Soliton and Shock Profiles in Electron-positron-ion Degenerate Plasmas for Both Nonrelativistic and Ultra-Relativistic Limits

1 Introduction

The propagation of the ion-acioustic (IA) waves is very important from the view of its vital role in unnderstanding the electrostatic disturbances in space and laboratory plasma. On the other hand, it is a common idea that electron-positron plasmas have presumably appeared in the early universe [1], [2] and are frequently encountered in active galactic nuclei [3] and pulsar magnetospheress [4], [5]. This electron-positron plasma is usually characterised as a fully ionised gas consisting of electrons and positrons of equal mass. The electron-positron plasmas are generated naturally by pair production in high-energy process in the vicinity of several astrophysical objects and laboratory experiments with finite life time [6]. Due to the long life time of positrons, most of the plasmas (astrophysical and laboratory) usually contains ions, in addition to the electrons and positrons [7]. It has also been shown that over a wide range of parameters, annihilation of electrons and positrons, which is the analog of recombination in plasma composed of ions and electrons, is relatively unimportant in classical [8], as well as in dense quantum plasmas [9] to study the collective plasma oscillations. Recently, there has been a great deal of interest in studying linear and nonlinear wave motions in such plasmas [10], [11]. The nonlinear studies have been focused in the nonlinear self-consistent structures [10], [11], [12] such as envelope solition and shokes. Soliton is self-reinforcing solitary wave which preserves its shape and speed, even after collisions with other solitary wave. When the dissipation is weak at the characteristic dynamical timescales of the system, it arises because of the balance between the effects of nonlinearity and dispersion. Korteweg–de Vries (KdV) equation is used to study the both solion and solitary waves. The properties of wave motion in an electron-positron-ion (e-p-i) plasma should be different from those in two-component electron-positron plasmas. Rizzato [13] and Berezhiani et al. [14] have investigated envelope solitons of electromagnetic waves in three-component e-p-i plasmas.

The ultradense degenerate electron-positron plasmas with ions are delivered to be found in compact astrophysical bodies like neutron stars and the inner layers of white dwarfs [9], [15], [16], [17] and also in the laser-matter interaction experiments [18], [19]. Therefore, the study of influence of quantum effects on dense e-p-i plasmas become important. A dense plasma is usually characterised as cold and degenerate such as the encountered in metals and semiconductors. However, it has been remarked that a hot fusion plasma such as that found in dense stellar objects e.g. neutrons stars and white dwarfs, may also be considered as quantum degenerate plasmas [20]. Quantum plasmas obey the Fermi–Dirac distribution leading to the Fermi pressure and new forces arising due to Bohm potential [20] play a vital role. In such environments, the electron-positron annihilation rate is higher then naturally expected. Thus, the electron and positron density are very high. The degenerate electron number density in such a compact object is so high (Table 1) 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 [17], [21], [22].

Several authors have theoretically investigated the collective effects in dense unmagnetised and magnetised e-p-i quantum plasmas under the assumption of low-phase velocity in comparison with electron or positron Fermi velocity [23], [24], [25]. It is found that Bohm potential leads to the wave dispersion due to quantum correlation of density fluctuations associated with wave-like nature of the charge carriers. In these studies, the authors have focused on the lower order quantum corrections appearing in the well-known classical modes. Nowadays, a number of authors have become interested to study the properties of matter under extreme conditions [26], [27], [28], [29]. Recently, a number of theoretical investigations have also been mate of the nonlinear propagation of electrostatic waves in degenerate plasma [30], [31], [32]. However, these investigations are based on the equation of state valid for the nonrelativistic limit. Some investigations have made of the nonlinear propagation of electrostatic waves in a degenerate dense plasma based on the degenerate electron equation of state valid for ultra-relativistic limit [33], [34], [35], [36].

Many scientist have considered unmegnetised nonrelativistic degenerate plasmas [30], [31] and ultra-relativistic degenerate plasmas [33], [35] while studying the nonlinear propagation of electrostatic waves. In this article, an attempt has been taken to develop a general equation [as shown in (6)] with the constants parameters [denoted in (7–9)], which can be used nonrelativistic and ultra-relativistic degenerate plasmas simultaneously. Earlier, Haider and Mamun [37] developed a general form of such a case and using the equation they studied nonlinear propagation of IA solitary wave with degenerate electrons. In this work, a more simplified general form of equation of state, to express the degenerate pressure for ultra-relativistic and nonrelativistic cases, has developed and applied in studying the ion-acoustic solitons and shokes in a degenerate dense plasma system in the absence of the magnetic field but containing inertial ion fluid and inertialess (comparing to ions) ultra-relativistic or nonrelativistic electrons and positrons. Standard KdV and KdV–Barger equation have derived with their constants to study soliton and shock profile, respectively, which might be relevant to interstellar compact object like white dwarf and neutron stars.

This article is organised as follows. A general equation for degenerate pressure of Chandrasekhar has been developed in Section 2. The basic governing equations in unmagnetised e-p-i plasma system are given in Section 3. The Solitary waves equation is derived and its solution in Section 4. The Shock waves equation is also derived as well as its solution in Section 5. A numerical analysis is discussed in Section 6 and finally, a brief discussion containing comperision is presented in Section 7.

2 General Equation for Degenerate Pressure of Chandrasekhar

According to the Chandrasekhar [26], [27], [28], the pressure for the nonrelativistic and ultra-relativistic degenerate electrons/fermions fluid can be expressed by the following equations, respectively as

Where, the values of K₁ and K₂ and ρ are given below

with h is the Plank’s constant, c is the speed of light, m is the mass of electron, n is the number density of plasma particle, μ is the molecular weight, and H is the mass of proton.

Now, generalising (1) and (2), we can write the following equation that can be used both nonrelativistic and ultra-relativistic limits

where γ=5/3 for nonrelativistic case and γ=4/3 for ultra-relativistic case and constant K is given by

with

3 Basic Equations

The nonlinear propagation of IA solitary and shock waves have been considered in a ultra cold degenerate plasma containing inertial ion fluid with inertia less electron and positron following ultra-relativistic or nonrelativistic limits. The one-dimensional nonlinear dynamics of electrostatic purturbations are governed by the following equations

Where n_i is the ion number density normalised by its equilibrium value n_i0, u_i is the ion fluid speed normalised by C_i=(m_ec²/m_i)^1/2 with m_e (m_i) being the rest mass of electron (ion), and c being the speed of light in vacuum. ψ is the electrostatic wave potential normalised by m_ec²/e with e being the magnitude of the charge of an electron, m_e is the mass of electron. The time variable (t) is normalised by ωpi−1=(mi/4πni0e2)1/2 and the space variables is normalised by λ_i=(z_im_ec²/4πn_i0e²)^1/2. In (12), n_s represents n_e (n_p) is the electron (positron) number density normalised by their equilibrium values n_e0 (n_p0), j=+ 1 for positron and j=−1 for electron and also β=(K/E)n0γ − 1. We have considered the viscuss term, i.e. coefficient of viscosity (η′) is equals to zero at the time of studying solitary waves.

Now, using equilibrium charge neutrality condition n_e0=n_i0+n_p0, one can write μ_e=1+μ_p where μ_e=n_e0/n_i0 and μ_p=n_p0/n_i0.

Rearranging (12), the value of n_e and n_p can be express, respectively, as

4 Solitary Wave

Introducing independent variables through the stretched coordinates [38], [39], [40], [41], [42], [43], [44], [45], to follow the reductive perturbation technique [38] to construct a weakly nonlinear theory for the electrostatic waves with a small but finite amplitude, as

ξ=ϵ1/2(x−V0t),τ=ϵ3/2t,

where ϵ is a small parameter measuring the weakness of the dispersion and V₀ is the unknown wave phase speed (to be determined later) is normalised by the ion-acoustic speed (C_i). It may be noted here that ξ is normalised by the Debye radius (λ_i), τ is normalised by the ion plasma period (ωpi−1). The perturbed quantities can be expanded about their equilibrium values in powers of ϵ as [38], [39], [40], [41], [42], [43], [44], [45]

Using the stretched coordinates and (16)–(18) in (10)–(13); and equating the coefficients of ϵ^3/2 from the continuity and momentum equation and coefficients of ϵ from Poisson’s equation, one can obtain the first-order continuity, momentum, and Poisson’s equation as

Comparing (20) and (21), the linear dispersion relation can be written as

To the next higher order of ϵ, i.e. equating the coefficients of ϵ^5/2 from continuity and momentum equation and coefficients of ϵ² from Poisson’s equation, one can write, respectively,

Now, using (23)–(25), KdV equation can be readily obtained as

where

Now, transforming the independent variables ζ and τ′ to ζ=ξ−U₀τ and τ′=τ and imposing the appropriate boundary conditions (viz. ψ⁽¹⁾→0, ∂ψ⁽¹⁾/∂ζ→0, ∂²ψ⁽¹⁾/∂ζ²→0 at ζ→± ∞), one can express the stationary solution [46], [47] of the KdV equation (26) as

This is to note here that U₀ is a constant speed, normalised by C_i, which describe the speed of the solitary structure. This is always a positive quantity and most of cases upper range is ≤ 1 (i.e. generally 0<U₀≤1).

The value of B is negligible compared to A, so we can write the amplitude (ψ_m) and the width (Δ) of the solitary wave denoted in equation (29) as

It is obvious from (30) and (31) that U₀ is a linear function of amplitude and inverse function of width, which mean that the profile of the faster solitary structures will be taller and narrower then slower one.

5 Shock Wave

Introducing the stretched coordinates in reductive perturbation method [38], [48], [49] to obtain KdV–Barger equation, as

ξ=ϵ1/2(x−V0t),τ=ϵ3/2t,η′=ϵ1/2η0,

and expanding the perturbed quantities about their equilibrium values in powers of ϵ as in (16)–(18) and equating the coefficients of the lowest order of ϵ from the continuity, momentum, and Poisson’s equation, one can obtain the first order of these equations as such as (19)–(21) and the linear dispersion relation as

which is same as the dispersion relation as solitary waves as in (22).

To the next higher order of ϵ, i.e. equating the coefficients of ϵ^5/2 from continuity and momentum equation and coefficients of ϵ² from Poisson’s equation, one can write, respectively,

Now, using (33)–(35), one can readily obtain the KdV–Barger equation as

where

Transforming the independent variables ζ and τ′ to ζ=ξ−U₀τ and τ′=τ, and imposing the appropriate boundary conditions as in the solitary waves, one can express the stationary solution [48], [49] of the KdV–Barger equation (36) as

where the amplitude (ψ_m as in equation (30) and the width Δ) are given by

It is found from (41) and (42) that U₀ is a linear function of amplitude and inverse function of width, as same as in the case of SWs, which mean that the profile of the faster shock waves will be taller and narrower then slower one.

6 Numerical Analysis

As (27) and (37) indicate that the value of A in both solitary and shock wave equations is same and in both case the amplitude is 3U₀/A, so the variation of amplitude of solitary and shock waves with depending parameters are same. Thus, the solitary and shock waves associate with a positive potential (ψ_m>0) if A>0 or a negative potential (ψ_m<0) if A<0. It is clear that A=0 corresponds to μp=(−3±2)/6 for ultra-relativistic limit and μ_p=−2/3 or −1/3 for nonrelativistic limit. It means that negative solitary and shock waves can be associate at μp<(−3±2)/6 for ultra-relativistic case and μ_p<−2/3 or −1/3 for nonrelativistic case. However, μ_p(=n_p0/n_i0) cannot be negative. So, the value of A is always positive and the solitary and shock waves always associated with positive potentials. Figure 1 shows the variation of amplitude (ψ_m) of the solitons and shocks with μ_p for U₀=0.1 and n₀=10³⁶ both ultra-relativistic (dotted line) and nonrelativistic (solid line) limits. As the value of μ_p increases, the amplitude decreases for both the cases but the amplitude for nonrelativistic case is always higher than the ultra-relativistic situation. Figure 2 shows the variation of amplitude of the solitary and shock waves with particle number density for both ultra-relativistic and nonrelativistic cases. From this figure, it can be said that there is critical value (≈8×10³⁵ for U₀=0.1 and μ_p=1) below which the amplitude of ultra-relativistic case is higher than for nonrelativistic cases for same value of particle number density and above this critical value the amplitude of nonrelativistic cases is higher.

Figure 1:

Showing the variation of amplitude of solitary and shock waves with μ_p with U₀=0.1 and n₀=10³⁶ for γ=4/3 (dotted curve) and γ=5/3 (solid curve).

Figure 2:

Showing the variation of amplitude of solitary and shock waves with ln n₀ with U₀=0.1 and μ_p=1 for γ=4/3 (dotted curve) and γ=5/3 (solid curve).

The variation of width of solitary waves with μ_p and ln n₀ is shown in Figures 3 and 4 for both ultra-relativistic and nonrelativistic cases. Increasing the value of μ_p, the width of the solitary waves decreases which means that positron number density makes the solitary waves narrower. It is also found that the width is higher for nonrelativistic cases than ultra-relativistic cases for the same μ_p. However, increasing the plasma number density the width increases for both the cases. Below the critical value, the width is higher for ultra-relativistic cases than nonrelativistic one, and after the critical value, the rate of increment is higher for nonrelativistic cases then ultra-relativistic one.

Figure 3:

Showing the variation of width of solitary waves with μ_p with U₀=0.1 and n₀=10³⁶ for γ=4/3 (dotted curve) and γ=5/3 (solid curve).

Figure 4:

Showing the variation of width of solitary waves with ln n₀ with U₀=0.1 and μ_p=1 for γ=4/3 (dotted curve) and γ=5/3 (solid curve).

7 Discussion

Soliton and shock profiles have been analysed in a collisionless electron-positron-ion plasma containing ultra-cold inertial ion fluid and ultra-cold inertia less degenerate electron and positron fluid by deriving KdV and KdV–Barger equation, respectively, by reductive perturbation method. The findings are summarised as follows:

1. In present plasma system, solitons and shocks are always associated with positive potentials and propagate with same amplitude.

2. Profile of the faster solitons and shocks are more spiky than slower one.

3. There must be a critical value of μ_p and n₀, considering other parameters constant, for which amplitudes of solitons and shocks are same for ultra-relativistic and nonrelativistic cases.

4. The amplitude and width of the solitons increases with increasing n₀ for both ultra-relativistic and nonrelativistic cases, which means that higher n₀ leads stronger soliton profile.

5. Below the critical value both the amplitude and width for ultra-relativistic case are higher than nonrelativistic one and above the critical value it is higher for nonrelativistic case.

6. Increasing the value of μ_p, the amplitude and width of solitons decrease for both ultra-relativistic and nonrelativistic cases, i.e. higher μ_p makes week soliton profile.

7. For the case of shocks the amplitude increases with n₀ and decreases with μ_p for both the cases as such as solitons but the width is a function of η₀, not n₀ and μ_p. So the width of the shocks is not same as solitons and independent of n₀ and μ_p and increases with increasing η₀ for both ultra-relativistic and nonrelativistic cases.

The ranges of the plasma parameters for astrophysical compact objects like white dwarfs and neutron stars are very wide. The plasma parameters for white dwarfs and neutron stars are shown in Table 1. The plasma parameters used in our present theoretical analysis correspond to white dwarfs. However, our present theoretical analysis can applied to neutron stars. Therefore, our present results may be useful for understanding the localised electrostatic disturbance in astrophysical compact objects, particularly, in white dwarf and neutron stars.