Ion-acoustic dressed solitons in electron–positron–ion plasmas

Physics Letters A 372 (2008) 3461–3466 www.elsevier.com/locate/pla

Ion-acoustic dressed solitons in electron–positron–ion plasmas R.S. Tiwari Department of Physics, University of Rajasthan, Jaipur-302004, India Received 25 May 2007; received in revised form 4 December 2007; accepted 4 February 2008 Available online 9 February 2008 Communicated by F. Porcelli

Abstract Expanding the Sagdeev potential to include fourth-order nonlinearities of electric potential and integrating the resulting energy equation, an exact soliton solution is determined for ion-acoustic waves in an electron–positron–ion (e–p–i) plasma system. This exact solution reduces to the dressed soliton solution obtained for the system using renormalization procedure in the reductive perturbation method (RPM), when Mach number (M) is expanded in terms of soliton velocity (λ) and terms up to order of λ2 are retained in the analysis. Variation of shape, velocity, width and product (P ) of amplitude (A) and square of width (W 2 ) for the KdV soliton, core structure, dressed soliton, and exact soliton are graphically represented for different values of fractional positron concentration (p). It is found that for a given value of the fractional positron concentration (p) and amplitude of soliton, the velocity of the dressed soliton is faster and width is narrower than the KdV or exact soliton, and agrees qualitatively with the experimental observations of Ikezi et al. for small amplitude solitons in the plasma free from positron component. Among all these structures, the product P (AW 2 ) is found to be lowest for the dressed soliton and it decreases as Mach number of soliton or fractional positron concentration in the plasma increases. © 2008 Elsevier B.V. All rights reserved. PACS: 52.27.-h; 52.35.-g

During the last decade the study of nonlinear wave phenomena in field free and magnetized electron–positron–ion (e–p–i) plasmas remained a subject of great interest [1–11]. Recent techniques in positron trapping have lead to room temperature plasmas of 107 positrons with a half life of 103 seconds [12], and has opened a new series of experiments ranging from electron–positron plasmas to positron–molecule interactions. Three components e–p–i plasma can also be created in laboratory and over a wide range of parameters, annihilation of electrons and positrons, is relatively unimportant [13]. At low temperatures of the order of 1 eV and electron density of 1012 cm−3 , the observed positron annihilation time is greater than 1 second, which is much larger than the characteristic time scale for the ion-acoustic wave. Most of the astrophysical and laboratory plasmas [12] as well as magnetospheres of neutron stars may have electrons, positrons, and ions as their constituents. Properties of wave

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motion in such plasmas are very much different from those in electron–positron plasmas [2,3]. Presence of positron component in conventional e–i plasma reduces the number density of ions and restoring force on electron fluid and hence the characteristics of linear waves as well as that of nonlinear structures are found to change considerably. For example, the small amplitude double layer [5] associated with the kinetic Alfven wave exists only when positron density at equilibrium is appreciably smaller than the ion density. Investigation of ion-acoustic envelope solitons in e–p–i plasmas [6] shows that critical wave number of modulational instability depends on relative concentration of positrons and ions present in the plasma and temperature ratio of electrons and positrons. Large amplitude ion-acoustic solitons in e–p–i plasmas, using the Sagdeev potential approach, have been studied by Popel et al. [1] and found that presence of positron component in a plasma reduces the amplitude of ion-acoustic soliton. Study of large amplitude solitons in magnetized e–p–i plasmas [8] predicts that presence of positron component increases the amplitude of solitary structures. Using numerical integration in the

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Sagdeev potential approach, large amplitude weakly dissipative dust-acoustic solitons are also studied by Popel et al. [14]. However, the analytical solution is only possible for small amplitude solitons, using either the Sagdeev potential approach or the reductive perturbation method (RPM) and up to the KdV soliton description both the approaches give rise the same soliton solution. In a recent investigation [15], using renormalization procedure in the RPM, we have studied the effect of positron component on characteristics of ion-acoustic dressed soliton in electron–ion plasmas and found that the contribution of higher order perturbations to the KdV soliton description considerably modifies the velocity and width of the soliton. The aim of the present Letter is to study the small amplitude ion-acoustic solitons in e–p–i plasma system including the effect of fourth order nonlinearities of electric potential in Sagdeev potential approach and to compare the results with those obtained by renormalization procedure in the RPM. We extend the work of Popel et al. [1] to include the effect of fourthorder nonlinearity of electric potential in the expansion of the Sagdeev potential and integrate the resulting energy equation with proper boundary condition required for a soliton structure. This procedure gives soliton solution in terms of coefficients of nonlinearities of second order (α1 ), third order (α2 ) and fourth order (α3 ) of electric potential in the expansion of the Sagdeev potential. Assuming, α3 < α2 and using the Taylor’s series expansion of Mach number in terms of soliton velocity (λ) and keeping the terms up to order of λ2 , this exact solution reduces to nonsecular dressed soliton solution [15]. Here we consider a collision less, un-magnetized, three components plasma consisting of electrons, positrons and singly charged ion species. The propagation of one-dimensional nonlinear ion-acoustic wave is governed by the following normalized basic equations [15]: ∂ ∂n + (nv) = 0, ∂t ∂x ∂v 1 ∂φ ∂v +v =− , ∂t ∂x β ∂x ∂ 2φ = eφ − pe−σ φ − (1 − p)n. ∂x 2

(1) (2) (3)

Here n, ne and np are respectively the number densities of ions, electrons and positrons in the plasma, normalized with respect to equilibrium electron density (ne0 ). v is the ion fluid velocity normalized with respect to ion sound speed, Cs = (βkTe e/m)1/2 , where β is determined from the linear dispersion relation for the ion-acoustic wave and is given by β = 1−p 1+pσ . The electric potential (φ) and the space variable (x) are normalized respectively in terms of characteristic potential 2 n , is the kTe /e, and the Debye length λD = ( εn0eokTe e )1/2 . p = npo eo fractional concentration of positron component in the plasma and σ = Te /Tp is the temperature ratio of electron and positron fluids. The parameter β introduced here, simplifies the comparison procedure of the soliton solution obtained by Sagdeev potential approach and the nonsecular dressed soliton solution determined by using the renormalization procedure in the RPM.

To obtain stationary soliton solution, from the basic equations (1)–(3), we introduce the usual transformation η = x − Mt.

(4)

Here M is the Mach number of soliton with respect to ionacoustic wave frame for the system. Following the same procedure as mentioned by Popel et al. [1], the energy equation for pseudo-particle in potential well can be written as   1 dφ 2 + ψ = 0, (5) 2 dη where the Sagdeev potential (ψ) is given by  p ψ = 1 − eφ + 1 − e−σ φ σ   1  2φ 2 + βM 2 (1 − p) 1 − 1 − . βM 2

(6)

Using the Taylor series expansion and keeping terms up to fourth-order nonlinearities of electric potential (φ) in (6), the energy equation (5) for the pseudo-particle can be expressed in the following form  2 dφ = α1 φ 2 − α2 φ 3 + α3 φ 4 . (7) dη Here (1 − p) , βM 2 pσ 2 − 1 1 − p + 2 4, α2 = 3 β M 3 pσ + 1 5 (1 − p) α3 = . − 12 4 β 3M 6 α1 = 1 + pσ −

(8a) (8b) (8c)

Integration of (7) with respect to η gives stationary soliton solution as 2(α1 /α2 ) , φ= (9a) 4α1 α3 1/2 (1 − 2 ) (2 cos h2 (Dη) − 1) + 1 α2

where  1/2 α1 D= . 4

(9b)

The soliton solution (9a) is similar to (26) of our earlier investigation [18] for an ion beam plasma system with isothermal electrons. Although, numerical analysis of solution (26) and dressed soliton solution (21), obtained for an ion beam plasma system [18], produced approximately the same results, but in that investigation, we could not analytically equate both the solutions. Moreover, such an analysis is not available in the literature. Hence, it may be of interest to equate analytically the soliton solution given by Sagdeev potential approach including the effect of fourth-order nonlinearity of electric potential, with the dressed soliton solution obtained by using renormalization in the RPM. Use of expansion of Sagdeev potential (ψ) in terms of the electric potential (φ) in (7), limits our analysis to small amplitude soliton only and this limit also exists when the RPM is used

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for the analysis. To compare (9a) with the dressed soliton solution (24), obtained in [15], we expand the Mach number (M) in terms of soliton velocity (λ) as M = 1 + λ in (8a)–(8c), and retain terms in the expansion such that each term on R.H.S of (7) is of fourth order in combined nonlinearities of λ and φ. This procedure yields    1  3 2 2 λ− λ , α1 = (1 + pσ ) 2λ − 3λ = (10a) b 2   (1 − p) 2aβ − 4λ , α2 = (10b) 3 β2 (1 − p) α3 = (10c) R, 12β 3 where   1 3(1 + pσ ) pσ 2 − 1 + , a= 2 1−p 1 + pσ

b=

1 2(1 + pσ )

and R=

(1 + pσ 3 ) (1 − p)2 − 15. (1 + pσ )3

Here a and b are respectively the coefficients of nonlinear and dispersive terms of the KdV equation, which can be obtained from the basic equations (1)–(3) by using the RPM. Comparison of the coefficients a and b with those obtained in [15], shows that the frame velocity (V ) does not appear here. This is due to the introduction of the parameter β, which normalizes the ion fluid velocity with respect to e–p–i plasma instead of e–i plasma. To compare the exact soliton solution (9a) with nonsecular dressed soliton solution [15] for an e–p–i plasma system, we have to retain terms up to the order of λ2 in (9a) and use (10a)– (10c) to write α1 3λ 9λ2 18λ2 = − + 2 , α2 a 2a a β   α3 R 6λ = , 1 + α2 8aβ 2 aβ   3 λR α1 α3 1/2 =1− + ···. 1−4 2 4 a2β 2 α2

(11a) (11b) (11c)

Substituting (11a) and (11c) in (9a) and keeping terms up to order of λ2 in the expansion, we can write the soliton solution as   3λ 9λ2 18λ2 9 λ2 R ˜ φ= − + 2 + sec h2 (Dη) a 2a 8 a3β 2 a β +

9 λ2 R ˜ tan h2 Dη. ˜ sec h2 Dη 8 a3β 2

(12a)

Including the contribution of λ2 term in (9b), we can express D˜ as 2   1/2  λ − 3λ2 1/2 α1 D˜ = (12b) = . 4 4b

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Keeping terms of order λ only soliton solution (12a) reduces to  1/2 λ 3λ , sec h2 Dη, D = φ= (13) a 4b which is the same as obtained by Popel et al. [1] and Nejoh [4], when their solutions are expressed in dimensionless form. The dressed soliton solution (12a) is exactly the same as one would obtain from the basic equations (1)–(3), employing the renormalization procedure in RPM, using the stretched co-ordinates in the analysis as ξ = ε 1/2 (x − t) and τ = ε 3/2 t . We may also use λ → Vλ , a → Va and β = V 2 in (12a) to obtain (24) of our previous investigation [15]. For conventional e–i plasma dressed soliton solution given by (12a) exactly matches with those obtained by Kodama and Taniuti [16] or Sugimoto and Kakutani [22] and for e–p–i plasma system with [15]. Hence, nonsecular dressed soliton solution can also be obtained including the effect of fourth-order nonlinearities in the expansion of Sagdeev potential for the system under consideration and does not need renormalization. This is because, when Sagdeev potential approach is used to determine a soliton solution, the boundary conditions initially imposed on dependent variables remain valid for all orders of expansion but in the RPM, we initially expand the dependent variables in power series of a small parameter and integrate the basic equations for each order of expansion of the dependent variables. This procedure produces secular behavior when solutions of the first-order variables are introduced in the evolution equation of the second-order variable to determine it and so on. Therefore, the renormalization procedure is needed for each order of expansion in the RPM. To remove the secular behavior for each order of expansion, shift in soliton velocity is introduced in the renormalization procedure, which changes the amplitude and width of the soliton, whereas Sagdeev potential approach gives nonsecular dressed soliton solution and higher order corrections in amplitude or width of the soliton appears automatically as a result of expansion of the Mach number in terms of λ. Following the analysis [22] for e–i plasma, we denote here the first and second terms of (12a) as   3λ 9λ2 18λ2 9 λ2 R ˜ − + 2 + sec h2 (Dη), φcore = (14a) a 2a 8 a3β 2 a β φcloud =

9 λ2 R ˜ tan h2 Dη. ˜ sec h2 Dη 8 a3β 2

(14b)

As positron concentration in plasma tends to zero, (14a) and (14b) reduce to that obtained in [22]. The contribution of cloud part to amplitude of the dressed soliton is zero and it moves stably with the core structure [24], hence its negative contribution to the core structure decreases the width of dressed soliton, which becomes less than that of the core structure or the KdV soliton. The effect of positron concentration on shape, amplitude, width and velocity of the KdV soliton, core structure, dressed soliton and exact soliton is discussed using the following figures. Fig. 1 represents the KdV soliton {φk (p)}, core {φc (p)} and cloud {φcloud (p)} structures, dressed soliton {φd (p)} and

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Fig. 1. Variation of the shape of the KdV soliton {φk (p)}, core {φcore (p)} and cloud {φcloud (p)} structures, dressed soliton {φd (p)} and exact soliton {φe (p)} with the variable η for p = 0, 0.10, σ = 1 and M = 1.2.

the small amplitude exact soliton solution {φe (p)} given by (13), (14a), (14b), (12a) and (9a) respectively, for the fractional positron concentration p = 0, 0.10, σ = 1 and Mach number (M = 1 + λ) = 1.2. Here we have used the given value of the Mach number of the soliton to calculate amplitude and width of the exact soliton but for the KdV soliton, core structure and dressed soliton corresponding value of λ is used for the numerical analysis. From Fig. 1, we find that as fractional concentration of positron component in the plasma increases, the amplitudes of these structures decreases, which agrees with the analysis of Popel et al. [1]. It can be noted from this figure that for a given amplitude, the width of the dressed soliton at half maxima is narrower than that of core structure for p = 0 and also for p = 0.10. As positron concentration in the plasma increases contribution of the cloud structure decreases. We also note that for the given value of the Mach number and positron concentration, the amplitude of the exact soliton is larger than the KdV soliton as well as that of dressed soliton. Hence for given amplitude of the soliton the velocity of the exact soliton should be small as compared with the KdV soliton or dressed soliton. Using (13) and (14a), we denote the half widths (W ) of the KdV soliton and core structure as: WKdV = D −1 and Wcore = D˜ −1 , which are at 0.4199 of their respective amplitudes. We have also calculated the half widths of the exact soliton (We ) and dressed soliton (Wd ) at 0.4199 of their amplitudes, using (9a) and (12a) respectively, to show their variation with respect to amplitude of the soliton and positron concentration in the plasma. Fig. 2 shows the variation of velocity of KdV soliton {λk (p)}, dressed soliton {λd (p)}, exact soliton {λe (p)} and width of the KdV soliton {Wk (p)}, core structure {Wc (p)}, dressed soliton {Wd (p)} and the small amplitude exact soliton {We (p)} as a function of its amplitude for different values of positron concentration p = 0, 0.20, 0.40 and σ = 1. We find

Fig. 2. Variation of velocity and width of the KdV soliton {λk (p), Wk (p)}, core structure {λc (p), Wc (p)}, dressed soliton {λd (p) = λc (p), Wd (p)} and exact soliton {λe (p), We (p)} for different values of fractional concentration of positron p = 0, 0.2, 0.4 and temperature ratio σ = 1.

that widths of these structures decrease as amplitude or positron concentration in plasma increases. We also note that for a given value of amplitude and positron concentration, the velocity of the dressed soliton is faster and its width is narrower than the

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KdV or exact soliton. For amplitude >0.1 and p = 0, the width of the exact soliton is less than the KdV soliton but its velocity is also less than that of the KdV soliton and does not match with experimental observations [23]. However, using the expansion of the Mach number in terms of soliton velocity in our exact soliton solution (9a), it reduces to dressed soliton solution, which agrees qualitatively with the experimental observations [23] for plasmas free from positron component. From (13), we can note that the product P = AW 2 for the KdV soliton is independent of amplitude or velocity and depends only on the parameters of the plasma. Fig. 3 represents the variation of the product P = AW 2 with Mach number (M) for the KdV soliton {Pk (p)}, core structure {Pc (p)}, dressed soliton {Pd (p)} and exact soliton {Pe (p)} for different values of fractional positron concentration p = 0, 0.05, 0.20, and 0.40 with σ = 1. For a given value of positron concentration (p) and temperature ratio (σ ), this product remains constant for the KdV soliton but increases continuously for the core structure and decreases for the dressed soliton. For the exact soliton this product initially decreases and then start to increase and becomes more than that for the KdV soliton. The product P , for exact soliton remains less than that for the core structure and it has lowest value for the dressed soliton. In Fig. 4, we have shown the variation of the product AW 2 for these structures with fractional positron concentration (p) for M = 1.2 and 1.3, having σ = 1. Since this product for the KdV soliton does not depend on the Mach number (M) or soliton velocity (λ), hence for both the values of Mach number (M), we are having a single curve. For a given value of the Mach number (M), as positron concentration increases, the product P for the KdV soliton, core structure and exact soliton decreases but for the dressed soliton

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Fig. 3. Variation of the product of P = AW 2 for the KdV soliton {Pk (p)}, core structure {Pc (p)}, dressed soliton {Pd (p)} and exact soliton {Pe (p)} with the Mach number of soliton for different fractional concentrations p = 0, 0.05, 0.20 and 0.40 with σ = 1.

Fig. 4. Variation of the product P = AW 2 for the KdV soliton {Pk (M)}, core structure {Pc (M)}, dressed soliton {Pd (M)} and exact soliton {Pe (M)} with positron concentration for the Mach number M = 1.2 and 1.3 with σ = 1.

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it increases initially with the increase in positron concentration and then starts to decrease. For a given fractional positron concentration as the Mach number increases the product P for core and exact soliton increases but for dressed soliton it decreases. A similar analysis can also be used to determine the dressed soliton solution including the effect of higher order nonlinearities in the expansion of Sagdeev potential for a dusty plasma, ion beam plasma and electron beam plasma systems. The determined ion-acoustic dressed soliton solutions exactly match with those obtained by using renormalization procedure in the RPM for a dusty plasmas [17], ion beam plasma system with isothermal electrons [18], nonisothermal electrons [19] and also for electron acoustic soliton in an electron beam plasma system [20]. The use of the Sagdeev potential technique to include higher order perturbation correction to characteristics of ion-acoustic soliton in an ion beam plasma system with nonisothermal electrons [19], gives the soliton solution in the form of square of R.H.S. of (9a) and neglecting the contribution of higher order perturbations, it reduces to the KdV soliton solution obtained by Abrol and Tagare [21]. In summary, it can be mentioned that Sagdeev potential approach also gives the same dressed soliton solution for an e–p–i plasma system as can be obtained by using renormalization procedure in the RPM. For the given values of fractional positron concentration and amplitude of soliton, velocity of dressed soliton is faster and its width is narrower as compared with the corresponding quantities for the KdV or exact soliton. The product AW 2 (where W is calculated at 0.4199 of soliton’s amplitude (A)) is also lowest for dressed soliton and decreases as Mach number or positron concentration in the plasma increases. The dressed soliton solution, obtained using the expansion of Mach number in terms of soliton velocity in our exact soliton solution (9a), resembles qualitatively with the experimental observations [23] for plasmas free from positron component.

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