2. Governing equations and model

We are interested in the formation and propagation of weakly magnetosonic envelope solitons in an astrophysical relativistic dense magnetoplasma. For this purpose, we solve the fluid and Maxwell’s equations together and obtain a set of equations describing the dynamics of the amplitude modulated waves. The inertialess electrons are assumed to be relativistic and degenerate, while the singly ionized positive ions are taken as classical, dynamic and magnetized. At equilibrium, the charge neutrality condition demands $n_{e}^{(0)}\approx n_{i}^{(0)}$ , where $n_{i}^{(0)}$ ( $n_{e}^{(0)}$ ) is the equilibrium ion (electron) number density. Such a dense plasma is subjected to a magnetic field along the $z$ -direction, i.e. $\boldsymbol{B}=(0,0,B^{(0)}+B)$ , where $B^{(0)}(B)$ is the external ambient (perturbed) magnetic field. Wave oscillations however propagate along the $x$ -axis $[\text{i.e. }\boldsymbol{{\rm\nabla}}=(\partial _{x},0,0)]$ . For nonlinear analysis, we employ the reductive perturbation technique to obtain an NLS equation and investigate the properties of the magnetosonic dark and grey solitons in a relativistic dense quantum plasma.

For relativistic and degenerate electrons, the degeneracy pressure law may be expressed (Chandrasekhar Reference Chandrasekhar1935a ,Reference Chandrasekhar b ) as:

where ${\it\eta}_{e}$ is the normalized relativity parameter, $m_{e}$ is the electron mass and $h$ is Planck’s constant. Expanding (2.1) in a Taylor series around an equilibrium density $n_{e}^{(0)}$ , we obtain (Masood & Eliasson Reference Masood and Eliasson2011) the following expression

or

where $P_{e}^{(0)}$ is the equilibrium pressure and ${\it\delta}n_{e}$ represents all the perturbations to the number densities (viz., ${\it\delta}n_{e}={\it\varepsilon}n_{e}^{(1)}+{\it\varepsilon}^{2}n_{e}^{(2)}+\cdots \,$ ), ${\it\gamma}_{e0}=(1+{\it\eta}_{e0}^{2})^{1/2}$ is the relativistic factor for electrons, ${\it\eta}_{e0}~(=p_{e}^{(0)}/m_{j}c)$ is the relativity parameter, $p_{e}^{(0)}=(3h^{3}n_{e}^{(0)}/8{\it\pi})^{1/3}$ is the electron Fermi momentum and ${\it\epsilon}_{Fe}~[=(3h^{3}n_{e}^{(0)}/8{\it\pi})^{2/3}/2m_{e}]$ is the electron Fermi energy. For the NR case, we assume ${\it\eta}_{e0}\ll 1$ and consequently (2.3) is reduced to

For UR case, viz., ${\it\eta}_{e0}\gg 1$ , we obtain from (2.3) as

It is important to note that we have incorporated a third-order expansion of the Taylor series to study the envelope excitations. The nonlinear dynamics of the magnetoacoustic waves in a degenerate magnetoplasma with relativistic electrons is governed by the following quantum magnetohydrodynamic equations (Haas Reference Haas2005):

and

where $\boldsymbol{J}~(=\sum _{j=e,i}q_{j}n_{j}\boldsymbol{v}_{j})$ is the plasma current density, $q_{j}=e(-e)$ represents the ionic (electronic) charge, $\boldsymbol{v}_{i}(\boldsymbol{v}_{e})$ is the ionic (electronic) fluid velocity, $m_{i}$ is the ionic mass and $\hbar~(=h/2{\it\pi})$ , is the reduced Planck’s constant. In order to derive the governing equations involving a one-fluid QMHD model, we combine (2.8) and (2.10) to obtain

Inserting (2.11) into (2.7) and (2.9), we have

and

It is important to note that the charge neutrality condition $(n_{e}\approx n_{i}=n)$ is imposed in the set of (2.6), (2.12) and (2.13) by assuming that the perturbation scale length, $({\it\lambda})$ , associated with the magnetosonic wave is larger than the characteristic scale length, $({\it\lambda}_{Fe})$ being termed the plasma approximation (Chen Reference Chen1983). The magnetosonic waves propagate along the $x$ -axis and are perpendicular to the magnetic field $\boldsymbol{B}(x,t)$ in the $z$ direction, therefore (2.6), (2.12) and (2.13) can be normalized in a one-dimensional form as:

and

with

In obtaining (2.14)–(2.16), we have used the normalizations: $\bar{x}\rightarrow x{\it\omega}_{pi}/C_{s}$ , $\bar{t}\rightarrow t{\it\omega}_{pi}$ , $\bar{B}\rightarrow B/B^{(0)}$ , $\bar{u}\rightarrow v_{ix}/C_{s}$ and $n\rightarrow n/n^{(0)}$ . Other parameters, such as the ion-plasma frequency, the ion-sound speed, ion-Alfvén speed and quantum parameter are given, respectively, by ${\it\omega}_{pi}=(e^{2}n^{(0)}/{\it\varepsilon}_{0}m_{i})^{1/2}$ , $C_{_{s}}=c(m_{e}/m_{_{i}})^{1/2}$ , $C_{A}=B^{(0)}/({\it\mu}_{0}m_{_{i}}n^{(0)})^{1/2}$ and $H=\hbar{\it\omega}_{pi}/m_{e}c^{2}$ .

(Note that (2.14)–(2.16) are, the limiting case of the QMHD equations used in Ref. (Wang et al. Reference Wang, Lü and Eliasson2013a ) for investigating the modulational instability in a Fermi–Dirac Pauli plasma.) To proceed further, we drop the bar $(-)$ notation from the physical variables for simplicity and introduce the following stretched coordinates (Asano, Taniuti & Yajima Reference Asano, Taniuti and Yajima1969; Taniuti & Yajima Reference Taniuti and Yajima1969; Kawahara Reference Kawahara1973; Sabry et al. Reference Sabry, Moslem, Shukla and Saleem2009)

where ${\it\varepsilon}\ll 1$ is a small dimensionless parameter and $V_{g}$ is the envelope group speed. A generalized state vector $\boldsymbol{S}$ representing the physical variables of (2.14)–(2.16), can be written as,

where

where $\boldsymbol{S}_{l}^{(0)}$ stands for the equilibrium state of the variables and $k({\it\omega})$ denotes the carrier wavenumber (frequency). The quantity $\boldsymbol{S}_{l}^{(m)}$ must be real and can be taken as $\boldsymbol{S}_{-l}^{(m)}=\boldsymbol{S}_{l}^{(m)\ast }$ , where the asterisk designates the complex conjugate.

Plugging (2.18) and (2.19) into (2.14)–(2.16), we find the first harmonic perturbations (i.e. $m=1$ and $l=1$ ) of the ion number density and velocity in terms of $B_{1}^{(1)}$ as,

and

Equation (2.23) shows a linear dispersion relation for the fast magnetoacoustic waves in a semi relativistic degenerate magnetoplasma. It is clear that the wave dispersion is significantly modified by the relativistic parameter ${\it\eta}_{e0}$ and the quantum Bohm potential through the parameters ${\it\alpha}$ and $H$ . If we restore the dimensions, then (2.23) can be expressed

For ${\it\eta}_{e0}\ll 1$ , (2.24) gives a dispersion of the form ${\it\omega}=k(C_{A}^{2}+2{\it\epsilon}_{Fe}/3m_{i}+\hbar^{2}k^{2}/4m_{e}m_{i})^{1/2}$ in the NR case. However, for the UR case, we assume ${\it\eta}_{e0}\gg 1$ and as a result, (2.24) may be written ${\it\omega}=k(C_{A}^{2}+2{\it\epsilon}_{Fe}/3m_{i}{\it\eta}_{e0}+\hbar^{2}k^{2}/4m_{e}m_{i})^{1/2}$ . By neglecting the quantum parameter $H=0$ and replacing the electron Fermi energy by its thermal energy ${\it\epsilon}_{Fe}\rightarrow K_{B}T_{e}$ , the Fermi sound speed approaches a classical sound speed, i.e. $C_{sF}~(=2{\it\epsilon}_{Fe}/3m_{i}(1+{\it\eta}_{e0}^{2})^{1/2})\rightarrow C_{s}$ , and hence our result (2.24) agrees with the classical result by ignoring the displacement current term in Faraday’s law for low-frequency waves (Chen Reference Chen1983).

The first harmonic perturbations in the second-order approximation, i.e. $m=2$ and $l=1$ , are given by:

and

Maintaining the compatibility condition, the group velocity is

The group velocity strongly depends upon the relativistic and quantum Bohm potential effects.

The second harmonic mode $(l=2)$ in the second-order approximation $(m=2)$ can be determined by collecting the higher order terms of the perturbation series in (2.14)–(2.16), as

and

with dimentionless coefficients

and

Equations (2.28)–(2.30) infer that the above harmonic perturbations are due to the nonlinear self-interactions of the carrier waves. Similarly, nonlinear self-interactions of the carrier envelope may initiate the generation of zeroth harmonics. Thus, to determine the strength of such harmonic modes, we take $(l=0)$ in the second-order $(m=2)$ expansion of (2.14)–(2.16) to obtain

and

with

and

The first harmonic perturbation of the third order can be determined by taking $m=3$ and $l=1$ in (2.14)–(2.16), which yields a set of equations containing the second- and third-order perturbed quantities, namely, $n_{1}^{(2)}$ , $B_{1}^{(2)}$ , $u_{1}^{(2)}$ , $n_{1}^{(3)}$ , $B_{1}^{(3)}$ and $u_{1}^{(3)}$ after some mathematical manipulation we arrive at the nonlinear Schrödinger (NLS) equation, which describes the nonlinear evolution of the quantum magnetosonic envelope solitons, as

For simplicity, we have assumed $B_{1}^{(1)}\equiv B$ . The dispersion and nonlinearity coefficients, respectively, can be expressed in the following way

and

where

and

It is important to mention that by taking double derivatives of (2.23), one must satisfy the condition that $P=\partial ^{2}{\it\omega}/\partial k^{2}\equiv \partial V_{g}/\partial k$ , where $V_{g}$ is given by (2.27).

3. Results and discussion

For numerical illustration, we investigate the effects of degeneracy pressure and quantum Bohm potential on the profiles of small amplitude magnetosonic dark/grey solitons in a semi relativistic collisionless dense magnetoplasma. As astrophysical objects, for example white dwarf, magnetars and neutron stars contain a dense plasma (Mahmood, Sadiq & Haque Reference Mahmood, Sadiq and Haque2013) of number density range $n^{(0)}=(10^{36}{-}10^{40})~\text{m}^{-3}$ and magnetic field strength $B_{z}^{(0)}=(10^{7}{-}10^{10})$  T, we therefore choose $n^{(0)}=(10^{36}{-}10^{37})~\text{m}^{-3}$ and $B_{z}^{(0)}=10^{8}$  T, for our numerical analysis. By taking the values of number density $n^{(0)}=10^{36}~\text{m}^{-3}$ , and $n^{(0)}=10^{37}~\text{m}^{-3}$ , with $B^{(0)}=10^{8}$  T, one can compute other plasma parameters such as the ratio of Alfvén-to-acoustic speed $C_{A}/C_{s}=0.31$ , 0.09 relativistic parameter ${\it\eta}_{e0}=1.19$ , 2.5 quantum parameter $H=0.0016$ , 0.005 the ratio of dispersion to nonlinearity coefficients $P/Q=-5\times 10^{-5}$ and $-$ 0.001 at the normalized wavenumber $k=1$ . As the electron Fermi energy at these number densities ( ${\it\epsilon}_{Fe}=5.8\times 10^{-14}$  and $2.7\times 10^{-13}$  J) exceeds the electron rest mass energy, the relativistic effects of the degenerate electrons cannot be ignored and as a result, the Chandrasekhar degeneracy pressure law becomes applicable. It is also worth mentioning that the Chandrasekhar degeneracy pressure is specifically derived for a spherical geometry which also represents the relativistic effects associated with a degenerate fluid (electrons). Thus, our model describes a three-dimensional plasma system in which the magnetosonic wave propagates in one dimension along the $x$ -axis (Rahman et al. Reference Rahman, Mc Kerr, El-Taibany, Kourakis and Qamar2015). Furthermore, we have included the third order in the Chandrasekhar pressure (2.3) by using Taylor expansion to investigate the envelope solitons. Because the NLS equation is based on the third-order expansion of the perturbations. Moreover, the effect of the spin magnetization energy at magnetic field strength $B^{(0)}=10^{8}$  T becomes very small compared to the electron Fermi energy. Therefore, the spin effects are neglected in the present analysis when ${\it\mu}_{B}B^{(0)}\ll {\it\epsilon}_{Fe}$ holds, where ${\it\mu}_{B}$ is the Bohr magneton. The plasma parameters, namely ${\it\omega}$ , $k$ , $V_{g}$ , $P$ and $Q$ etc., are dimensionless. For clarity, we will from this point associate a bar $(-)$ sign to the corresponding parameters.

To examine the impact of the relativistic effect and the associated quantum Bohm potential for degenerate electrons, we plot the normalized $\bar{{\it\omega}}{-}\bar{k}$ relation of the magnetosonic waves and solve (2.23) numerically in figure 1(a). The variation of the number density $n^{(0)}=10^{36}~\text{m}^{-3}$ (solid curve), $n^{(0)}=5\times 10^{36}~\text{m}^{-3}$ (thin curve) and $n^{(0)}=10^{37}~\text{m}^{-3}$ (dashed curve) with fixed magnetic field strength $B^{(0)}=10^{8}$  T, increase the relativistic and quantum parameters which leads to a modification to the angular wave frequency as a function of the (normalized) wavenumber. Figure 1(b) represents the modification due to density variations on the group velocity $(\bar{V}_{g})$ associated with modulated magnetosonic waves. Moreover, taking the quantum parameter $H=0$ , the wave dispersion is significantly affected and as a result, the wave frequency (group velocity) varies linearly (remains constant) with respect to $\bar{k}$ see figure 1(c,d).

Figure 1.

The normalized angular frequency $\bar{{\it\omega}}~(={\it\omega}/{\it\omega}_{pi})$ and group velocity $\bar{V}_{g}~(=V_{g}/C_{s})$ of magnetosonic waves are plotted against the carrier wavenumber $\bar{k}~(=kC_{s}/{\it\omega}_{pi})$ in (a) and (b), respectively, with variations in relativistic and quantum diffraction parameters as, ${\it\eta}_{e0}=1.19$ , $H=0.001$ (solid line), ${\it\eta}_{e0}=2.04$ , $H=0.003$ (thin line), ${\it\eta}_{e0}=2.5$ and $H=$ $0.005$ (dashed line). The same physical quantities are plotted in (c) and (d) for $H=0$ .

To proceed further, we analyse the stability/instability criteria for envelope magnetosonic solitons in a semi relativistic degenerate plasma with different plasma parameters. For this, we have plotted the dispersion $(\bar{P})$ and nonlinearity $(\bar{Q})$ coefficients, as well as their ratio $(\bar{P}/\bar{Q})$ in figure 2. Note that the coefficient $\bar{P}$ always bears a positive sign, whereas $\bar{Q}$ is negative, hence the condition $\bar{P}/\bar{Q}<0$ holds. Such a system is favourable for modulationally stable waves, e.g. dark and grey solitons. Since the relativistic, and quantum parameters lead to an enhancement of the ratio $(\bar{P}/\bar{Q})$ , it physically affects the spatial extent of the envelope solitons (as discussed in the latter paragraph). Equations (2.40) and (2.41) also infer that the coefficients $\bar{P}$ and $\bar{Q}$ depend on the strength of the external magnetic field, which in turn affects the ratio $\bar{P}/\bar{Q}$ , as shown in figure 2(d). It may be noticed that the magnetic field strength effectively decreases the magnitude of $P/Q$ ratio at large wavenumber and consequently, reduces the width of the envelope solitons. It has been confirmed analytically that, even if it is assumed $H=0$ (which is not valid in this model), all of the terms on the left-hand side in (2.29) are cancelled out making $(B_{1}^{(1)})^{2}=0$ , as obtained earlier by Wang et al. (Reference Wang, Li, Dai, Chen and Zhang2013b ) (see (2.17)). However, this will lead to (2.39) with second-order secular terms in the form $B_{2}^{(2)}$ whose analytical solution is not possible. Consequently, in our present investigations, we cannot take $H=0$ .

Figure 2.

The normalized dispersion $(\bar{P})$ , nonlinearity $(\bar{Q})$ coefficients and ratio $(\bar{P}/\bar{Q})$ are depicted in (a), (b) and (c), respectively, versus the wavenumber $(\bar{k})$ for ${\it\eta}_{e0}=2.5$ , $H=0.005$ (solid line), ${\it\eta}_{e0}=3.2$ , $H=0.007$ (thin line), ${\it\eta}_{e0}=3.7$ and $H=0.009$ (dashed line) and (d) taking ${\it\eta}_{e0}=2.5$ , $H=0.005$ , $\bar{P}/\bar{Q}$ is plotted for $B^{(0)}=10^{8}$  T (solid line) $B^{(0)}=2\times 10^{8}$  T (thin line) $B^{(0)}=3\times 10^{8}$  T (dashed line).

Figure 3(a) depicts the real part of the solution $\bar{B}=\sqrt{\bar{{\it\rho}}}\exp (\text{i}{\it\theta})$ using (2.44), showing the propagation of a dark soliton in a normalized spatial coordinate $\bar{{\it\zeta}}$ . A dark soliton is a zero amplitude (void) region at $\bar{{\it\zeta}}=0$ with a non-zero amplitude on both sides of $\bar{{\it\zeta}}=0$ forming a uniform structure travelling in a nonlinear, dispersive medium. The profile of a dark (as well as grey) soliton exhibits rapid oscillations of the carrier wave inside a wave packet. Figure 3(b) shows the above solution for higher values of the relativistic and quantum parameters viz., ${\it\eta}_{e0}=4.4$ and $H=0.01$ . It can be seen that the increase in relativistic and quantum parameters cause an enhancement in the width of the dark soliton because the dispersion to nonlinearity $\bar{P}/\bar{Q}$ ratio is affected (as discussed earlier).

Figure 3.

The profile of a dark soliton is shown in normalized spatial coordinates $\bar{{\it\zeta}}({\it\zeta}{\it\omega}_{pi}/C_{s})$ for (a) ${\it\eta}_{e0}=2.5$ , $H=0.005$ and (b) ${\it\eta}_{e0}=4.4$ , $H=0.01$ .

A grey envelope soliton may also be observed in the modulationally stable regime when $\bar{P}/\bar{Q}<0$ . The profile of a grey soliton is shown in figure 4(a). Note that the grey soliton is excited with a locally reduced and non-zero wave amplitude in a semi relativistic dense plasma. The intensification of the relativistic and quantum parameters affects the grey soliton in a similar fashion to the dark soliton (see figure 4 b). Recall that the $\bar{P}/\bar{Q}$ ratio reduces when the magnetic field strength is increased. Finally, it is concluded that the magnetic field strength is favourable for the formation of narrower envelope dark/grey solitons in a relativistically degenerate plasma.

Figure 4.

The profile of a grey soliton is plotted versus the spatial coordinate $(\bar{{\it\zeta}})$ for (a) ${\it\eta}_{e0}=4.4$ , $H=0.011$ and (b) ${\it\eta}_{e0}=5.5$ , $H=0.016$ .

4. Summary

To summarize, we have presented the propagation characteristics of small amplitude quantum magnetosonic dark/grey envelope solitons in a semi relativistic degenerate collisionless magnetoplasma whose constituents are relativistically degenerate electrons and inertial and non-degenerate cold ions. Employing the one-fluid QMHD equations and the reductive perturbation technique, we have obtained an NLS equation, which admits modulationally stable waves under the condition $\bar{P}/\bar{Q}<0$ , where $\bar{P}$ and $\bar{Q}$ represent the dispersion and nonlinearity coefficients associated with the NLS equation. The solutions of the NLS equation show modulationally stable excitations, namely, dark/grey solitons. It is deduced that the electron relativistic degeneracy and quantum diffraction parameters significantly affect the dispersion-to-nonlinearity ratio $(\bar{P}/\bar{Q})$ and hence lead to an increase in the spatial width of the envelope soliton. However, the impact of the magnetic field strength has the opposite effect on envelope solitons. Our results are important for understanding the formation and propagation characteristics of magnetosonic dark and grey solitons in a semi relativistic degenerate plasma, e.g. white dwarf, magnetars and neutron stars, where the relativistic degeneracy pressure plays an important role in stabilization.