Modeling of Modified Electron-acoustic Solitary Waves ... - Springer Link

Journal of the Korean Physical Society, Vol. 65, No. 12, December 2014, pp. 2045∼2052

Modeling of Modified Electron-acoustic Solitary Waves in a Relativistic Degenerate Plasma M. R. Hossen∗ and A. A. Mamun Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh (Received 16 September 2014) The modeling of a theoretical and numerical study on the nonlinear propagation of modified electron-acoustic (mEA) solitary waves has been carried out in an unmagnetized, collisionless, relativistic, degenerate quantum plasma (containing non-relativistic degenerate inertial cold electrons, both non-relativistic and ultra-relativistic degenerate hot electron and inertial positron fluids, and positively-charged static ions). A reductive perturbation technique is used to derive the planar and the nonplanar Korteweg-de Vries (K-dV) equations, which admit a localized wave solution for the solitary profile. The solitary wave’s characteristics are found to have been influenced significantly forin the non-relativistic and the ultra-relativistic limits. The mEA solitary waves are also found to have been significantly modified due to the effects of the degenerate pressure and the number densities of this dense plasma’s constituents. The properties of the planar K-dV solitary wave are quite different from those of the nonplanar K-dV solitary wave. The relevance of our results to astrophysical objects (like white dwarfs and neutron stars), which are of scientific interest, is briefly mentioned. PACS numbers: 52.27.Ny, 52.35.Fp Keywords: Modified electron-acoustic waves, Solitary waves, Degenerate pressure, Relativistic effects, Compact objects DOI: 10.3938/jkps.65.2045

I. INTRODUCTION The electron-positron (e-p) plasma is being extensively studied because of its wide range of applications and its importance in the understanding of many astrophysical environments such as those in the inner regions of accretion discs [1], in the cores of white dwarfs [2, 3], in the magnetospheres of neutron stars [4], in active galactive cores [5] and even in solar flares [6]. The presence of a positron component in a conventional electron-ion (e-i) plasma reduces the number density of ions and the restoring force on the electron fluid; hence, the characteristics of the linear waves and of the nonlinear structures are found to change considerably. For that reason, the study of an e-p plasma is momentous for comprehending the behavior of both astrophysical [1–6] and laboratory plasmas [7–9]. In present days, there has been a great deal of interest in understanding the basic properties of matter under extreme conditions [10–15]. White dwarfs and neutron stars are examples of matter under extreme conditions. These interstellar compact objects are contracted significantly, and, as a result, the density of their interiors becomes extremely high to provide non-thermal pressure ∗ E-mail:

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via degenerate fermion/electron pressure and particleparticle interactions. The observational evidence and theoretical analysis imply that these compact objects, which support themselves against gravitational collapse by using the cold degenerate fermion/electron pressure, are of two categories [16, 17]. The interior of the first category is close to a dense solid (ion lattice surrounded by degenerate electrons, and possibly other heavy particles or dust). One examples of this kind of star is a white dwarf, which is supported by the pressure of degenerate electrons. The interior of the second category is close to a giant atomic nucleus: a mixture of interacting nucleons and electrons, and possibly other heavy elementary particles and condensates or dust. One example of this kind of star is a neutron star, which is supported by the pressure due to a combination of nucleon degeneracy and nuclear interactions. These unique states (extreme conditions) of matter occur by significant compression of the interstellar medium. The degenerateplasma’s number density in such a compact object is so high (e.g. the degenerate-plasma’s number density can be of the order of 1030 cm−3 and 1036 cm−3 or more in white dwarfs and neutron stars, respectively) [18–21] that the electron Fermi energy is comparable to the electron mass energy and the electron speed is comparable to the speed of light in vacuum. The equation of state for degenerate fermions was mathematically clarified by

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Journal of the Korean Physical Society, Vol. 65, No. 12, December 2014

Chandrasekhar [22,23] for two limits, that is to say, the non-relativistic and the ultra-relativistic limits. In case of the aforementioned interstellar compact objects, the degenerate pressure for the ion fluid can be given by the following equation: Pi = Ki nα i ,

(1)

where α=

3 π 13 π2 3 5 Ki = Λc c 3 5 3 m 5

(2)

for the non-relativistic limit (where Λc = π/mc = 1.2 × 10−10 cm, and is the Planck constant divided by 2π). However, for the electron fluid, Pe = Ke nγe ,

(3)

and for the positron fluid, Pp = Kp nγp ,

(4)

where for non-relativistic limit [22–26] γ=α=

5 Ke = Kp 3

(5)

and for the ultra-relativistic limit [22–26] 4 3 γ= Ke = Kp = 3 4

π2 9

13

c

3 c. 4

(6)

The study of quantum plasmas has gained interest due to its wide-ranging applications in different astrophysical environments (viz. white dwarfs, neutron stars, etc.) [27,28], in laser-produced plasmas [9], in nano-structured materials [29] and quantum-wells [30], and in high-gain free-electron lasers [31, 32]. The linear and the nonlinear ion-acoustic waves in an unmagnetized quantum ep-i plasma have been investigated theoretically by Ali et al. [33]. In particular, they found that an increase in positron concentration decreased the amplitude and the width of the solitary waves. They also observed that the positron Fermi temperature to electron Fermi temperature increased the amplitude and width of the solitary waves. Khan and Haque [34] studied nonlinear structures in dissipative quantum e-p-i plasmas and showed that the height of the shocks could be increased by increasing the quantum diffraction parameter (H), the positron concentration, and the dissipation parameter. AkbariMoghanjoughi [35] considered a small amplitude ionacoustic solitary waves in a degenerate Thomas-Fermi ep-i magnetized plasma and rigorously investigated the effects of relativistic degeneracy. Pakzad [36] investigated quantum ion-acoustic shock waves in dissipative warm plasma with Fermi electrons and positrons and showed that these structures are affected by the plasma parameters. Khan and Mirza [37] theoretically investigated the formation and the propagation of shocks and solitons in an unmagnetized, ultradense plasma containing a degenerate Fermi gas of electrons and positron and a

classical ion gas by employing the Thomas-Fermi model. They observed that the effects of the ion temperature, positron concentration and dissipation are significantly depended on the nonlinear structures. The nonlinear propagation of fast and slow magnetosonic perturbation modes in an ultra-cold, degenerate (extremely dense) e-p plasma (containing non-relativistic, ultra-cold, degenerate electron and positron fluids) was rigorously investigated by El-Taibany et al. [38]. They found that the basic features of the electromagnetic solitary structures, which were found to exist in such a degenerate e-p plasma, were significantly modified by the effects of the degenerate electron and positron pressures. By considering the non-relativistic and ultra-relativistic degenerate quantum plasma. Hossen et al. [39–42] rigorously investigated the salient features of the solitary and shock structures. Mannan et al. [43,44] considered EA waves and rigorously investigated the basic characteristics of solitary, shock, and double layer structures. Zobaer et al. [45,46] also considered a quantum e-p-i plasma and theoretically investigated the basic features of solitary, shock, and double layer structures for the planar case. There are several cases of practical importance where planar geometry does not work, so one would have to consider a nonplanar geometry. The notable examples where nonplanar geometry plays a vital role are white dwarfs, neutron stars, black holes, circumstellar disks, dark molecular clouds, cometary tails, etc. To the best of our knowledge, no theoretical investigation has been made to study the extreme conditions of matter in both the non-relativistic and the ultra-relativistic limits by considering such quantum e-p-i plasma. Therefore, in our present work, we attempt to study the basic features of planar and nonplanar modified electron-acoustic (mEA) solitary waves by deriving the modified Burgers equations in a dense plasma containing non-relativistic degenerate inertial cold electrons, both non-relativistic and ultra-relativistic degenerate hot electron and inertial positron fluids, and positively-charged static ions. This manuscript is organized as follows. In Section II, the basic sets of equations for the system is investigated. In Section III, the nonplanar (K-dV) equation is derived. In Section IV, the small, but finite, amplitude solitary waves are numerically analyzed. A brief discussion is presented in Section V.

II. BASIC EQUATIONS We consider a nonplanar (cylindrical and spherical) geometry and nonlinear propagation of mEA waves in an unmagnetized, collisionless, dense plasma containing non-relativistic degenerate inertial cold electrons, both non-relativistic and ultra-relativistic degenerate hot electron and inertial positron fluids, and positively-charged static ions. Hence, at equilibrium, we have nc0 + nh0 = np0 + ni0 , where nc0 , nh0 , np0 , and ni0 are the number

Modeling of Modified Electron-acoustic Solitary Waves · · · – M. R. Hossen and A. A. Mamun

densities of cold electrons, hot electrons, positrons, and ions, respectively. The nonlinear dynamics of the electrostatic waves propagating in such a degenerate dense plasma system is governed by the following normalized equations 1 ∂ ν ∂ns + ν (r ns us ) = 0, ∂t r ∂r ∂uc ∂uc ∂φ K1 ∂nα c + uc − + = 0, ∂t ∂r ∂r nc ∂r ∂up ∂up ∂φ K1 ∂nα p + up + + = 0, ∂t ∂r ∂r np ∂r ∂nγ ∂φ − K2 h = 0, nh ∂r ∂r ∂nγp ∂φ − K2 = 0, np ∂r ∂r 1 ∂ ν ∂φ (r ) = −ρ, rν ∂r ∂r ρ = (1 + σ − μ)np − nc + (1 − δ − μ)nh + μ,

(7) (8) (9) (10) (11) (12) (13)

where ν = 0 for a one-dimensional planar geometry and ν = 1 (2) for a nonplanar cylindrical (spherical) geometry ns (s = c, h, p) is the plasma species number density normalized by its equilibrium value ns0 , us is the plasma species ion fluid speed normalized by Ci = (mh c2 /mc )1/2 with mh (mc ) being the hot electron (cold electron) rest mass, c is the speed of light in vacuum, and φ is the electrostatic wave potential normalized by mh c2 /e with e being the magnitude of the charge of an electron. Here μ (ni0 /nc0 ) is the ratio of the number density of positivelycharged ions to that of the cold electrons, σ (nh0 /nc0 ) is the hot electron to cold electron number density ratio, and δ (np0 /nc0 ) is the positron to cold electron number density ratio at equilibrium. The time variable (t) is normalized by ωpc = (4πn0 e2 /mc )1/2 , and the space variable (r) is normalized by λs = (me c2 /4πn0 e2 )1/2 . 2 The relativistic constants are K1 = nα−1 c0 Kc /mh c and γ−1 2 γ−1 2 K2 = nh0 Kh /mh c = np0 Kp /mh c .

(1)

∂ns ∂τ

(2)

Now, we derive the K-dV equation by employing the reductive perturbation technique to examine the electrostatic perturbations propagating in this dense plasma system, and we introduce the stretched coordinates [47] as follows: ζ = −1/2 (r + Vp t), 3/2

τ =

(15)

where Vp is the wave’s phase speed (ω/k, with ω being the angular frequency and k being the wave number of the perturbation mode), and is a smallness parameter measuring the weakness of the dispersion (0 < < 1). We then expand the parameters ns , us , φ, and ρ in power series of : 2 (2) ns = 1 + n(1) s + ns + · · ·,

us =

u(1) s (1)

+

+ φ

(16)

+ · · ·,

(17) (18)

ρ = ρ(1) + 2 ρ(2) + · · ·.

(19)

Now, expressing Eqs. (7)-(13) (using Eqs. 14 and 15) in terms of ξ and τ , and substituting Eqs. (16)(19) into (7)-(13), one can easily develop different sets of equations in various powers of . To the lowest order in , (1) (1) we have uc = Vp φ(1) /(Vp2 − K1 ), up = −Vp φ(1) /(Vp2 − (1)

(1)

(1)

K1 ), nc = −φ(1) /(Vp2 − K1 ), nh = np = φ(1) /K2 , (2+σ−μ)K2 + K1 , where K1 = αK1 and K2 = Vp = μ+δ−1 (2+σ−μ)K2 γK2 . The relation Vp = + K1 represents μ+δ−1 the dispersion relation as well as the phase speed for the modified electron-acoustic (mEA)-type electrostatic waves in this degenerate plasma under consideration. To the next higher order in , we obtain a sets of equations

(1)

∂φ(2) (α − 2) (1) 2 ∂uc ∂uc ∂ − u(1) + − K1 [n(2) (nc ) ] = 0, c c + ∂ζ ∂ζ ∂ζ ∂ζ 2

− Vp

∂up ∂up ∂ ∂φ(2) (α − 2) (1) 2 − u(1) − − K1 [n(2) (np ) ] = 0, + p ∂ζ ∂ζ ∂ζ ∂ζ p 2

(2)

2 u(2) s 2 (2)

+ · · ·,

φ = φ

− Vp

(2)

(14)

t,

∂ns ∂ (2) νus (1) − [u + n(1) = 0, s us ] − ∂ζ ∂ζ s Vp τ

(1)

∂up ∂τ

III. DERIVATION OF THE NONPLANAR K-DV EQUATION

− Vp

(1)

∂uc ∂τ

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(20)

(1)

(21)

(1)

∂φ(2) ∂ (γ − 2) (1) 2 (2) − K2 nh + (nh ) = 0, ∂ζ ∂ζ 2 (2) ∂φ (γ − 2) (1) 2 ∂ (2) − K2 n + (np ) = 0, ∂ζ ∂ζ p 2

(23) (24)

(22)

(2)

(2) n(2) c − (1 − δ − μ)nh − (1 + σ − μ)np =

∂ 2 φ(1) ∂ζ 2

(25)

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Now, combining Eqs. (20)-(24), we deduce a nonplanar K-dV equation ∂φ(1) ∂φ(1) ∂ 3 φ(1) νφ(1) + Aφ(1) +B = 0, (26) + 3 ∂τ ∂ζ ∂ζ 2τ where A=

2 (Vp2 − K1 ) 3Vp2 + K1 (α − 2)(2 + σ − μ) 3 2Vp (μ − σ) (Vp2 − K1 ) (γ − 2)(μ + δ − 1) + , (27) K2 2 2

B=

(Vp2 − K1 ) . 2Vp (μ − σ)

(28)

IV. NUMERICAL ANALYSIS AND RESULTS In this section, we now look at the Eq. (26) with the (1) term νφ2τ , which is due to the effects of the nonplanar (cylindrical and spherical) geometry. An exact analytic solution to Eq. (26) is not possible. However, for clear understanding, we first briefly discuss about the stationary solitary wave solution for equation (26). We should (1) note that for a large value of τ , the term νφ2τ is negli-

Fig. 1. (Color online) Nonlinear solitary waves are shown for different values of ν when cold electron, hot electron and positron fluids are non-relativistic degenerate with parameters u0 = 0.8, σ = 0.2, δ = 0.5, and μ = 0.9.

(1)

gible (i.e., νφ2τ → 0). Thus, in our numerical analysis, we start with a large value of τ (viz. τ = −9), and at this large (negative) value of τ , we choose the stationary solitary wave solution of Eq. (26) [without the term νφ(1) 2τ ] as our initial pulse. The stationary solitary wave solution of the standard K-dV equation is obtained by considering a frame ξ = ζ − u0 τ (moving with speed u0 ), and the solution is φ(1) (ν→0) = φm sech2 (

ξ ), Δ

(29)

Fig. 2. (Color online) Nonlinear solitary waves are shown for different values of ν when cold electron fluids are nonrelativistic degenerate and hot electron and positron fluids are ultra-relativistic degenerate with parameters u0 = 0.8, σ = 0.2, δ = 0.5, and μ = 0.9.

where the amplitude φm = 3u0 /A, and the width Δ = (4B/u0 )1/2 .

(30)

In case of solitary waves, the nonlinearity plays a predominant role because it’s caused by the balance between nonlinearity and dispersion. This is a unique phenomenon in that the degree of nonlinearity is proportional to the potential of the plasma system as a highly nonlinear medium causes a high electrostatic potential. Obviously from the Eq. (30) the height of the amplitude of the solitary structures is directly proportional to the solitary wave moving with u0 and inversely proportional to A. On the other hand, Eq. (30) also implies that the width of these solitary structures is directly proportional to the constant B and inversely proportional to the solitary wave moving with u0 . Our present investigation deals with the properties of cylindrical and spherical mEA solitary structures in a relativistic degenerate plasma (containing non-relativistic

degenerate inertial cold electrons, both non-relativistic and ultra-relativistic degenerate hot electron and inertial positron fluids, and positively-charged static ions), and we have derived the nonplanar K-dV equation. We have then analyzed the basic properties of the cylindrical and the spherical mEA solitary structures. The nonplanar K-dV equation derived here is valid only for the limits A = 0, A > 0 and A < 0 [48, 49]. Also, the initial condition that we used in our numerical results had the form of the stationary solutions (Eq. 29) at τ = −9 (negative value of τ ) [3, 50]. One can also consider the positive value of τ [11,51]. The positive or the negative values of τ just indicate the time where the stationary solitary pulse is first assumed to be observed. The conditions for the existence of cylindrical and spherical solitary structures and their basic characteristics are found to be significantly modified in the presence of non-relativistic degenerate inertial cold electrons, both non-relativistic


Fig. 3. (Color online) Time evolution of solitary wave profiles created in a plasma when cold electron, hot electron and positron fluids are non-relativistic degenerate with parameters u0 = 0.8, σ = 0.2, δ = 0.5, μ = 0.9, and ν = 1.

Fig. 4. (Color online) Time evolution of solitary wave profiles created in a plasma when cold electron fluids are nonrelativistic degenerate and hot electron and positron fluids are ultra-relativistic degenerate with parameters u0 = 0.8, σ = 0.2, δ = 0.5, μ = 0.9, and ν = 1.

and ultra-relativistic degenerate hot electron and inertial positron fluids, and positively-charged static ions. The mEA waves are modified more when the e-p plasma is non-relativistic degenerate (α = γ = 53 ) than when e-p plasma is ultra-relativistic degenerate (α = 53 ; γ = 43 ). The ranges (u0 = 0.01 − 1, μ = 0.2 − 0.9, σ = 0.1 − 0.5, and δ = 0.1 − 0.6) [52–56] of plasma parameters used in this numerical analysis are very wide and correspond to space and laboratory plasma situations. The solitary profiles for the planar and the nonplnar geometris (both non-relativistic and ultra-relativistic cases) are presented in Figs. 1 and 2. The strength of the solitary is found to be maximum for the spherical geometry, intermediate for the cylindrical geometry, and it is minimum for the planar geometry. Again, the steepness of the solitary front follows the same trend as the strength of the solitary front. Also, it is mentioned that the increase in steepness tells us that the change

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Fig. 5. (Color online) Nonlinear solitary waves are shown for different values of σ when cold electron, hot electron and positron fluids are non-relativistic degenerate with parameters u0 = 0.8, ν = 1, δ = 0.5, and μ = 0.9. The solid curves represent the non-relativistic case, and the dashed curves represent the ultra-relativistic case.

Fig. 6. (Color online) Nonlinear solitary waves are shown for different values of σ when cold electron fluids are nonrelativistic degenerate and hot electron and positron fluids are ultra-relativistic degenerate with parameters u0 = 0.8, ν = 2, δ = 0.5, and μ = 0.9. The solid curves represent the non-relativistic case, and the dashed curves represent the ultra-relativistic case.

in parameters downstream of the soliatry wave is drastic while the amplitude of the solitary wave determines its strength. The structures of the solitary profile in nonplanar geometries differs significantly from the structure of the solitary profile in the planar geometry due to the (1) presence of the νφ2τ term in Eq. (26). Figures 3 and 4 show the time evolutions of the solitary structures in cylindrical and spherical geometry, respectively. Clearly, as time decreases, the amplitude of the solitary waves in cylindrical and spherical geometry increases (due to the decrease in the nonlinearity coefficient A) significantly. In spherical geometries, the solitary wave propagates faster than it does in cylindrical geometries owing to the fact that the solitary wave diverge at large values of the

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Fig. 7. (Color online) Nonlinear solitary waves are shown for different values of δ when cold electron, hot electron and positron fluids are non-relativistic degenerate with parameters u0 = 0.8, ν = 1, σ = 0.2, and μ = 0.9. The solid curves represent the non-relativistic case, and the dashed curves represent the ultra-relativistic case.

Fig. 9. (Color online) Nonlinear solitary waves are shown for different values of μ when cold electron, hot electron and positron fluids are non-relativistic degenerate with parameters u0 = 0.8, ν = 1, σ = 0.2, and δ = 0.5. The solid curves represent the non-relativistic case, and the dashed curves represent the ultra-relativistic case.

Fig. 8. (Color online) Nonlinear solitary waves are shown for different values of δ when cold electron fluids being nonrelativistic degenerate and hot electron and positron fluids are ultra-relativistic degenerate with parameters u0 = 0.8, ν = 2, σ = 0.2, and μ = 0.9. The solid curves represent the non-relativistic case, and the dashed curves represent the ultra-relativistic case.

Fig. 10. (Color online) Nonlinear solitary waves are shown for different values of μ when cold electron fluids being nonrelativistic degenerate and hot electron and positron fluids are ultra-relativistic degenerate with parameters u0 = 0.8, ν = 2, σ = 0.2, and δ = 0.5. The solid curves represent the non-relativistic case, and the dashed curves represent the ultra-relativistic case.

potential in the case of spherical geometry. The increase is more pronounced in spherical geometry than it is in cylindrical geometry, which means that a density compression can be more effectively obtained in a spherical geometry. The cylindrical and spherical solitary structures at different values of σ are shown in Figs. 5 and 6 for both the non-relativistic and the ultra-relativistic cases. This shows that the amplitude of solitary structures increases with increasing value of σ. Actually, this happens because incredibly decreases both the predominant and the weak dissipative coefficients and the nonlinearity coefficient. Figures 7 and 8 shows the solitary structures at different δ in cylindrical and spherical geometries for both

the non-relativistic and the ultra-relativistic cases. The height and the steepness of the solitary profile are found to decrease with increasing value of δ. These happens because of the driving force of the EA wave, as the driving force for the EA wave is provided by the electron’s inertia. Actually, an increase in the positron concentration (depopulation of electrons) causes a decrease in the driving force, which is provided by the electron’s inertia, and consequently the solitary wave enervates. Figures 9-10 show the variations of cylindrical and spherical solitary structures at different μ for both the non-relativistic and the ultra-relativistic cases. The height and the steepness of the solitary profile are observed to decrease with increasing value of μ.


V. DISCUSSION A study of the nonlinear mEA solitary waves in a degenerate, dense, plasma system consisting of non-relativistic degenerate inertial cold electrons, both non-relativistic and ultra-relativistic degenerate hot electron and inertial positron fluids, and positivelycharged static ions has been carried out. The degenerate pressure is provided by the hot electrons and positron fluids whereas the inertia is provided by the cold electron and positron. The positively-charged static ions participate only in maintaining the quasi-neutrality condition at equilibrium. By using a reductive perturbation method we derived the nonplanar K-dV equation to describe the dynamics of such solitary waves. We observed that the solitary wave’s strength and the propagation speed of solitons were maximum for the spherical, intermediate for the cylindrical, and minimum for the planar geometries. We also observed that a decreasing nonlinear coefficient enhanced the solitary wave’s strength. Here, we should stress that the results of the present investigation should be useful for understanding the nonlinear features of localized electrostatic disturbances in some space and astrophysical plasma systems, particularly, white dwarfs, non-rotating neutron stars, etc. [2,3,24,25], where two distinct groups of relativistic electrons, relativistic positrons and static ions are the dominant plasma species.

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