982 1985
[34] B Ghosh S Chandra and S N Paul ldquoAmplitudemodulation ofelectron plasmawaves in a quantumplasmardquoPhysics of Plasmasvol 18 no 1 Article ID 012106 2011
[35] H R Pakzad ldquoIon acoustic solitary waves in plasma withnonthermal electron and positronrdquo Physics Letters A GeneralAtomic and Solid State Physics vol 373 no 8-9 pp 847ndash8502009
[36] T Kakutani and N Sugimoto ldquoKrylov-Bogoliubov-Mitr-opolsky method for nonlinear wave modulationrdquo The Physicsof Fluids vol 17 pp 1617ndash1625 1974
(iii) The value of the critical wave number and the char-acteristics of brightdark envelope solitons dependsignificantly on the nonthermal parameter (120573) iontemperature (120590
119894) and positron concentration (120594)
Finally we would like to mention that the results pre-sented in this paper may be useful to explain modulationalinstability and envelope soliton excitations of IAWs in someastrophysical and space environments where 119890-119901-119894 plasmaswith nonthermal electrons are present
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the reviewers for varioussuggestions and helpful comments in bringing the paper tothe present form
References
[1] H R Miller and P J Witta Active Galactic Nuclei SpringerBerlin Germany 1978
[2] F C Michel ldquoTheory of pulsar magnetospheresrdquo Reviews ofModern Physics vol 54 no 1 pp 1ndash66 1982
[3] F C Michel Theory of Neutron Star Magnetosphere ChicagoUniversity Press Chicago Ill USA 1991
[4] M I Barns Positron Electron Pairs in Astrophysics AmericanInstitute of Physics New York NY USA 1983
[5] W K Misner S Thorne and J A Wheeler GravitationFreeman San Francisco Calif USA 1973
[6] M J Rees G W Gibbons S W Hawking and S SiklasedsTheEarly Universe Cambridge University Press Cambridge UK1983
[7] E Tandberg-Hanssen and A Gordon Emslie The Physics ofSolar Flares CambridgeUniversity Press Cambridge UK 1988
[8] A Cairns R Bingham R O Dendy C M C Nairn P KShukla and A A Mamun ldquoIon sound solitary waves withdensity depressionsrdquo Journal of Physics IV France vol 5 no C6pp 43ndash48 1995
[9] S I Popel S V Vladimirov and P K Shukla ldquoIon-acousticsolitons in electron-positron-ion plasmasrdquo Physics of Plasmasvol 2 no 3 pp 716ndash719 1995
[10] Y N Nejoh ldquoThe effect of the ion temperature on large ampli-tude ion-acoustic waves in an electron-positron-ion plasmardquoPhysics of Plasmas vol 3 no 4 pp 1447ndash1451 1996
[11] H R Pakzad ldquoIon acoustic solitary waves in plasma withnonthermal electron positron and warm ionrdquo Astrophysics andSpace Science vol 323 no 4 pp 345ndash350 2009
[12] S Ghosh and R Bharuthram ldquoIon acoustic solitons and doublelayers in electron-positron-ion plasmas with dust particulatesrdquoAstrophysics and Space Science vol 314 no 1-3 pp 121ndash127 2008
[13] N Jehan W Masood and A M Mirza ldquoPlanar and nonplanardust acoustic solitary waves in electronpositron-ion- dust plas-masrdquo Physica Scripta vol 80 no 3 Article ID 035506 2009
[14] R A Cairns A A Mamun R Bingham and P K ShuklaldquoIon acoustic solitons in a magnetised plasma with nonthermalelectronsrdquo Physica Scripta vol 63 pp 80ndash86 1996
[15] B Ghosh S Banerjee and S N Paul ldquoEffect of non-thermalelectrons andwarmnegative ions on ion-acoustic solitarywavesinmulti-component drifting plasmardquo Indian Journal of Pure andApplied Physics vol 51 no 7 pp 488ndash493 2013
[16] B Ghosh S N Paul C Das I Paul and S Banerjee ldquoElectro-static double layers in amulticomponent drifting plasma havingnonthermal electronsrdquo Brazilian Journal of Physics vol 43 no1-2 pp 28ndash33 2013
[17] P O Dovner A I Eriksson R Bostrom and B Holback ldquoFrejamultiprobe observations of electrostatic solitary structuresrdquoGeophysical Research Letters vol 21 no 17 pp 1827ndash1830 1994
[18] R Bostrom G Gustafsson B Holback G Holmgren HKoskinen and P Kintner ldquoCharacteristics of solitary waves andweak double layers in the magnetospheric plasmardquo PhysicalReview Letters vol 61 no 1 pp 82ndash85 1988
[19] R A Cairns A A Mamun R Bingham et al ldquoElectro-static solitary structures in non-thermal plasmasrdquo GeophysicalResearch Letters vol 22 no 20 pp 2709ndash2712 1995
[20] M Salahuddin H Saleem and M Saddiq ldquoIon-acoustic enve-lope solitons in electron-positron-ion plasmasrdquo Physical ReviewE vol 66 no 3 Article ID 036407 2002
[21] T S Gill C Bedi and A S Bains ldquoEnvelope excitations of ionacoustic solitary waves in a plasma with superthermal electronsand positronsrdquo Physica Scripta vol 81 no 5 Article ID 0555032010
[22] GMurtaza andM Salahuddin ldquoModulational instability of ionacoustic waves in a magnetised plasmardquo Plasma Physics vol 24no 5 pp 451ndash456 1982
[23] Yashvir T N Bhatnagar and S R Sharma ldquoNonlinear ion-acoustic waves and solitons in warm-ion magnetized plasmardquoPlasma Physics and Controlled Fusion vol 26 no 11 article 004pp 1303ndash1310 1984
[24] J K Chawla M K Mishra and R S Tiwari ldquoModulationalinstability of ion-acoustic waves in electron-positron-ion plas-masrdquoAstrophysics and Space Science vol 347 pp 283ndash292 2013
[25] T K Baluku andM A Hellberg ldquoIon acoustic solitary waves inan electron-positron-ion plasma with non-thermal electronsrdquoPlasma Physics and Controlled Fusion vol 53 no 9 Article ID095007 2011
[26] A E Dubinov and M A Sazonkin ldquoNonlinear theory of ion-acoustic waves in an electron-positron-ion plasmardquo PlasmaPhysics Reports vol 35 no 1 pp 14ndash24 2009
[27] S Mahmood S Siddiqui and N Jehan ldquoModulational instabil-ity of ion acousticwavewithwarm ions in electron-positron-ionplasmasrdquo Physics of Plasmas vol 18 no 5 Article ID 0523092011
[28] A S BainsN S Saini andT SGill ldquoModulational instability ofion-acoustic soliton in electron-positron-ion plasma with dustparticulatesrdquo Astrophysics and Space Science vol 343 no 1 pp293ndash299 2013
[29] P Eslami M Mottaghizadeh and H R Pakzad ldquoModulationalinstability of ion acoustic waves in e-p-i plasmas with electronsand positrons following a q-nonextensive distributionrdquo Physicsof Plasmas vol 18 no 10 Article ID 102313 2011
[30] J Zhang Y Wang and L Wu ldquoModulation instability of ionacoustic waves solitons and their interactions in nonthermalelectron-positron-ion plasmasrdquo Physics of Plasmas vol 16 no6 Article ID 062102 2009
[31] T S Gill A S Bains N S Saini and C Bedi ldquoIon-acousticenvelope excitations in electron-positron-ion plasma with non-thermal electronsrdquo Physics Letters A vol 374 no 31-32 pp3210ndash3215 2010
8 Journal of Astrophysics
[32] B Ghosh S N Paul C Das and I Paul ldquoModulationalinstability of high frequency surface waves on warm plasmahalf-spacerdquo Canadian Journal of Physics vol 90 no 3 pp 291ndash297 2012
[33] B Ghosh and K P Das ldquoModulational instability of electronplasma waves in a cylindrical wave guiderdquo Plasma Physics andControlled Fusion vol 27 no 9 pp 969ndash982 1985
[34] B Ghosh S Chandra and S N Paul ldquoAmplitudemodulation ofelectron plasmawaves in a quantumplasmardquoPhysics of Plasmasvol 18 no 1 Article ID 012106 2011
[35] H R Pakzad ldquoIon acoustic solitary waves in plasma withnonthermal electron and positronrdquo Physics Letters A GeneralAtomic and Solid State Physics vol 373 no 8-9 pp 847ndash8502009
[36] T Kakutani and N Sugimoto ldquoKrylov-Bogoliubov-Mitr-opolsky method for nonlinear wave modulationrdquo The Physicsof Fluids vol 17 pp 1617ndash1625 1974
[32] B Ghosh S N Paul C Das and I Paul ldquoModulationalinstability of high frequency surface waves on warm plasmahalf-spacerdquo Canadian Journal of Physics vol 90 no 3 pp 291ndash297 2012
[33] B Ghosh and K P Das ldquoModulational instability of electronplasma waves in a cylindrical wave guiderdquo Plasma Physics andControlled Fusion vol 27 no 9 pp 969ndash982 1985
[34] B Ghosh S Chandra and S N Paul ldquoAmplitudemodulation ofelectron plasmawaves in a quantumplasmardquoPhysics of Plasmasvol 18 no 1 Article ID 012106 2011
[35] H R Pakzad ldquoIon acoustic solitary waves in plasma withnonthermal electron and positronrdquo Physics Letters A GeneralAtomic and Solid State Physics vol 373 no 8-9 pp 847ndash8502009
[36] T Kakutani and N Sugimoto ldquoKrylov-Bogoliubov-Mitr-opolsky method for nonlinear wave modulationrdquo The Physicsof Fluids vol 17 pp 1617ndash1625 1974
![Page 1: Research Article Modulation Instability of Ion-Acoustic Waves ...downloads.hindawi.com/journals/jas/2014/785670.pdfsech 7 ]! \ 8 , where ] istheenvelopespeedand\ is the spatial width](https://reader033.documents.pub/reader033/viewer/2022060813/6091a552e6cbc06c1c66a863/html5/thumbnails/1.jpg)
![Page 2: Research Article Modulation Instability of Ion-Acoustic Waves ...downloads.hindawi.com/journals/jas/2014/785670.pdfsech 7 ]! \ 8 , where ] istheenvelopespeedand\ is the spatial width](https://reader033.documents.pub/reader033/viewer/2022060813/6091a552e6cbc06c1c66a863/html5/thumbnails/2.jpg)
![Page 3: Research Article Modulation Instability of Ion-Acoustic Waves ...downloads.hindawi.com/journals/jas/2014/785670.pdfsech 7 ]! \ 8 , where ] istheenvelopespeedand\ is the spatial width](https://reader033.documents.pub/reader033/viewer/2022060813/6091a552e6cbc06c1c66a863/html5/thumbnails/3.jpg)
![Page 4: Research Article Modulation Instability of Ion-Acoustic Waves ...downloads.hindawi.com/journals/jas/2014/785670.pdfsech 7 ]! \ 8 , where ] istheenvelopespeedand\ is the spatial width](https://reader033.documents.pub/reader033/viewer/2022060813/6091a552e6cbc06c1c66a863/html5/thumbnails/4.jpg)
![Page 5: Research Article Modulation Instability of Ion-Acoustic Waves ...downloads.hindawi.com/journals/jas/2014/785670.pdfsech 7 ]! \ 8 , where ] istheenvelopespeedand\ is the spatial width](https://reader033.documents.pub/reader033/viewer/2022060813/6091a552e6cbc06c1c66a863/html5/thumbnails/5.jpg)
![Page 6: Research Article Modulation Instability of Ion-Acoustic Waves ...downloads.hindawi.com/journals/jas/2014/785670.pdfsech 7 ]! \ 8 , where ] istheenvelopespeedand\ is the spatial width](https://reader033.documents.pub/reader033/viewer/2022060813/6091a552e6cbc06c1c66a863/html5/thumbnails/6.jpg)
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