Nonlinear structures: explosive, soliton and shock in a quantum electron-positron-ion magnetoplasma
aa r X i v : . [ phy s i c s . p l a s m - ph ] J u l Physics of Plasmas
Nonlinear structures: explosive, soliton and shock in a quantumelectron-positron-ion magnetoplasma
R. Sabry ,a , W. M. Moslem ,b , F. Haas ,c , S. Ali ,d and P. K. Shukla ,e Institut f¨ur Theoretische Physik IV,Fakult¨at f¨ur Physik und Astronomie,Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany and National Center for Physics, Quaid-I-Azam University Campus, Islamabad, Pakistan (Received 14 October 2008)
Abstract
Theoretical and numerical studies are performed for the nonlinear structures (explosive, solitonsand shock) in quantum electron-positron-ion magnetoplasmas. For this purpose, the reductive per-turbation method is employed to the quantum hydrodynamical equations and the Poisson equation,obtaining extended quantum Zakharov-Kuznetsov equation. The latter has been solved using thegeneralized expansion method to obtain a set of analytical solutions, which reflect the possibilityof the propagation of various nonlinear structures. The relevance of the present investigation tothe white dwarfs is highlighted. . INTRODUCTION Numerous investigations [1, 2, 3, 4, 5] relating to wave phenomena, have been studiedin dense quantum plasmas, are of fundamental importance for understanding collective in-teractions in superdense astrophysical environments [6], in high intense laser-solid densityexperiments [7], in ultracold plasmas [8], in microplasmas [9], and in micro-electronic de-vices [10]. New characteristics of quantum plasma arise due to the pressure law describingthe fermionic behavior of the charged carriers, quantum forces associated with the electrontunneling, as well as the Bohr magnetization involving the electron 1/2 spin. The quantumBohm potential produces modifications in the dispersions of collective modes at quantumscales. The latter are strongly effected by the plasma number densities and Fermi tempera-tures. It is well-known that quantum mechanical effects become relevant when the thermalde Broglie wavelength of the charged particles is equal or larger than the average interpar-ticle distance. In particular, quantum behavior of the electrons reaches much easily due toless mass compared to ions.In recent years, many theoretical and numerical analysis [11, 12, 13, 14, 15] have beencarried out to investigating the new features of plasmas with quantum corrections by usingboth the Schr¨odinger-Poisson and the Wigner-Poisson systems. In this context, Manfredi [11]reported different approaches to model the collisionless electrostatic dense quantum plasmas.Haas et al . [12] investigated the linear and nonlinear properties of the quantum ion-acoustic(QIA) waves in dense quantum plasmas by employing the quantum hydrodynamical (QHD)equations for inertialess electrons and mobile ions. They examined that the quantum Bohmpotential modifies the linear wave dispersion and affects strongly the QIA solitary waves.Shukla and Eliasson [13] presented the numerical study of the dark solitons and vortices inquantum electron plasmas. Moslem et al. [14] investigated the quantum dust-acoustic doublelayers in a multi-species quantum dusty plasma. It was found that both compressive andrarefactive double layers can only exist for positively charged dust particles. Later, Ali et al .[16] studied the QIA waves in a three-component plasma, comprised of electrons, positrons,and ions. They employed the reductive perturbation method and pseudo-potential approachfor the small and arbitrary amplitude nonlinear QIA waves, respectively. It was shown thatthe amplitude and width are significantly altered due to the quantum statistics and quantumtunneling effects. Misra et al. [17] considered the nonlinear propagation of electron-acoustic2aves in a nonplanar quantum plasma, consisting of two groups of electrons: the inertialcold electrons and inertialess hot electrons as well as the stationary ions. They obtained thebright and dark solitons depending strongly upon the presence of cold electrons.The laboratory and dense astrophysical quantum plasmas can be confined by an externalmagnetic field. Therefore, the effect of the magnetic field has to taken into account, espe-cially for astrophysical observations (such as white dwarfs, neutron stars, magnetars, etc.)where the high magnetic field plays an important role in the formation and stability of theexisting waves. Several authors have considered the effect of magnetic field in different quan-tum plasma models. For example, Haas [18] introduced a three-dimensional QHD modelfor dense magnetoplasmas and established the conditions for an equilibrium in the idealquantum magnetohydrodynamics (QMHD). Ali et al. [19] employed the QMHD equationspresenting a fully nonlinear theory for ion-sound waves in a dense Fermi magnetoplasma.It was revealed that only subsonic ion-sound solitary waves may exist. Shukla and Stenflo[20] derived the dispersive shear Alfv´en waves in a quantum magnetoplasma, incorporat-ing the strong electron and positron density fluctuations. The shear Alfv´en modes acquireadditional dispersion due to quantum corrections. Later, Ali et al. [21] have been carriedout for the low-frequency electrostatic drift-like waves in a nonuniform collisional quantummagnetoplasma. It was shown that the modes become unstable and can cause cross-fieldanomalous ion-diffusion.Three decades ago, Zakharov and Kuznetsov [22] derived an equation for nonlinear ion-acoustic waves in a magnetized plasma containing cold ions and hot isothermal electrons.The Zakharov-Kuznetsov (ZK) equation has also been derived for different physical systemsand scenarios [23, 24]. Nonlinear wave solution for ZK equation can produce an instabilityin a three-dimensional system as discussed in Refs. [25, 26]. Moslem et al . [27] extendedthe work for a three-dimensional nonlinear ion-acoustic waves in a quantum magnetoplasma,highlighting the bending instability of the solitary wave solution of the quantum ZK equa-tion. Recently, Masood and Mushtaq [28] studied obliquely propagating electron-acousticwaves in a two-electron population quantum magnetoplasma and examining the effects ofnonlinearity at quantum scales.In the present paper, we shall investigate the possible nonlinear structures (soliton, ex-plosive and shock pulses) of the QIA waves in a collisionless electron-positron-ion magneto-plasma using the QHD equations. By means of computational investigations, we examine3he effect of the positron concentration, the quantum diffraction and the quantum statisticaleffects on the profiles of the nonlinear excitations. The paper is organized as follows: Thebasic equations governing the dynamics of the nonlinear QIA waves are presented and theextended quantum ZK equation describing the system is derived in Sec II. In Sections IIIand IV, we apply the generalized expansion method to solve the extended quantum ZKequation. A set of analytical solutions is obtained, and then used to investigate numericallythe effect of positrons and the quantum parameters on the nonlinear excitations. The resultsare summarized in section V. II. BASIC EQUATIONS AND DERIVATION OF THE EXTENDED QUANTUMZK EQUATION
We consider a dense magnetoplasma whose constituents are the electrons, positrons, andsingly charged positive ions. The plasma is confined in an external magnetic field H = H b z , where b z is the unit vector along the z − axis and H is the strength of the magnetic field.We assume that the quantum plasma satisfies the condition T F e,p ≫ T F i , and obeys theelectron/positron pressure law P e,p = m n e,p V F e,p / n e,p , where V F e,p = (2 K B T F e,p /M ) / isthe electron/positron Fermi thermal speed, K B is the Boltzmann constant, T F e,p ( T F i ) isthe electron/positron (ion) Fermi temperature, M is the electron and positron mass, n e,p is the electron/positron number density, with the equilibrium value n e,p . The nonlinearpropagation of the QIA waves is governed by the dimensionless hydrodynamics equations as ∂n i ∂t + ∇ . ( n i u i ) = 0 , (1) ∂ u i ∂t + u i . ∇ u i = −∇ φ + u i × b z , (2)Ω ▽ φ = n e − n p − n i , (3) n e = µ e (cid:18) φ + H e ▽ √ n e √ n e (cid:19) , (4)and n p = µ p (cid:18) − σφ + σH e ▽ √ n p √ n p (cid:19) , (5)where n i , u i , and φ are the ion number density, the ion fluid velocity, and the electrostaticpotential, respectively. Since, the ion mass is much larger than the electron/positron mass,4ne can ignore the quantum effects of the ions in Eq. (2). The statistical and diffractioneffect for the system can be seen through the nondimensional parameters σ (= T F e /T F p ) and H e (= eH ~ / c √ M i M K B T F e ) , respectively, where ~ is the Planck constant dividedby π , M i ( M ) is the ion (electron/positron) mass, and c is the speed of lightin vacuum. Here, Ω(= ω ci /ω pi ) , where ω ci (= eH /m i c ) and ω pi ( = p πe n i /M i )are the ion gyrofrequency and the ion plasma frequency, respectively. n i is theequilibrium ion density. Equations (4) and (5) reveal that the electrons and positronsdo not follow the Boltzmann law contrary to the classical plasma. The physical quantitiesappearing in Eqs. (1)–(5) have been appropriately normalized: n e,i,p → n e,i,p /n i , u i → u i /C s , t → tω ci , ∇ → ∇ ρ s , and φ → eφ/ K B T F e , where ρ s (= C s /ω ci ) is the ion-soundFermi gyroradius and C s (= p K B T F e /M i ) is the ion-sound Fermi speed.Before going to the nonlinear developments, it is necessary to examine the condition forneglecting the source term in the continuity equation due to annihilation of plasma species.The details are given in the Appendix.To investigate the propagation of QIA waves, we expand the dependent variables n e,i,p , u i , and φ about their equilibrium values in power of ǫ,n i = 1 + ǫn i + ǫ n i + ǫ n i + ...,n e,p = µ e,p + ǫn e,p + ǫ n e,p + ǫ n e,p + ...,u ix,y = ǫ u ix,y + ǫ u ix,y + ǫ u ix,y + ..., (6) u iz = ǫu iz + ǫ u iz + ǫ u iz + ...,φ = ǫφ + ǫ φ + ǫ φ + ..., where ǫ is a keeping order parameter proportional to the amplitude of the perturbation.Following the reductive perturbation method [29], we express the independent variables intoa moving frame in which the nonlinear wave moves at a phase-speed of λ (normalized withthe ion-sound Fermi speed C s ) as X = ǫx, Y = ǫy, Z = ǫ ( z − λt ) and T = ǫ t. (7)The neutrality condition at equilibrium reads µ e = 1 + µ p , where µ e = n e /n i and µ p = n p /n i . Subistituting (6) and (7) into Eqs. (1)–(5), we obtain the lowest-order in ǫ as5 i = 1 λ φ , u ix = − ∂φ ∂Y ,u iy = ∂φ ∂X , u iz = 1 λ φ , (8) n e = µ e φ , n p = − σµ p φ , along with the phase speed rule λ = (cid:18)
11 + µ p (1 + σ ) (cid:19) / . (9)It is clear here that the phase speed λ of the QIA waves is affected by the quantum statisticaleffect and by the positron concentration µ p . To the next-order in ǫ , we have n i = 43 λ φ + 1 λ φ , u ix = λ ∂ φ ∂X∂Z − ∂φ ∂Y ,u iy = λ ∂ φ ∂Y ∂Z + ∂φ ∂X , u iz = 12 λ φ + 1 λ φ , (10) n e = − µ e (cid:0) φ − φ (cid:1) , n p = − σµ p (cid:0) σφ + 2 φ (cid:1) , while the Poisson equation gives Qφ = 0 , (11)where Q = [( σ − µ p λ − λ − λ . Since φ = 0, therefore Q should be at least of the order of ǫ. Therefore, Qφ becomes of theorder of ǫ ; so it should be included in the next order of the Poisson equation. The next-order in ǫ gives a system of equations. Solving this system with the aid of Eqs. (8)-(10), wefinally obtain the extended quantum ZK equation as ∂ϕ∂T + (cid:0) A ϕ + B ϕ (cid:1) ∂ϕ∂Z + C ∂ ϕ∂Z + D ∂∂Z (cid:18) ∂ ∂X + ∂ ∂Y (cid:19) ϕ = 0 , (12)where we have replaced φ by ϕ for simplicity. The nonlinear and dispersion coefficients are6iven, as A = λ + 3 − ( σ − µ p λ λ ,B = − σ + 1) µ p λ + λ − λ ,C = 18 λ (cid:0) − ( µ p + 1) H e − σ H e µ p (cid:1) ,D = C + 12 λ The extended quantum ZK equation (12) constitutes the final outcome of this model. Theanticipated balance between dispersion and nonlinearity (which contain the quantum me-chanical effects) within the extended quantum ZK equation may give rise to different non-linear structures. Some of these solutions will recover in the next section.
III. EXACT SOLUTIONS OF THE EXTENDED QUANTUM ZK EQUATION
To obtain the possible analytical solutions of Eq. (12), we assume that ξ = L X X + L Y Y + L Z Z − ϑT, (13)where L X , L Y and L Z are the direction cosines and ϑ is the QIA wave speed to be determinedlater. Using (13) into (12), we obtain − ϑϕ ′ + A ϕ ϕ ′ + B ϕ ϕ ′ + γϕ ′′′ = 0 , (14)where A = AL Z , B = BL Z and γ = CL Z + DL Z ( L X + L Y ). According to the generalizedexpansion method [30] the solution of Eq. (14) can represent by ϕ = a + a ω, (15)with dωdξ = k (cid:0) c + c ω + c ω + c ω + c ω (cid:1) / , (16)where a , a , c , c , c , c and c are arbitrary constants to be determined later and k = ± ω . Equating the coefficients of different powers of ω , we obtain an overdetermined7ystem of algebraic equations which can be solved with the help of symbolic manipulationpackage Mathematica to give three Jacobi elliptic doubly periodic type solutions as ϕ = − A B + k s γ c m B (2 m −
1) cn (cid:18)r c (2 m − ξ (cid:19) , with c = − c m (1 − m ) c (2 m − , c > , c < , (17) ϕ = − A B + k s γ c B (2 − m ) dn (cid:18)r c (2 − m ) ξ (cid:19) , with c = c (1 − m ) c (2 − m ) , c > , c < , (18)and ϕ = − A B + k s γ c m B ( m + 1) sn (cid:18)r − c ( m + 1) ξ (cid:19) , with c = c m c ( m + 1) , c < , c > , (19)where m is a modulus of the Jacobian elliptic function and c = c = 0. As m →
1, theJacobi doubly periodic solutions (17) and (18) degenerate to the bell-shapped solitary wave ϕ = − A B + k r γ c B sech ( √ c ξ ) , (20)where the arbitrary constant c vanishes. Again, as m → ϕ = − A B + k r γ c B tanh r − c ξ ! , (21)where c = c / c . In the solutions (17)-(21), the QIA wave speed ϑ = ( − A / B + 2 γc )where c = A / γB .Furthermore, the generalized expansion method provides us with further analytical solu-tions of the extended quantum ZK equation (12) as ϕ = − c c + k p c − c c cosh (cid:0) √ c ξ (cid:1) , with c = c = 0 , c = ϑγ , c = − A γ , c = − B γ , (22)8nd ϕ = − A B k coth s − A γB ξ , with ϑ = − A B and B < . (23) IV. PARAMETRIC ANALYSIS FOR WHITE DWARFS
It is clear that the propagation speed of the QIA wave is modified by the effect of thequantum statistical effect σ and by the presence of positrons µ p . As σ and µ p increase, thepropagation speed of the QIA wave will decrease. The dependence of the nonlinear structuresamplitude and width on the equilibrium positron number density ( µ p ) and quantum effects σ and H e is more perplex. First, it is important to note that changing µ p leads to a changein the phase-speed ( λ ) of the QIA waves [see Eq. (9)] , as well as the electron concentration(via the charge-neutrality condition µ e = 1+ µ p ). Since the electron (positron) Fermitemperature depends upon the equilibrium electron (positron) number density, it can also beaffected by µ p through the charge-neutrality condition. As a result, the quantum statistical( σ ) and diffraction ( H e ) effects will vary with the positron concentration µ p .Based upon the above findings, we shall now investigate the effects of the relevant phys-ical quantities, namely the positron concentration µ p on the profiles of the QIA nonlinearstructures. We have used, as a starting point, a typical set of plasma parameter valuesfor white dwarfs [11] (in the absence of positrons; µ p = 0), namely: n e = 4 × cm − ,T F e = 4 . × K , ω ci = 1 . × s − and ω pi = 2 . × s − . However, once thepositrons species density is determined, the values of T F e , λ and H e are subsequently com-puted, according to the above formulae, which also determine A, B, C and D. In the plots,we shall change the positrons concentration, which leads to recalculate all the physical pa-rameters again. Obviously, by varying the positron concentration, we simultaneously modifyall the parameter values used in the plots below.
A. Solitary and Explosive/Blowup Excitations
It may be appropriate to point out that the analytical solutions in Sec. III have beenobtained for different arbitrary constants k, c , ...c . One of them is the localized solution922), which is a bell-shapped solitary wave solution. Recall that the arbitrary constant k can be either +1 or −
1. For k = −
1, a positive solitary pulse can propagate and for k = +1 , a negative solitary pulse exist. Note that we have executed the negative solitarypulse since it is not physically correct in the the present model. Figure 2 depicts the QIAsolitary pulse for different values of positron concentration µ p , which now determines T F e,p (and the ratio σ ) through the charge-neutrality condition. It is found that the amplitude ofthe soliton pulse decreases by increasing µ p , resulting an increase (decrease) of the electronFermi temperature T F e (quantum diffraction effect H e ). Physically, the increase of T F e leadsto an increase of the electron Fermi energy (viz. K B T F e = E F e ≡ ( ~ / m )(3 π n e ) / ), andas a result the ion Fermi energy should decrease to conserve the energy law. The decreaseof the ion Fermi energy decreasing the nonlinearity of the system and hence the height ofthe soliton pulse shrinks.It may be interesting to note that for certain values of plasma parameters the solitarypulse convert to an explosive/blowup excitation as shown in Fig. 3. The blowup excitationindicates that an instability in the system can produce due to the effect of the nonlinearity(which in our case depends on the positron concentration µ p and the quantum statisticaleffects σ ). On the other hand, the magnitude of some quantities (e.g. temperature, pressure,density, etc.) leads to prejudice the balance between the dispersion and the nonlinearity [31].Therefore, the amplitude may increase to very high values, which gives rise to increasingthe electric potential and then accelerate the moving particles.It is important to notice that Eq. (23) is an explosive/blowup solution, i.e. the potential ϕ infinitely grows at a finite point (for any fixed X, Y, Z → X , Y , Z ), there exist an ξ atwhich the solution (23) blowup and thereby we regard the latter as an explosive solution asdepicted in Fig. 4. B. Shock/Double Layer Excitation
For the shock/double layer solution [32], the boundary condition ϕ ( ξ ) → ξ → ∞ must satisfy. Applying the last boundary condition into Eq. (21), we obtain the doublelayer solution as ϕ = ϕ m [1 + tanh ( W D ξ )] , (24)10here the amplitude of the double layers is ϕ m = − A / B , the width is W D = p − γB /A . Here ϑ (= − A / B ) is the shock wave speed. Notice that B < W D real. The numerical analysis in Fig. 5,however, shows that for small positron concentration µ p the dominant situation correspondsto B < , so the double layers may exist. For large positron concentration µ p , double layerscannot occur, since B > . Typically, we have used the plasma density value for whitedwarf [11] via n i = 2 × cm − and assume that L z = 0 . , which leads to the fact that fornegative B (i.e., formation of double layers) the positron concentration n p must less than1 . × cm − . Also, it noted that the narrow range of µ p [corresponding to B < . Generally speaking, one can also note from Eq.(24) that the nature of the double layer depends on the sign of A , i.e. for A > ϕ m > A < ϕ m < A is usually greater thanzero and then only positive double layers can exist.Equation (24) describes the double layer potential, which has a well-know profile (cf. Fig.6). This profile may change due to vary of physical parameters. The dependence of doublelayer characteristics on the positron concentration µ p [which determines T F e,p , H e and σ through the charge-neutrality condition] is depicted in Fig. 7. It is obvious that an increasein the positron concentration µ p shrinks the double layers width but the amplitude increasesby increasing µ p . It important to note here that in Ref. [33], the soliton excitation in e-p-i magnetoplasma was investigated but the present work investigates soliton, shockand explosive excitations in e-p-i magnetoplasma. Therefore, the present modelstudies another two nonlinear structures, which did not discuss in Ref. [33].Also, in Ref. [27], the authors used the extended Conte’s truncation methodto obtain the solitary, explosive, and periodic solutions of the QZK equation.Note that this method gives solitary and explosive excitations described byequation (25) and periodic excitation described by equation (26). Thus, theextended Conte’s truncation method cannot predict the shock formation, whichmay arise due to the presence of weakly double layers. In the present work,we have used generalized expansion method. The later succeeded to describesoliton, explosive, as well as shock excitations. Therefore, the present method an be considered as a powerful tool to deal with more general nonlinear partialdifferential equations. V. SUMMARY
To summarize, we have presented the properties of the nonlinear structures QIA wavesin a very dense Fermi plasma, composed of the electrons, positrons and positive ions. Byemploying the reductive perturbation method, an extended quantum ZK equation is derived.The latter has been solved using the generalized expansion method to obtain a set of ana-lytical solutions, which reflects the possibility of propagation of various nonlinear structures(viz. explosive, soliton and shock pulses). We have numerically examined the effects of thepositron concentration (which changes the quantum statistics and quantum diffraction pa-rameters through the charge-neutrality condition) on the electrostatic potential excitations,by varying relevant physical parameters. It is found that the amplitudes and widths of thenonlinear structures are significantly affected by the positron concentration, quantum statis-tical, and quantum tunneling effects. Also, for certain plasma parameters the solitary pulsetransforms to blowup pulse. Finally , we stress that this investigation should be useful forunderstanding the features of the nonlinear structures QIA waves in an electron-positron-ionplasma, such as those in the superdense white dwarfs and in the intense laser-solid matterinteraction experiments. Appendix: The necessary condition to neglect the annihilationprocess
To neglect the annihilation process, the following inequality must satisfy1 ω pe << T ann , (A1)where (1 /ω pe ) is the electron plasma period and T ann is the annihilation time. For nonrela-tivistic plasma, the time of annihilation reads [34] T ann = 43 σ T n e c (cid:20) Θ1 + 6Θ (cid:21) , (A2)where σ T (= 6 . × − cm ) is the cross section and Θ(= K B T /mc ) is the temperaturerange, which satisfy the inequality [34] 12 < Θ < , (A3)where α (= 7 . × − ) is the Fine-structure constant. Equation (A3) can be rewrittenin terms of temperature as 3 × < T (K) < . × (A4)Inserting Eq. (A2) into (A1), we obtainΘ > . × − n / . (A5)Using Eq. (A3) and (A5), one can calculate the range of the density where the annihilationcan be ignored 3 . × < n e (cid:0) cm − (cid:1) < . × . (A6)The quantum effects become important for certain values of density ( n e,p ) and temperature( T e,p ). The quantum condition n e,p λ B > T e,p . × − n / e,p . (A7)Using Eq. (A6) with (A7), one can calculate the range of temperature in quantum plasmaas 3 . × < T e,p (K) < . × . (A8)It is clear that the range for neglecting annihilation is well satisfied for white dwarf [see Ref.[11]]. Therefore, the present model can be applicable to the dense white dwarf. Acknowledgments
R.S. acknowledges the financial support from the Egyptian Government under the Post-doctoral Research Program. The work of W.M.M. was partially supported by Ruhr-Universit¨at Bochum through the Framework of the HGF Impulse and Networking Fund/13Z-J¨ulich (Project Number: S080200W). W.M.M. also thanks Professor R. Schlickeiser forhis hospitality. F.H. thanks the financial support from the Alexander von Humboldt Stiftung(Bonn, Germany).(a) Also at: Theoretical Physics Group, Department of Physics, Faculty of Science,Mansoura University, Damietta Branch, New Damietta 34517, Egypt. Electronic mail:[email protected] and [email protected](b) Present address: Department of Physics, Faculty of Science-Port Said, Suez CanalUniversity, Egypt. Electronic mail: [email protected] and [email protected](c) Also at: Universidade do Vale do Rio dos Sinos-UNISINOS, Av. Unisinos, 950,93022-000 Sao Leopoldo RS, Brazil. Electronic mail: [email protected](d) Electronic mail: shahid [email protected](e) Also at: Department of Physics, Ume˚a University, SE-90187 Ume˚a, Sweden. Elec-tronic mail: [email protected] 14
1] B. Shokri and A. A. Rukhadze, Phys. Plasmas , 3450 (1999); B. Shokri and A. A. Rukhadze, ibid. , 4467 (1999).[2] G. Manfredi and M. Feix, Phys. Rev. E , 6460 (1996); N. Suh, M. R. Feix, and P. Bertrand,J. Comput. Phys. , 403 (1991).[3] L. G. Garcia, F. Haas, L. P. L. de Oliveira, and J. Goedert, Phys. Plasmas , 012302 (2005).[4] P. K. Shukla and B. Eliasson, Phys. Rev. Lett. , 096401 (2007); M. Marklund and G.Brodin, ibid. , 025001 (2007); D. Shaikh and P. K. Shukla, ibid. , 125002 (2007).[5] D. Pines, J. Nucl. Energy C: Plasma Phys. , 5 (1961); P. K. Shukla, L. Stenflo, and R.Bingham, Phys. Lett. A , 218 (2006).[6] Y. D. Jung, Phys. Plasmas , 3842 (2001); M. Opher, L. O. Silva, D. E. Dauger, V. K. Decyk,and J. M. Dawson, ibid. , 2454 (2001).[7] M. Marklund and P. K. Shukla, Rev. Mod. Phys. , 597 (2006); D. Kremp, Th. Bornath, M.Bonitz, and M. Schlanges, Phys. Rev. E , 4725 (1999).[8] T. C. Killian, Nature (London) , 298 (2006).[9] K. Becker, K. Koutsospyros, S. M. Yin et al ., Plasma Phys. Control. Fusion , B513 (2005).[10] P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, “Semiconductor Equations”, (Springer-Verlag, New York, 1990).[11] G. Manfredi, Fields Inst. Commun. Series , 263 (2005).[12] F. Haas, L. G. Garcia, J. Goedert, and G. Manfredi, Phys. Plasmas , 3858 (2003).[13] P.K. Shukla and B. Eliasson, Phys. Rev. Lett. , 245001 (2006).[14] W. M. Moslem, P. K. Shukla, S. Ali, and R. Schlickeiser, Phys. Plasmas , 042107 (2007).[15] P. K. Shukla and S. Ali, Phys. Plasmas , 114502 (2005); S. Ali and P. K. Shukla, ibid. ,022313 (2006); S. Ali and P. K. Shukla, Eur. Phys. J. D , 319 (2007); A. Mushtaq andS. A. Khan, Phys. Plasmas 14, 052307 (2007). [16] S. Ali, W. M. Moslem, P. K. Shukla, and R. Schlickeiser, Phys. Plasmas , 082307 (2007).[17] A. P. Misra, P. K. Shukla, and C. Bhowmik, Phys. Plasmas , 082309 (2007).[18] F. Haas, Phys. Plasmas , 062117 (2005).[19] S. Ali, W. M. Moslem, P. K. Shukla, and I. Kourakis, Phys. Lett. A
606 (2007).[20] P. K. Shukla and L. Stenflo, New J. Phys. , 111 (2006); P. K. Shukla and L. Stenflo, J.Plasma Phys. , 605 (2006).[21] S. Ali, N. Shukla, and P. K. Shukla, Europhys. Lett. , 45001 (2007).
22] V. E. Zakharov and E. A. Kuznetsov, Soviet Phys. JETP , 285 (1974).[23] S. Munro and E. J. Parkes, J. Plasma Phys. , 305 (1999); S. Munro and E. J. Parkes, ibid. ,411 (2001). A. A. Mamun and R. A. Cairns, ibid. , 175 (1996); J. Das, A. Bandyopadhyay,and K. P. Das, ibid. , 587 (2006).[24] A. A. Mamun, Astrophys. Space Sci. , 507 (1999); S. K. El-Labany and W. M. Moslem,Phys. Scr. , 416 (2002); S. K. El-Labany, W. M. Moslem, and F. M. Safi, Phys. Plasmas , 082903 (2006).[25] M. A. Allen and G. Rowlands, J. Plasma Phys. , 413 (1993); M. A. Allen and G. Rowlands, ibid. , 63 (1995); E. J. Parkes and S. Munro, ibid. , 695 (2005).[26] E. Infeld and G. Rowlands, “Nonlinear waves, Solitons and Chaos”, (Cambridge UniversityPress, Cambridge, 2000).[27] W. M. Moslem, S. Ali, P. K. Shukla, X. Y. Tang and G. Rowlands, Phys. Plasmas , 082308(2007).[28] W. Masood and A. Mushtaq, Phys. Plasmas , 022306 (2008).[29] H. Washimi and T. Taniuti, Phys. Rev. Lett. , 996 (1966).[30] R. Sabry, M. A. Zahran and E. Fan, Phys. Lett. A , 93 (2004).[31] M. N. Rosenbluth and R. Z. Sagdeev, ”Hand Book of Plasma Physics”, vol. 2 (North-HollandPhysics Publishing, Amsterdam, 1984); S. Chen, Ph.D. thesis, University of Simon Fraser(2000).[32] R. Z. Sagdeev, Rev. Modern Phy. 51, 11 (1979); M. Raadu and J. J. Rasmussen,Astrophy. Space Sci. 144, 43 (1988). [33]
S. A. Khan and W. Masood, Phys. Plasmas 15, 062301 (2008). [34] R. Svensson, The Astrophys. J. , 335 (1982). igure Captions Figure 1 (color online):
Three-dimensional profile of the solitary pulse [given by Eq. (22)]. A positive solitarypulse for k = − µ p = 0 . σ = 1 . . H e = 0 . , T = 0, Y = 0 . L x = 0 . L z = 0 . Figure 2 (color online):
Two-dimensional profile of the solitary pulse [given by Eq. (22)]. A positive solitarypulse for k = −
1. For curve A, µ p = 0 . σ = 2 .
08, Ω = 0 .
01, and H e = 0 . µ p = 0 . σ = 1 .
75, Ω = 0 . H e = 0 . µ p = 1, σ = 1 . . H e = 0 . T = 0, X = Y = 0 . L x = 0 .
01, and L z = 0 . Figure 3 (color online):
Three-dimensional profile of the explosive/blowup pulse [given by Eq. (22)]. A positiveexplosive pulse for k = − µ p = 0 . σ = 1 .
9, Ω = 0 . H e = 0 . , T = 0, Y = 0 . L x = 0 .
01, and L z = 0 . Figure 4:
Three-dimensional profile of the explosive/blowup pulse [given by Eq. (23)], for µ p =0 . σ = 7 .
37, Ω = 0 . H e = 0 . , T = 0, Y = 0 . L x = 0 .
01, and L z = 0 . Figure 5:
The nonlinear coefficient B is depicted against the positron density n p for n i = 2 × cm − and L Z = 0 . . Recall that for n p < . × cm − the nonlinear coefficient B < Figure 6:
Three-dimensional profile of the shock pulse [given by Eq. (24)], for µ p = 0 . σ = 7 . . H e = 0 . , T = 0, Y = 0 . L x = 0 .
01, and L z = 0 . Figure 7:
Two-dimensional profile of the shock pulse [given by Eq. (24)]. For curve A, µ p = 0 . σ = 7 .
6, Ω = 0 . H e = 0 .
008 and for curve B, µ p = 0 . σ = 7 .
37, Ω = 0 .
03, and H e = 0 . . Here, T = 0, X = Y = 0 . L x = 0 .
1, and L z = 0 .
2. Recall that the narrowrange of µ p will not affect on the ion-gyrofrequency Ω ..