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Electromagnetic solitary structures in dense electron–positron–ion magnetoplasmas
This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2010 Phys. Scr. 82 065508 (http://iopscience.iop.org/1402-4896/82/6/065508) View the table of contents for this issue, or go to the journal homepage for more
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PHYSICA SCRIPTA
Phys. Scr. 82 (2010) 065508 (9pp)
doi:10.1088/0031-8949/82/06/065508
Electromagnetic solitary structures in dense electron–positron–ion magnetoplasmas W Masood1,2 , S Hussain1,3 , H Rizvi1 , A Mushtaq1,4 and M Ayub5 1
TPPD, PINSTECH, PO Nilore, Islamabad, Pakistan National Center for Physics (NCP), Pakistan 3 DPAM, PIEAS, PO Nilore, Islamabad, Pakistan 4 School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia 5 Government College University (GCU), Lahore, Pakistan 2
E-mail:
[email protected] (W Masood)
Received 18 August 2010 Accepted for publication 7 October 2010 Published 29 November 2010 Online at stacks.iop.org/PhysScr/82/065508 Abstract The linear and nonlinear propagation characteristics of low-frequency obliquely propagating magnetoacoustic waves in dense electron–positron–ion magnetoplasmas are studied in this paper by using the quantum magnetohydrodynamic (QMHD) model. A quantum Kadomtsev–Petviashvili (KP) equation is derived by using the reductive perturbation technique. The dependence of the fast and slow magnetoacoustic solitary waves on the positron concentration, the obliqueness parameter θ and the magnetic field is also investigated. The present investigation may have relevance to dense astrophysical environments where the quantum effects are expected to dominate. PACS numbers: 52.25.Xz, 52.27.Ep, 52.35.Sb, 52.35.Fp (Some figures in this article are in colour only in the electronic version.)
Dirac–Maxwell approaches, which govern the statistical and hydrodynamic behaviors of plasma particles at quantum scales. These quantum models are equivalent to the fluid and kinetic models of classical plasma physics. The basic ingredients of the quantum hydrodynamic (QHD) model are derived from the classical fluid concept of plasmas, and the resulting transport equations are expressed in terms of conservation laws for particles, momentum and energy. In the quantum hydrodynamic model, an additional term, the so-called ‘Bohm potential’, is introduced into the equation of motion of charged particles, which allows particle tunneling through potential barriers. The classical fluid equations are retrieved in the limit that the quantum effects go to zero in conformity with Bohr’s correspondence principle. The fundamental modes of a plasma in an obliquely propagating magnetic field are the fast and slow magnetoacoustic waves from the magnetohydrodynamics (MHD) standpoint, besides the Alfvén wave that is obtained along with them. Nonlinear magnetoacoustic waves have been the subject of investigation of several authors on account of
1. Introduction Quantum plasmas have attracted significant interest on account of their applications in several physical systems such as dusty plasmas [1, 2], dense astrophysical environments [3] (e.g. white dwarfs and neutron stars), microelectronic devices [4], laser-produced plasmas [5] and nonlinear quantum optics [6, 7]. It is well known that classical plasma, which is characterized by high temperature and low density, becomes a dense plasma because the plasma condenses at sufficiently low temperatures. When this happens, the distance between two nearest-neighbor plasma particles becomes comparable to or smaller than the de Broglie wavelength associated with the particles. Under these circumstances, the plasma behaves like a Fermi gas, and it becomes imperative to include the quantum mechanical effects in the study of the behavior of charged particles [8–11]. Several approaches have been used to study quantum plasmas, among which the ones most commonly used are the Schrödinger–Poisson, the Wigner–Poisson and the 0031-8949/10/065508+09$30.00
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© 2010 The Royal Swedish Academy of Sciences
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their significance for space and fusion plasmas, where they are used in particle acceleration and heating experiments [12–16]. Low-frequency Alfvén and magnetosonic waves have been suggested to be responsible for ion acceleration in quasi-parallel shocks [17] and form a part of the shock front structure itself [18, 19]. The magnetoacoustic waves are also believed to be the precursors that steepen to form a shock front in quasi-perpendicular shocks such as the Earth’s bow shock [20]. Low-frequency waves are widely present in solar wind and were found to contain a considerable part of the shock energy [21, 22]. De Vito and Pantano [23] studied the propagation of two-dimensional nonlinear magnetoacoustic waves in a cold plasma and showed that the propagation characteristics of these waves are governed by the Kadomtsev–Petviashvili (KP) equation. Later, Shah and Bruno [24] extended De Vito and Pantano’s work in a warm plasma and showed that the propagation of both slow and fast modes was also governed by the same equation. The electron–positron (e–p) plasmas have been found to manifest different behavior as opposed to typical electron–ion (e–i) plasmas [25, 26]. An interesting feature of e–p plasma vis-à-vis the usual e–i plasma is that it has the same mass and charge of constituents as an e–p plasma. The e–p plasmas are considered to be the dominant constituent in the pulsar magnetosphere [27]. They could also be the major component of the relativistic jets that stream from the nuclei of quasars and active galaxies. They are believed to have existed in the early universe [28] and are also found in the center of our own galaxy [29]. Since in many astrophysical environments, there exist a small number of ions along with the electrons and positrons, it is important to study the linear and nonlinear behavior of plasma waves in an electron–positron–ion (e–p–i) plasma. Much research has been carried out to study e–p and e–p–i plasmas over the last few years [30–36]. For instance, Berezhiani and Mahajan [33] and Mahajan et al [34] have analytically shown that large-amplitude localized structures can be generated in an unmagnetized e–p plasma with a small fraction of ions. Popel et al [35] studied ion-acoustic solitons in e–p–i plasma and showed that the presence of positrons reduced the amplitude of ion-acoustic solitons. Nejoh [36] explored the effect of ion temperature on the large-amplitude ion-acoustic waves in e–p–i plasma and observed that ion temperature decreased the amplitude and increased the maximum Mach number of an ion-acoustic wave. Hasegawa et al [37] investigated the waves propagating perpendicular to the magnetic field theoretically as well as numerically and found that the low-frequency mode solitary pulse can emit high-frequency mode solitons if the amplitude of the original pulse is large and the ion density is low. Very recently, Sabry et al [38] investigated the nonlinear wave modulation of planar and nonplanar cylindrical and spherical ion-acoustic envelope solitons in a collisionless, unmagnetized e–p–i plasma with two-electron temperature distributions by employing the reductive perturbation method. These authors reported a modulation instability period for the cylindrical and spherical wave modulation that was nonexistent in the planar geometry. Nonlinear electromagnetic waves have been paid relatively less attention in comparison with the electrostatic waves in dense plasmas. Masood and Mushtaq [39]
investigated the linear properties of obliquely propagating magnetoacoustic waves (both fast and slow) in multicomponent e–p–i and dust–electron–ion (d–e–i) quantum magnetoplasma following Haas’s quantum magnetohydrodynamic (QMHD) model [40] and found that the propagation characteristics of fast and slow magnetoacoustic waves in both of these multi-component plasmas were significantly modified due to the quantum diffraction term. The major purpose of both [39] and [40] was to highlight the significance of the quantum Bohm potential term, and the classical expression for pressure was used, which in itself was a very simplifying approximation. Later, Masood et al [41] showed that, in general, the quantum statistical term dominates the quantum Bohm potential term and both the effects should be incorporated into the study of the quantum behavior of the system. Masood [42] incorporated all these considerations into his work on nonlinear magnetoacoustic shock waves and studied the linear and nonlinear propagation characteristics of low-frequency magnetoacoustic waves in e–i quantum magnetoplasmas and found that the quantum Bohm potential by increasing the number density, obliqueness angle θ, magnetic field and resistivity affected the shock structure of the magnetoacoustic wave potential. In all these works, the spin effects were ignored using the assumptions spelled out unequivocally in this paper earlier. Recently, the spin effects have been included in studies by Marklund et al [43] and Brodin and Marklund [44]. These authors have studied the nonlinear propagation characteristics of small-amplitude electromagnetic spin solitary structures and highlighted the effects brought about by the inclusion of spin effects. In this paper, we study the obliquely propagating two-dimensional magnetoacoustic waves in an e–p–i quantum magnetoplasma. The linear dispersion relation and the nonlinear KP equation are derived by using the small-amplitude expansion method. The organization of the paper is as follows. In section 2, the governing equations and the linear dispersion relation for the system under consideration are presented. The nonlinear KP equation is derived in section 3 and its solution is presented using the tanh method. In section 4, the numerical results are presented and discussed. Finally, in section 5, we recapitulate the main findings of this paper.
2. Basic equations and formulation We present here a linear and nonlinear investigation of the magnetoacoustic waves in an e–p–i quantum magnetoplasma. We consider a Cartesian system in which the background magnetic field B0 lies in the (x, y) plane making a small angle θ with the x-axis, and the propagation in the nonlinear regime is considered in the (x, z) plane. The basic set of equations used in this paper is the one-fluid QMHD equations. The one-fluid QMHD model for e–p–i plasma can be developed by starting with the usual QHD fluid equations for electrons, positrons and ions with both statistical and diffraction terms. We treat ions as classical particles since they are three orders of magnitude more massive than electrons and positrons. We ignore the spin effects of charged particles assuming that µB B0 K B T , where µB = eh¯ /2mc is the magnitude of the Bohr magneton and h¯ is Planck’s constant divided by 2π. 2
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It has been theoretically shown by Brodin et al [45] that while studying the electromagnetic perturbations, an additional condition i.e. µB B0 m i V A2 needs to be met besides µB B0 K B T in order to ignore the electron spin effects. The basic set of equations of the effective one-fluid QMHD model for e–p–i plasmas is then given by dvi eE e = + (vi × B) , dt mi mic 2√ ∇ ne e ∇ Pe h¯ 2 0 = −eE − (ve × B) − + ∇ √ , c ne 2m e ne " √ # ∇ Pp ∇ 2 np e h¯ 2 0 = eE + vp × B − + ∇ . √ c np 2m p np
Using equations (1), (7) and the quasi-neutrality condition n i + n p ' n e , we obtain the following normalized effective one-fluid momentum equation: β (1 + σ ) p n p 2/3 dvi 1 n e 2/3 = ∇n p − β ∇n i (∇ × B) × B − dt ni 1 − p ni ni
(1)
+
(11) √ √ where p = n p0 /n i0 , Vqs = K B TFe /m i , V A = B0 / 4πm i n i0 , β = Vqs2 /V A2 (the ratio of Fermi to magnetic pressure), q He = h¯ 2 2i /m e m i V A4 is a dimensionless parameter that measures the strength of quantum diffraction effects and i is the ion gyrofrequency. Now, on eliminating E between equations (1) and (8), the normalized magnetic field induction equation reads as ∂B dvi , = ∇ × (vi × B) − ∇ × (12) ∂t dt
(2)
(3)
In equations (2) and (3), two quantum effects, namely quantum diffraction and quantum statistics, are included. The quantum diffraction effects appear because of the wave nature of particles in a quantum plasma, which is taken into account by the term proportional to h¯ 2 , also known as the Bohm potential. However, quantum pressure is obtained by using quantum statistics which takes into account the fermionic nature of electrons. For a three-dimensional Fermi gas, electron and positron pressures are defined by [46]
and the normalized ion continuity equation is ∂n i + ∇ · (n i vi ) = 0. ∂t
2 2/3
Pe =
h¯ 2 3π 5m e
n e5/3 ,
h¯ 2 3π 2 Pp = 5m p
(13)
The following normalizations for the QMHD equations have been used here:
(4)
2/3 n 5/3 p .
2√ 2√ ∇ ne ∇ ne He2 n e p He2 n p ∇ √ ∇ √ + , 1 − p ni ne 1 − p ni ne
r=
(5)
r i , VA
vi =
vi , VA
B = B/B0 ,
t = i t,
n = n/n 0 .
The ion continuity equation is ∂n i + ∇ · (n i vi ) = 0 ∂t
3. Derivation of the KP equation (6)
By following the standard procedure of Washimi and Tanuiti [47], we introduce the following stretched variables:
and Maxwell’s equations are ∇ ×B =
4π j, c
(7)
1 ∂B c ∂t
(8)
qs n s vs .
(9)
∇ ×E = − X
χ = z,
(15)
where is a small expansion parameter and λ is the phase velocity of the wave. We expand the field quantities in terms of the smallness parameter . For n, vix , viy , Bx and B y , we have the following form: 1 n1 n2 n vix u1 u2 0 2 viy (16) = 0 + v1 + v2 + · · · , B x 0 0 cos θ By sin θ B y1 B y2
s
Here, vi is the ion fluid velocity, whereas n s (n so ) is the perturbed (unperturbed) particle density of s species. s = e, p, i stands for electrons, positrons and ions, respectively. E is the electric field vector, B is the magnetic field vector, j is the current density and (d/dt) = (∂/∂t) + (vi · ∇) is the hydrodynamic derivative. To derive the basic governing equations of the QMHD model, we substitute ve from equation (9) into equation (2) to obtain 1 c n i vi − ∇ ×B ×B en e c 4π 2√ 2 ∇ ne ∇ Pe h¯ − + ∇ √ . n e e 2em e ne
(14)
τ = 3/2 t,
and j=
ξ = 1/2 (x − λt),
and for viz and Bz viz w1 w2 = 3/2 + 5/2 . Bz Bz1 Bz2
E=−
(10) 3
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p
p
Figure 1. (a) Variation of the linear dispersion propagation characteristics of fast and slow magnetoacoustic waves with obliqueness θ . The other parameters are n e0 = 1.5 × 1026 cm−3 , p = 0.7 and B0 = 1010 G. (b) Variation of the linear dispersion propagation characteristics of fast and slow magnetoacoustic waves with positron concentration. The other parameters are n e0 = 1.5 × 1026 cm−3 , θ = 15◦ and B0 = 1010 G.
where β1 = β(1 + σ ) p/(1 − p). Equation (19) describes the linear dispersion characteristics of magnetoacoustic waves propagating obliquely making an angle θ with the external magnetic field in a quantum e–p–i plasma. In expression (19), the positive sign corresponds to the fast quantum magnetoacoustic mode, whereas the negative sign corresponds to the slow quantum magnetoacoustic mode. Figure 1(a) shows the variation of the fast and slow quantum magnetoacoustic modes as a function of the obliqueness parameter θ. It is observed that the fast mode increases and the slow mode decreases with increasing obliqueness. Figure 1(b) depicts the variation of the fast and slow modes with increasing positron concentration. It is found that the fast mode increases and the slow mode decreases with increasing positron concentration. This is because the slow mode has a predominantly acoustic character (as it turns into acoustic mode at θ = 0◦ ), whereas the fast mode, which has both acoustic and Alfvénic character (the dispersion relation for the pure fast magnetoacoustic mode at θ = 90◦ has both acoustic and Alfvénic contributions), increases with increasing positron concentration, though the increase is not very substantial. It is worth mentioning here that the present results are discussed for low β plasmas and hence the observed effects at the pure fast mode in the case of low β plasmas has a predominantly Alfvénic character.
Substituting equations (14) and (16) into equations (11), (13) and collecting terms of lowest order in , i.e. ( 3/2 ), λn i1 = u 1 , λ
∂ B y1 2 ∂n i1 2 ∂n p 1 ∂u 1 = sin θ + β + β1 , ∂ξ ∂ξ 3 ∂ξ 3 ∂ξ
λ
∂ B y1 ∂v1 = cos θ , ∂ξ ∂ξ λ
∂ B y1 ∂u 1 ∂v1 = sin θ − cos θ , ∂ξ ∂ξ ∂ξ
(17)
(18)
∂n p1 ∂u 1 ∂v1 −λ = − sin2 θ + sin θ cos θ . ∂ξ ∂ξ ∂ξ The normalized dispersion relation is obtained as λ2 =
1 2 2 1 + β + β1 sin2 θ 2 3 3
v u u × 1 ± t1 −
8 β 3
1+
2 β 3
+
cos2
θ
2 β 3 1
sin2 θ
2 ,
(19)
4
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In the next higher order of i.e. 2 , we obtain −λ −λ
7 1 λ2 − cos2 θ − β1 λ3 sin3 θ + λ2 sin2 θ + β 9 3 λ2 sin θ 2 2 + 2 λ2 − β1 sin θ − β λ2 − cos2 θ 3 3
∂ B y1 2 ∂n p1 2 ∂n i1 ∂w1 ∂ Bz1 = cos θ − sin θ − β1 − β , ∂ξ ∂ξ ∂χ 3 ∂χ 3 ∂χ
∂ Bz1 w1 ∂ 2 v1 = cos θ +λ 2 . ∂ξ ∂ξ ∂ξ
(20)
−2 λ2 − cos2 θ sin3 θ − 2 cos2 θ λ sin3 θ 2 − λ2 − cos2 θ sin θ − 2 cos2 θ sin θ − β1 3 3 3 2 2 2 +4 sin θ λ − cos θ + 4 sin θ cos θ
From the terms of order 5/2 , we obtain the following: ∂ B y2 2 ∂n p2 2 ∂n i1 ∂u 2 + sin θ + β1 + β ∂ξ ∂ξ 3 ∂χ 3 ∂χ ∂ B y1 ∂ B y1 ∂u 1 ∂u 1 =− − u1 − B y1 + sin θ n 1 ∂τ ∂ξ ∂ξ ∂ξ −λ
−
∂n p1 2 ∂n p1 7 + β1 n p1 + β1 n i1 9 ∂χ 3 ∂χ
C=
2 cos4 θ sin2 θ 2 + β1 2 − β1 cos2 θ sin2 θ, 3 λ − cos2 θ 3
∂ 2u1 ∂ ∂ 2 w1 (u 1 B y1 ) − λ 2 + λ ∂ξ ∂ξ ∂ξ ∂χ ∂n 2 ∂u 2 ∂n 1 ∂ ∂w1 + =− − (u 1 n 1 ) − ∂ξ ∂ξ ∂τ ∂ξ ∂χ
2 2 4 4 (λ2 − cos2 θ ) sin θ − β1 sin2 θ − β 3 9 9 λ2 2 2 1 − λ2 021 + λ2 − β1 sin θ− β λ 01+λ 2 3 3 λ sin θ (λ −cos2 θ ) # " 2 sin2 θ cos2 θ + β1 01 + 2 sin θ 01 , 3 λ2 − cos2 θ
D= −
∂n p2 ∂u 2 ∂v2 + sin2 θ − sin θ cos θ ∂ξ ∂ξ ∂ξ ∂(n p1 v1 ) ∂(n u p1 1 ) = sin θ cos θ − sin2 θ ∂ξ ∂ξ −λ
−
∂n p1 ∂ 2 Bz1 . +sin θ cos θ ∂τ ∂ξ 2
2 λ2 cos2 θ sin θ 2 βλ2 cos2 θ λ4 cos2 θ − β1 2 − λ2 − cos2 θ 3 λ − cos2 θ 3 λ2 − cos2 θ 2 2 2 − (Hem + Hpm )λ2 sin2 θ + Hem (λ2 − cos2 θ )
∂ B y2 ∂ B y1 ∂u 2 ∂v2 ∂w1 −λ + sin θ − cos θ =− − sin θ ∂ξ ∂ξ ∂ξ ∂τ ∂χ
−λ
2 1 2 2 cos θ λ − β λ2 − cos2 θ 2 λ sin θ 3
1 2 2 2 2 − 2 cos θ λ − β λ − cos2 θ , λ sin θ 3
1 He2 ∂ 3 n p1 ∂n i1 (1 + p)He2 ∂ 3 n p1 − βn i1 + + 3 3 ∂χ 1− p ∂ξ 1 − p ∂ξ 3 ∂ B y2 ∂ B y1 ∂v2 ∂v1 ∂v1 −λ −cos θ =− −u 1 −cos θ n 1 ∂ξ ∂ξ ∂τ ∂ξ ∂ξ
−
2
(21)
where By eliminating the second-order quantities (i.e. with subscript 2) and terms containing Bz1 and w1 from equation (20) using equations (19) and (21), we obtain ∂ ∂ξ
∂ 2 B y1 ∂ B y1 ∂ B y1 ∂ 3 B y1 + n = 0. + l B y1 +m ∂τ ∂ξ ∂ξ 3 ∂χ 2
2 2 (λ2 − cos2 θ ) 01 = sin θ + β1 sin θ + β , 3 3 λ2 sin θ
(22)
The nonlinear evolution equation (22) is the KP equation that describes the propagation of both fast and slow magnetoacoustic solitary waves in a quantum magnetoplasma. The coefficients l, m and n in equation (22) are given as l = B/A, m = C/A and n = D/A, where A, B, C and D are given by
2
λ2 − cos2 θ + sin θ
He2 , 1− p
2 Hpm =
p He2 . 1− p
It should be noted that quantum statistical effects modify all the coefficients of the solitary wave; however, the quantum Bohm potential affects only the dispersion in the dominant direction (i.e. the one with the coefficient m). The solution of the KP equation using the tanh method [48, 49] (i.e. equation (22)) is
2 2 A = 2λ3 − λ(1 + β) + cos2 θ(1 − β), 3 3λ 4 λ2 − cos2 θ B= − 3 λ2 sin θ
2 Hem =
2
8 (ξ, χ , τ ) =
3U sech2 [ζ ], l
(23)
where ζ = k(ξ + χ − U τ ) and U = 4k 2 m, with k being the dimensionless nonlinear wave number.
2 − β1 λ2 − cos2 θ sin θ − sin θ λ2 − cos2 θ 3 5
Phys. Scr. 82 (2010) 065508
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W Masood et al
m
m
p
n
n
p
p Figure 2. Variation of the coefficients of nonlinearity, l, predominant dispersion, m, and weak dispersion, n, for both fast and slow magnetoacoustic modes with increasing positron concentration, p. The other parameters are n e0 = 1.5 × 1026 cm−3 , θ = 15◦ and B0 = 1010 G.
In figures 2 and 3, the variation of the coefficients of nonlinearity, l, predominant dispersion, m, and weak dispersion, n, with increasing the positron concentration and obliqueness parameter for both the fast and the slow waves is shown. It is observed that although the general trend is the same, the variation is different for the fast and slow magnetoacoustic modes. In figure 4, we study the variation of fast and slow modes with increasing positron concentration. It is found that both the fast and slow modes show rarefactive solitary structures. It is worth mentioning here that the increase in positron concentration decreases the density dip for slow magnetoacoustic solitary waves; however, the fast mode does not get affected by the positron concentration at all.
4. Results and discussion In this section, we numerically investigate the dependence of the quantum electromagnetic magnetoacoustic wave potential on various parameters in dense e–p–i plasmas. In the high-density plasmas found in dense astrophysical objects such as neutron stars and white dwarfs, the plasma densities are enormous and quantum effects may be important. For illustration, parameters are chosen that are representative of the plasma in dense astrophysical bodies, i.e. n 0 ∼ 1026 –1028 cm−3 and B0 ∼ 109 –1011 G [50, 51]. A graphical analysis of the quantum magnetoacoustic solitary profile given by equation (22) is presented by plotting the electromagnetic wave potential B y1 against different parameters affecting the wave. 6
Phys. Scr. 82 (2010) 065508
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m
n
m
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W Masood et al
Figure 3. Variation of the coefficients of nonlinearity, l, predominant dispersion, m, and weak dispersion, n, for both fast and slow magnetoacoustic modes with increasing obliqueness parameter, θ. The other parameters are n e0 = 1.5 × 1026 cm−3 , p = 0.7 and B0 = 1010 G.
In figure 5, we investigate the variation of fast and slow modes with increasing obliqueness parameter θ. It is again observed that both the fast and the slow modes show rarefactive solitary structures and the density dip is greater in the case of fast magnetoacoustic solitary waves in comparison with the slow waves. Note that for both the fast and the slow magnetoacoustic solitary waves, the electromagnetic wave potential decreases with increasing obliqueness parameter θ; however, the change is more pronounced for slow waves for smaller values of θ, which is in agreement with the character of slow and fast modes expounded in detail in the discussion of linear dispersion for both these modes. Finally, in figure 6, we study the variation of fast and slow modes with increasing magnetic field strength. Once again, it is observed that both the fast and the slow modes show rarefactive solitary structures and the density dip is greater in the case of fast magnetoacoustic solitary waves in comparison with the slow waves. Note that the electromagnetic wave potential decreases with increasing magnetic field for the fast
mode; however, the converse is true for the slow modes. It can be easily seen from equation (22) that the expression for the quantum Bohm potential involves magnetic field and therefore the variation in magnetic field indirectly represents the change in the wave potential with the quantum Bohm potential term.
5. Conclusion We have presented here a study of linear and nonlinear obliquely propagating nonlinear magnetoacoustic waves in dense e–p–i magnetoplasmas using the QMHD model. In this regard, the nonlinear quantum KP equation for obliquely propagating fast and slow magnetoacoustic solitary waves has been presented, which admits solitary wave solutions. The variation of the fast and slow magnetoacoustic solitary wave profiles with the quantum Bohm potential by increasing the magnetic field strength, the obliqueness parameter θ and the positron concentration has also been investigated. It is observed that the fast and slow magnetoacoustic modes 7
Phys. Scr. 82 (2010) 065508
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Figure 6. Variation of the fast and slow magnetoacoustic potential B y1 with increasing magnetic field, i.e. B0 = 5.0 × 109 G (dotted line), B0 = 7.5 × 109 G (dashed line) and B0 = 1.0 × 1010 G (solid line). The other parameters are n e0 = 2.0 × 1026 cm−3 , p = 0.7 and θ = 15◦ .
Figure 4. Variation of the fast and slow magnetoacoustic potential B y1 with positron concentration, p.
for both fast and slow waves has been shown, and it is observed that although the general trend is the same, the variation is different for the fast and the slow magnetoacoustic modes. It is found that for increasing positron concentration, both the fast and slow modes manifest rarefactive solitary structures; however, the fast magnetoacoustic mode does not get affected by the variation in positron concentration. It is found that with increasing obliqueness parameter θ, both the fast and the slow modes show rarefactive solitary structures; however, the density dip is greater in the case of fast magnetoacoustic solitary waves in comparison with the slow waves. Finally, it is observed that the electromagnetic wave potential decreases with increasing magnetic field (which is related to the quantum Bohm potential) for the fast mode; however, the converse is true for the slow modes. The present investigation may be beneficial for understanding the linear and nonlinear propagation of low-frequency electromagnetic solitary waves in dense astrophysical environments.
References Figure 5. Variation of the fast and slow magnetoacoustic potential B y1 with the obliqueness parameter, θ .
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admit rarefactive solitary wave structures in dense e–p–i plasmas. The variation of the coefficients of nonlinearity, l, predominant dispersion, m, and weak dispersion, n, with increasing positron concentration and obliqueness parameter 8
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