PHYSICS OF PLASMAS 17, 102315 共2010兲
Ion-acoustic nonlinear periodic waves in electron-positron-ion plasma J. K. Chawlaa兲 and M. K. Mishra Department of Physics, University of Rajasthan, Jaipur 302004, India
共Received 11 August 2010; accepted 23 September 2010; published online 29 October 2010兲 Ion-acoustic nonlinear periodic waves, namely, ion-acoustic cnoidal waves have been studied in electron-positron-ion plasma. Using reductive perturbation method and appropriate boundary condition for nonlinear periodic waves, the Korteweg–de Vries 共KdV兲 equation is derived for the system. The cnoidal wave solution of the KdV equation is discussed in detail. It is found that the frequency of the cnoidal wave is a function of its amplitude. It is also found that the positron concentration modifies the properties of the ion-acoustic cnoidal waves. The existence regions for ion-acoustic cnoidal wave in the parameters space 共p , 兲, where p and are the positron concentration and temperature ratio of electron to positron, are discussed in detail. In the limiting case these ion-acoustic cnoidal waves reduce to the ion-acoustic soliton solutions. The effect of other parameters on the characteristics of the nonlinear periodic waves is also discussed. © 2010 American Institute of Physics. 关doi:10.1063/1.3501138兴 I. INTRODUCTION
In recent years, there has been great deal of interest in the study of linear and nonlinear wave phenomena in electron-positron-ion plasmas. The electron-positron plasmas are thought to be generated naturally by pair production in high energy processes occurring in many astrophysical environments such as the early universe,1,2 neutron stars, active galactic nuclei,3 or pulsar magnetosphere,4,5 and in solar atmosphere.6 The electron positron plasmas have also been created in the laboratory.7–9 Greaves et al.10 have reported that advances in the positron trapping technique have led to room-temperature plasmas of 107 positrons with lifetime of 103 s. Because of long lifetime of the positrons most of the astrophysical6,11 and laboratory plasmas12–14 become an admixture of electrons, positrons, and ions. Therefore, the study of electron-positron-ion 共EPI兲 plasmas is important to understand the behavior of both astrophysical and laboratory plasmas. During the past few years nonlinear wave propagating in EPI plasmas has received a great deal of attention in investigating the nonlinear structures. The envelope solitons associated with the electromagnetic waves in EPI plasmas have been studied by Rizzato15 and Berezhiani et al.16 Ionacoustic solitons in EPI plasmas have been reported by Popel et al.17 Nejoh18 studied the effects of the ion temperature on large amplitude ion-acoustic waves in an EPI plasma. Large amplitude solitary electromagnetic waves in EPI plasmas have been studied by Verheest and Cattaert.19 The double layers associated with the kinetic Alfven wave in a magnetized EPI plasma have been studied by Kakati and Goswami.20 Ion-acoustic envelope solitons in EPI plasmas have been studied by Salahuddin et al.21 Tiwari et al.22 studied the effects of positron density and temperature on ionacoustic dressed solitons in an EPI plasma. In a recent paper, Mishra et al.23 have studied small amplitude ion-acoustic a兲
Present address: Government College, Thanagazi, Alwar, Rajasthan, India. Electronic mail:
[email protected].
1070-664X/2010/17共10兲/102315/6/$30.00
double layers in multicomponent plasma with positrons. However, ion-acoustic cnoidal waves in EPI plasma have not been studied so far. The study of nonlinear periodic waves24–28 in field free or magnetized plasma as well as in other dispersive media29,30 has been a subject of considerable interest. Nonlinear periodic waves expressed in terms of Jacobian elliptical-functions, such as sn, cn, and dn waves, are finding important applications in diverse areas of physics. One of them is the nonlinear transport phenomena. Nonlinear periodic dn waves are believed to be generated in the defocusing region of the ionospheric plasma.31 Kauschke and Schlüter32 found that periodic signals observed at the plasma edge in their experiment33 can be well described by the cnoidal waves. Schamel34 studied the nonlinear periodic wave solutions for small amplitude Langmuir waves, Konno et al.35 studied the ion-acoustic cnoidal wave solution of the Korteweg–de Vries 共KdV兲 equation. Yadav et al.36 studied ion-acoustic nonlinear periodic waves in a magnetized plasma with two electron species. Cnoidal waves have also been experimentally observed in water,37,38 in the photorefractive bismuth titanate crystal,39 and in the LiNbO3-共Sio film兲 structure.29 The study of the finite amplitude waves in a weakly relativistic plasma with cold ions and two electron components40 inferred that the phase velocity of the cnoidal waves decreases as the relativistic effect increases. Moreover in the presence of higher temperature electrons together with the relativistic effect, the amplitude of the cnoidal waves increases. The study of ion-acoustic periodic waves in the two electron temperature plasma41 shows that the KdV equation with negative coefficient of the nonlinear term also gives rise to the cnoidal wave solution, which reduces to the rarefactive soliton solution in the limiting case as modulus共m2兲 → 0. Recently, Yadav and Sayal42 have investigated obliquely propagating cnoidal waves in a magnetized dusty plasma with variable dust charge. Yadav et al.43 have
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J. K. Chawla and M. K. Mishra
also studied the dust acoustic nonlinear periodic waves in a dusty plasmas with charge fluctuation. The aim of this paper is to study ion-acoustic nonlinear periodic waves, namely, ion-acoustic cnoidal waves in EPI plasma. The salient feature is to demonstrate the existence of ion-acoustic nonlinear periodic waves in EPI plasma. Using the reductive perturbation method, the KdV equation is derived for the system. The cnoidal wave solution of the KdV equation is discussed in detail. It is found that the frequency of the cnoidal wave is a function of its amplitude. The existence regions for ion-acoustic cnoidal wave in the parameters space 共p , 兲, where p and are the positron concentration and temperature ratio of electron to positron, are discussed in detail. The effects of other parameters on the characteristics of the nonlinear periodic waves are also discussed. It is found that in the limiting case the cnoidal wave solution reduces to ion-acoustic soliton solution. The plan of the paper is as follows. In Sec. II, the normalized fluid equations for the system have been presented. Using the reductive perturbation method the KdV equation has been derived in Sec. III. In Sec. IV cnoidal wave solution of the KdV equation has been obtained. Section V is devoted to the discussion and main conclusions have been summarized in Sec. VI. II. BASIC EQUATIONS
We consider a multicomponent plasma consisting of hot positrons, cold ions, and hot isothermal electrons. The nonlinear behavior of ion-acoustic waves may be described by the following set of normalized fluid equations:
n + 共nu兲 = 0, t x
共1兲
u u +u =− , x t x
共2兲
2 = ne − pn p − 共1 − p兲n, x2
共3兲
ne = exp共兲,
共4兲
n p = exp共− 兲.
共5兲
For small ne = 1 + +
2 3 + + ... , 2 6
n p = 1 − +
2 2 3 3 − + ... , 2 6
共6兲
共7兲
where p=
n p0 and = 共Te/T p兲. ne0
It may be noted that in the present case we have assumed that the positron annihilation time is larger than the inverse of the characteristics frequency of the ion-acoustic wave.
Under such condition, one can assume that annihilation of positrons with electrons is negligible and the effect of annihilation can be neglected. In the above equations, n and v are the density and fluid velocity of the ion species, n p is the positron density. ne0, n p0, and n0 are the equilibrium densities of electron component, positron component, and ion component, respectively. is the electrostatic potential and p is fractional concentration of positron with respect to electron in the equilibrium. In Eq. 共3兲, the electron and positron density distributions are considered to be of the Maxwell Boltzmann type. In the above Eqs. 共1兲–共5兲, velocity 共u兲, potential 共兲, time 共t兲, and space coordinate 共x兲 have been normalized with respect to ion-acoustic speed Cs, thermal potential Te / e, inverse of the and Debye length D ion-plasma frequency −1 pi , = 共0Te / n0e2兲1/2 respectively. Ion density 共n兲, electron density 共ne兲, and positron density 共n p兲 are normalized with their corresponding equilibrium densities, i.e., no, neo, and n po, respectively. T p and Te are temperatures of positron and electron fluid, respectively.
III. DERIVATION OF THE KdV EQUATION
To derive the KdV equation from the basic set of equations, viz. Eqs. 共1兲–共5兲, we introduce the following stretching of coordinates 共兲 and 共兲 as:
= 1/2共x − V0t兲
共8a兲
= 3/2t,
共8b兲
and
where is a small parameter and V0 is the phase velocity of the wave to be determined later. Now we expand the field quantities in the following form:
冤冥 冤冥 冤 冥 冤 冥 冤 冥
1 n2 n3 n n1 2 3 u = 0 + u1 + u2 + u3 + . . . . 0 1 2 3
共9兲
On substituting the expansion 共9兲 into Eqs. 共1兲–共5兲, using Eqs. 共8a兲 and 共8b兲, and equating terms with the same powers of , we obtain a set of equations for each order in . The set of equations at the lowest order, i.e., − V0
n1 u1 + = 0,
共10兲
− V0
u1 1 + = 0,
共11兲
1共1 + p兲 − 共1 − p兲n1 = 0.
共12兲
Integrating Eqs. 共10兲 and 共11兲 with respect to for a continuous wave which can have finite perturbation even at → ⫾ ⬁, we get a relationship among the first order perturbed quantities as
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Phys. Plasmas 17, 102315 共2010兲
Ion-acoustic nonlinear periodic waves…
冉
冊
1 + p 1 , 1−p
u1 = V0
冉
冊
1 + p 1 + c1共兲. 1−p
共13兲
Now to derive the KdV equation, substituting the expression for u2 / from Eq. 共17兲 into Eq. 共16兲 and using firstorder solutions; i.e., Eqs. 共13兲 and 共14兲, we get
共14兲
n2 2 1 3 1 2 1 1 2 1 c1 = + 1 + c1 + + . V30 V40 V30 V20 V20
Here c1 is integration constant which is independent of and may depend on the variable . For the existence of a nontrivial solutions of the first order quantities, from Eqs. 共13兲 and 共14兲, we require V0 =
冉
1−p 1 + p
冊
1/2
共15兲
,
where V0 is the phase velocity of the cnoidal wave in the ion-acoustic wave frame. Now taking next higher-order equations
Now differentiating Eq. 共18兲 with respect to and substituting for n2 / from Eq. 共24兲, we get the equation
3 + a + c1 +b 3 =0
n2 n1 u2 共n1u1兲 + + + = 0,
共16兲
a=
− V0
u2 u1 u1 2 + + u1 + = 0,
共17兲
b=
共18兲
Multiplying Eq. 共16兲 by V0 and then subtracting it from Eq. 共17兲 and on using first-order solutions, we get 1 2 1 1 c1 u2 V0 n2 1 = + − 3 1 + . 2 2V0 2V0 2V0
共19兲
共20兲
Substituting the above value of n2 into Eq. 共19兲, we get the following equation:
冋
册
共21兲
Integrating above equation with respect to , we get
册
1 2 1 共1 − p2兲 2 1 2 − 2 1− + + c2共兲. V0 共1 + p兲 V0 V0 2V0共1 + p兲 2 共22兲
In the above equation, c2共兲 is the second integration constant which is independent of but may depend on . In the derivation of Eq. 共22兲, the periodic boundary condition implies that
c1 = 0. Therefore c1 is independent of and .
V 03 . 2共1 − p兲
共26兲
共27兲
In Eq. 共25兲 is used in place of 1.
IV. SOLUTION OF KDV EQUATION
For the steady state solution of the KdV Eq. 共25兲, we consider
where u1 is a constant velocity. Integrating Eq. 共25兲 twice with respect to , we obtain
冉 冊
1 d 2 d
V共兲 =
3
冋
册
V0共1 − p2兲 3 − , 2V0 2共1 + p兲
2
+ V共兲 = 0,
共28兲
where the Sagdeev potential V共兲 is given by
u2 1 2 1 共1 − p2兲 1 2 − = + 共 兲 V0 V0 共1 + p兲 V20 1 1 1 c1 1 − . 3 + 2V0共1 + p兲 2V0
冋
= − u 1 ,
Equations 共18兲 implies that 1 2 1 共1 + p兲 12 共1 − p2兲 . + − n2 = 2 2 共1 − p兲 共1 − p兲 共1 − p兲 2
共25兲
which is the required KdV equation and describes the evolution of the first order perturbed potential 共1兲. The coefficients a and b are given by
− V0
12 2 1 2 = 共1 + p 兲 + 共1 − p 兲 − 共1 − p兲n2 . 2 2 2
u2 =
共24兲
共23兲
a 3 u 2 − + 0 − 21 E20 . 6b 2b
共29兲
0 and E0 are, respectively, the charge density and electric field when vanishes, and u = u1 − c1 . A cnoidal wave solution of Eq. 共28兲 is given by
= ␣2 + 共␣1 − ␣2兲cn2共D,m兲,
共30兲
where cn is the Jacobian elliptical function. The parameter m, called modulus, and D can be written in terms of three real zeros of Sagdeev potential, ␣1, ␣2, and ␣3, as m2 =
␣2 − ␣1 ␣3 − ␣1
while the quantity D is obtained as
共31兲
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Phys. Plasmas 17, 102315 共2010兲
J. K. Chawla and M. K. Mishra
冑
共␣1 − ␣3兲a . 12b
共32兲
For the EPI plasma considered here, the velocity 共u1兲 of the cnoidal wave in the ion-acoustic wave frame is determined as a u1 = c1 + 共␣1 + ␣2 + ␣3兲. 3
共33兲
The ␣1, ␣2, and ␣3 are such that ␣1 ⬎ ␣2 ⬎ ␣3 for the EPI plasma; for which a ⬎ 0. The amplitude A of the ion-acoustic cnoidal wave is given by Eq. 共30兲 A = ␣1 − ␣2 .
共34兲
The wavelength of the cnoidal wave is defined as 共35兲
D = 2K共m兲,
where K共m兲 is the first kind of complete elliptic integral. The ␣1, ␣2, and ␣3 can also be expressed in terms of modulus m, the amplitude A of the cnoidal wave, by using the conservation condition of particle density
冕
共n − 1兲dn = 0.
共36兲
0
To evaluate Eq. 共36兲, we assume that n2 and highest order terms are very much less than n1. Therefore from Eqs. 共13兲, 共30兲, 共31兲, and 共36兲, we obtain u1 = c1 −
再
冎
Aa 1 关2 − H共m兲兴 − 1 , 3 m2
共37兲
where H共m兲 = E共m兲 / k共m兲 and E共m兲 is the complete integral of the second kind from Eq. 共37兲, we get 3共u1 − c1兲
A=
再冉 冊 1
a
m2
关2 − 3H共m兲 − 1兴
冎
共38兲
.
The frequency f = V / of the cnoidal wave can be given by Eqs. 共31兲, 共32兲, 共35兲, and 共36兲 as f=
冉 冊
aA V − 2K共m兲m 12b
1/2
共39兲
.
The velocity V of the cnoidal wave in the laboratory frame is given by V = V0 + u1, using Eqs. 共15兲 and 共37兲, we have V=
冉
1−p 1 + p
冊
1/2
+ c1 −
再
冎
Aa 1 关2 − H共m兲 − 1兴 . 3 m2
共40兲
In the limiting case, m = 1 共when E0 = 0 = 0兲, ␣2 = ␣3 = 0, ␣1 = −A, and cn = sec h. In this case, the cnoidal wave solution 共30兲 is reduced to the ion-acoustic soliton solution
= m sec h2共/␦兲,
共41兲
where the amplitude of the soliton m = 3u1 / a and width of the soliton ␦ = 共4b / u1兲1/2.
FIG. 1. Plot of Sagdeev potential V共兲 vs potential using Eq. 共22兲, having p = 0.1, = 0.1, u = 0.015, and E0 = 0.007, 0 = −0.002 共-------兲, and E0 = 0, 0 = 0 共—兲.
V. DISCUSSION
The coefficients a and b in the KdV Eq. 共25兲 are functions of p and , therefore from Eqs. 共26兲 and 共27兲, it may be noted that the amplitude and the frequency of the ionacoustic cnoidal wave are modified by the positron concentration and temperature ratio of electron to positron. From Eq. 共39兲, it may also be noted that the frequency of the cnoidal wave increases as the amplitude of the cnoidal wave increases. In Fig. 1, we have plotted the variation in Sagdeev potential V共兲 with respect to potential using Eq. 共29兲 for two sets of parameter values. The dashed curve in Fig. 1 shows the Sagdeev potential V共兲 corresponding to the cnoidal wave, whereas the continuous curve in Fig. 1 having E0 = 0 and 0 = 0 represents the Sagdeev potential V共兲 corresponding to a soliton structure. A comparison of these curves clearly shows that there is a drastic change in the variation of Sagdeev potential in two cases. In the case of cnoidal wave, the three real zeros of the Sagdeev potential are ␣1, ␣2, and ␣3 共␣1 ⬎ ␣2 ⬎ ␣3兲. It may be noted from the Fig. 1, that the Sagdeev potential corresponding to cnoidal wave does not become zero at = 0. On the other hand, in the case of soliton, the Sagdeev potential V共兲 becomes zero at = 0. In the case of cnoidal wave, physically one can say that this Sagdeev potential is such that the pseudoparticle oscillates periodically back and forth in the potential well between points ␣1 and ␣2 and cannot reach point ␣3 due to potential barrier. In physical space, the potential of the cnoidal wave oscillates between two upper values of the real zeros of the Sagdeev potential ␣1 and ␣2, as shown by dashed curve of Fig. 1. This implies that the potential structure is repeated and the distance between repetitions of the wave shape corresponds to one wavelength. In Fig. 2 curves show the variation of Sagdeev potential V共兲 with potential for the fixed values of parameters 共, u, E0, and 0兲 as taken in Fig. 1, but having different value of positron concentration 共p兲. A comparison of Figs. 1 and 2 shows that for
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Phys. Plasmas 17, 102315 共2010兲
Ion-acoustic nonlinear periodic waves…
FIG. 2. Plot of Sagdeev potential V共兲 vs potential using Eq. 共22兲, having p = 0.2, = 0.1, u = 0.015, and E0 = 0.007, 0 = −0.002 共-------兲, and E0 = 0, 0 = 0 共—兲.
the fixed values of parameters 共, u, E0, and 0兲, if we increase the positron concentration 共p兲, the amplitude of the cnoidal wave slightly increases. In Fig. 3, we have plotted the phase curves using Eqs. 共28兲 and 共29兲. The dashed curve in Fig. 3 represents the phase curve for the cnoidal wave solution. In the dashed phase curve of Fig. 3, it may be noted that when 兩E0兩, 0 ⫽ 0; the phase curve is repeated on the same path and one complete cycle corresponds to a wavelength in the physical space. This implies that whenever the pseudoparticle’s velocity becomes zero 共i.e., 共d / d兲 = 0兲, the potential force reflects it back since −关dV共兲 / d兴 does not vanish and therefore it oscillates between two points. Therefore, the potential of the cnoidal wave oscillates between the two upper values of the real zeros of the Sagdeev potential 共see Figs. 1 and 2兲. The continuous curve in Fig. 3 with E0 = 0 and 0 = 0 represents the phase curve for the soliton. This curve clearly
FIG. 4. Phase curves using Eqs. 共21兲 and 共22兲, having p = 0.2, = 0.1, u = 0.015, and E0 = 0.007, 0 = −0.002 共-------兲, and E0 = 0, 0 = 0 共—兲.
shows that it starts from the origin, circling anticlockwise around the positive axis, and again stops at the origin, entering from the upper side. In Fig. 4, we have plotted the phase curve for the fixed values of parameters 共, u, E0, and 0兲 as taken in Fig. 3 but having different value of positron concentration 共p兲 with p = 0.2. VI. CONCLUSIONS
We have studied nonlinear periodic waves, namely, KdV cnoidal waves, in EPI plasma. In the limiting case, the ionacoustic cnoidal waves reduce to ion-acoustic soliton. It is also found that the frequency 共f兲 of the ion-acoustic cnoidal wave depends on its amplitude. As the amplitude of the cnoidal wave increases, its frequency also increases. We found that the positron concentration affects the characteristic of the cnoidal wave. It is also found that for a given set of parameter values, as we increase the positron concentration, amplitude of the cnoidal wave increases. The present investigation predicts findings on the existence of the nonlinear periodic waves, namely, ion-acoustic cnoidal waves as well as localized solitary structures in the presence of cold ions, hot positrons, and electrons. In actual situations, ion-acoustic wave events associated with positrons are frequently observed in the solar atmospheres. Hence, referring to the present studies, we think that these results will be helpful in understanding nonlinear periodic waves associated with ion-acoustic waves in space where the positrons exist. Although, we have not been referring to any specific observations, the present theory is applicable to analyzing nonlinear periodic waves, namely, ion-acoustic cnoidal waves associated with positron which may occur in space, active galactic nuclei, and solar atmosphere. W. Misner, K. S. Throne, and J. A. Wheeler, Gravitation 共Freeman, San Francisco, 1973兲, p. 763. M. J. Rees, in The Very Early Universe, edited by G. W. Gibbons, S. W. Hawking, and S. Siklas 共Cambridge University Press, Cambridge, 1983兲. 3 H. R. Miller and P. J. Witta, Active Galactic Nuclei 共Springer-Verlag, Berlin, 1987兲, p. 202. 1
2
FIG. 3. Phase curves using Eqs. 共21兲 and 共22兲, having p = 0.1, = 0.1, u = 0.015, and E0 = 0.007, 0 = −0.002 共-------兲, and E0 = 0, 0 = 0 共—兲.
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