Authors: Rupak Dey, Gadadhar Banerjee, Amar Prasad Misra, Chandan Bhowmik
Categories: Article, Astrophysical plasmas, Plasma physics
Source: Scientific Reports
The theory of ion-acoustic solitons in nonrelativistic fully degenerate plasmas and nonrelativistic and ultra-relativistic degenerate plasmas at low temperatures is known. We consider a multi-component relativistic degenerate electron-positron-ion plasma at finite temperatures. Specifically, we focus on the intermediate region where the particle’s thermal energy (kBT) and the rest mass energy (mc2) do not differ significantly, i.e., kBT∼mc2. However, the Fermi energy (kBTF) is larger than the thermal energy and the normalized chemical energy (ξ=μ/kBT) is positive and finite. Two different parameter regimes with β≡kBT/mc2<1 and β>1, relevant for astrophysical plasmas, are defined, and the existence of small amplitude ion-acoustic solitons in these regimes are studied, including the critical cases where the known KdV (Korteweg–de Vries) theory fails. We show that while the solitons with both the positive (compressive) and negative (rarefactive) potentials coexist in the case of β<1, only compressive solitons can exist in the other regime (β>1). Furthermore, while the rarefactive solitons within the parameter domains of β and ξ can evolve with increasing amplitude and hence increasing energy, the energy of compressive solitons reaches a steady state.
**Subject ** Plasma physics, Astrophysical plasmas
Nonlinear propagation of solitary waves in electron-positron-ion (e-p-i) plasmas has received significant research interests for understanding the electrostatic as well as electromagnetic disturbances in various plasma environments^1–5^. The dynamics of collective processes in degenerate dense e-p-i plasmas, which are frequently present in laser-solid interaction experiments^6,7^, as well as in dense astrophysical objects, such as those in active galactic nuclei^8^, white dwarfs^9^, pulsar magnetosphere^10^, the early universe, neutron star^11^, quasars, accretion discs and sun atmosphere^12^, modifies the existing features of nonlinear waves. In degenerate plasmas, physical parameters including the density, magnetic field and the particle temperature can play significant roles in the evolution of electrostatic and electromagnetic waves^13^. Depending on whether the Fermi energy is much larger than, close to or much smaller than the rest mass energy, degenerate species like electrons and positrons in plasmas may be nonrelativistic, relativistic, or ultra-relativistic. Thus, it is desirable to have a pressure law and the particle distribution that can efficiently and accurately represent the relevant physics of relativistic degenerate plasmas. In this context, the characteristics of linear and nonlinear electrostatic waves in relativistic degenerate e-p-i plasmas have been the focus of various studies over the last few years^2,14–16^.
In degenerate e-p-i plasmas, as the electrons and positrons are extremely dense, their inter particle distance is comparable to the corresponding thermal de-Broglie wavelength and so they obey the Fermi Dirac (FD) distribution instead of the Maxwell-Boltzmann distribution. The FD distribution is mostly used when the thermodynamic temperature Tj of j-th species particle is comparable to the corresponding Fermi temperature TFj (j=e for electrons and j=p for positrons). In particular, the limiting conditions Tj≫TFj and Tj≪TFj, respectively, correspond to the non-degenerate and completely degenerate states. However, in most real situations, either Tj
The primary elements of white dwarf stars are carbon, oxygen, and completely ionized helium and the typical particle number density is roughly of the order of 1032m-3 or more. For these kinds of extremely dense astrophysical objects where relativistic temperatures are common and particles’ velocities approach those of light, the relativistic effects are crucial. Thus, the nonlinear effects in such relativistic degenerate plasmas at finite temperature are able to provide interesting new insights of localization of electrostatic waves. To the best of our knowledge, no effort has been made to study the characteristics of ion-acoustic solitons in relativistic degenerate e-p-i plasmas at finite temperature, especially in the intermediate regimes where the particle’s Fermi energy does not significantly differ from the particle’s thermal energy and the rest mass energy and the chemical energy is positive and larger than the thermal energy. We ought to mention that the mechanism for the excitation of ion-acoustic solitons, to be discussed in the manuscript, is not related to the quantum electrodynamics in which the Schwinger limit may be applicable. Also, this study does not consider the mechanism of electron-positron pair creation and annihilation in a strong electromagnetic field that may be of the order of the Schwinger field strength or beyond. In astrophysical environments, e.g., inside a white dwarf, the mean energies of electrons and positrons increase as the stellar mass increases so that for a sufficiently massive white dwarf, the electrons and positrons can be relativistic.
It is to be noted that although high-density electron-positron pairs are efficiently produced in laser-matter interactions^6,7^ or observed in astrophysical environments^24^, there are also issues with the electron-positron annihilation rate compared to the time scale of oscillations, especially when plasmas are in local thermodynamic and chemical equilibrium. It has been shown that such annihilation rate can significantly drop with decreasing values of the particle temperature Tj and freezes out at Tj=16KeV≈1.85×108 K^24^. In the environments of white dwarfs with central temperature Tj∼107-1010K and particle number density ∼1028-1035cm-3, the electron-positron annihilation time can remain longer than the ion plasma period. So, for the excitation of ion-acoustic waves in such high density plasmas, the electron-positron annihilation can be safely neglected.
There are two basic approaches used to investigate the nonlinear evolution of electrostatic solitary the reductive perturbation technique^25^ and the Sagdeev pseudopotential approach^4,26,27^. In the former, the Korteweg–de Vries (KdV) equation is derived to describe the evolution of one-dimensional solitons. However, the KdV equation may not be valid when the nonlinear coefficient, say A vanishes or tends to vanish. In these critical situations, the higher-order nonlinear corrections are considered to derive the modified KdV (mKdV) equation (in the case of A=0) and/or the Gardner equation (in the case of A≃0)^28^. Numerous investigations have been made to study the nonlinear propagation of mKdV and Gardner solitons in multi-component plasmas with Maxwellian/non-Maxwellian particle distributions^29–33^.
In this work, our aim is to study the existence of ion-acoustic solitons in an intermediate regime of relativistic degenerate plasmas at finite temperature where the Fermi energies of electrons and positrons do not significantly differ from their thermal energies and the rest mass energy. The manuscript is organized in the following way. Section “The model” presents the modeling of relativistic degenerate e-p-i plasmas at finite temperatures. It demonstrates the basic set of fluid equations for the nonrelativistic classical thermal ions and relativistic degenerate electrons and positrons at finite temperatures. Using the Fermi-Dirac distribution, the number densities of electrons and positrons are also derived in two particular cases, namely kBTj
We consider the nonlinear excitation of electrostatic waves at ionic time scale in an unmagnetized multi-component plasma with relativistic flow of degenerate electrons and positrons at finite temperature and nonrelativistic classical singly charged positive thermal ions. The dynamics of relativistic electrons and positron fluids in one-dimensional geometry is given by^34–36^
where d/dt≡∂/∂t+vj∂/∂x and we have assumed that the time scale of variation of the pressure is much smaller than that of the electron and positron density fluctuations, i.e., (1/Pj)(dPj/dt)≫(1/nj)(dnj/dt). The symbols qj, nj, vj, Pj, and γj=1/1-vj2/c2, respectively, denote the particle’s charge, the fluid number density in the laboratory frame such that nj/γj is the proper number density, the fluid velocity, the total relativistic degeneracy pressure at finite temperature, and the Lorentz factor for the j-th species particles [j=e(p) for electrons (positrons)]. Also, ϕ is the electrostatic potential, qj=-e(e) for electrons (positrons) with e denoting the elementary charge, and Hj is the relativistic enthalpy per unit volume of each fluid species j, which includes the rest mass energy density, the internal energy density and the relativistic pressure.
The equations for the classical thermal ion fluids are
mi, ni, vi, and Ti, respectively, denote the mass, number density, velocity, and the thermodynamic temperature of ions, and kB is the Boltzmann constant. The above set of equations (1)–(4) are closed by the following Poisson equation.
It is imperative to make some assumptions on the fluid equations of electrons and positrons without loss of generality in the physics of ion-acoustic oscillations and normalize the physical quantities for brevity. For the excitation of ion-acoustic waves (IAWs), the inertial effects of relativistic electrons and positrons (with mass m), compared to those of ions, can be neglected due to Hjm≪mi^37^. This is valid for equilibrium number density, nj0≪3.6×1039 cm-3. Also, because of their heavy inertia and slow time scale of oscillations, compared to those of electrons and positrons, the ions are assumed to be classical, nonrelativistic and nondegenerate^37^. Furthermore, at the ionic time scale, the time variations of the electron and positron pressures can be assumed to be small, i.e., (γj2vj/c2)(∂/∂t)≪(∂/∂x). This is justified since the phase velocity of ion-acoustic waves can be shown to be well below the speed of light c in vacuum (See section “Linear analysis” for clarification) and the particle velocity vj does not exceed c. Thus, Eqs. (1)–(5) reduce, in dimensionless forms, to
In Eqs. (6) to (10), different physical quantities are normalized as follows. The number density nj is normalized by its unperturbed value nj0 (j=i for ions, j=e for electrons and j=p for positrons), the electrostatic potential ϕ is normalized by kBTe/e, the electron (positron) relativistic pressure Pe (Pp) is normalized by ne0KBTe (np0KBTp), and the ion fluid velocity vi is normalized by the ion-acoustic speed cs=(1/mi)dpe/dne0. Here, the suffix 0 denotes the value calculated at equilibrium. Furthermore, the time (t) and the space (x) variables are normalized by the ion plasma period ωpi-1=4πni0e2/mi-1/2 and the effective Debye length λD (=cs/ωpi) respectively. Also, σi=Ti/Te, σp=Tp/Te, αe=1/1-δ, αp=δ/1-δ, and δ=np0/ne0 such that αe=1+αp (The charge neutrality condition at equilibrium).
In the interior of stellar compact objects such as those of neutron stars and white dwarfs, electrons and positrons can have a relativistic speed and an arbitrary degree of degeneracy. There are different equations of state to model the degenerate matter of these compact stars. One particular, which efficiently represents the relevant physics, especially of white dwarfs, is the Chandrasekhar equation of state at finite temperature^9^.
We consider the following expression for the electron/positron number density that follows from the Fermi-Dirac statistics^1,9^.
where ħ is the Planck’s constant divided by 2π and μj is the chemical potential energy for electrons and positrons without the rest mass energy. Also, Ej(pj)=c2pj2+m2c4 is the relativistic energy and pj the relativistic momentum of j-th species particle.
Equation (11) can be put into the following alternative form^9,38^
where Fk is the relativistic Fermi-Dirac integral of order k, given by,
in which βj=kBTj/mc2 is the relativity parameter, tj=Ej(pj)/kBTj, and ηj=μj/kBTj is the normalized chemical potential energy.
The degeneracy pressure of the j-th species particle at finite temperature (Tj≠0K) is given by^9,38^
which can be expressed as
Next, using Eqs. (12) and (15), it can be shown that
This result, when applied to the momentum balance equations for inertialess electrons and positrons, i.e.,
gives μj=-qjϕ+μj0, where μj0 is the value of μj at ϕ=0. So, we must replace μj by -qjϕ+μj0 ( or -qjϕ+μj for brevity, keeping in mind that μj is now the value at ϕ=0) in Eqs. (12) and (15) to obtain the following modified expressions for the density nj and the pressure Pj.
where η~j=(μj-qjϕ)/kBTj is the normalized electrochemical potential energy. The energy Ej, appearing in Eq. (11), is now modified to εj≡Ej+qjϕj, implying that both the free and trapped particles are to be taken into account in the potential well. Here, particles having εj>0 and εj<0 are referred as the free and trapped particles respectively, and the trapping occurs for εj=0^1^. Since μj has the rest mass energy of electrons or positrons removed, it is the kinetic chemical potential for which the rest mass appears explicitly in the positron chemical potential, i.e., μp=-μe-2mc2, which gives
where ξj=μj0/kBTj≡μj/kBTj (since we have replaced μj0 by μj for brevity) is the degeneracy parameter for electrons (j=e) and positrons (j=p) at equilibrium (i.e., the value of η~j at ϕ=0). Furthermore, ξj satisfies the following harsh condition at zero relativistic and zero electrostatic potential energies (εj=0):
Typically, in the non-relativistic regime of a Fermi gas at finite temperature, βj≡kBTj/mc2≪1, whereas in the ultra-relativistic regime, we have βj≫1. However, these limiting cases have been considered in the literature but in nonrelativistic electron-ion plasmas^39^. So, we are interested in the intermediate regime in which the particle’s Fermi energy and the thermal energy do not differ significantly, i.e., TFj>Tj and the particle’s thermal energy is close to the rest mass energy, i.e., either βj<1 or βj>1.
Thus, evaluating the integrals Fk(η~j,βj) in Eq. (18) and following the method by Landau and Lifshitz^12^, we obtain the following expression for the number densities of electrons and positrons (in dimensionless forms) in two different cases of βj<1 and βj>1.
where the coefficient Aj is given by
Also, ϕe=ϕ, ϕp=-ϕ/σp, and we have assumed ηj>1. The validity of this restriction of ηj will be justified later in section “Physical regimes of ion-acoustic waves”. Thus, we have the generalized expression for the ion-acoustic speed as
where
It may be necessary to compare the new expression of the number density nj [Eq. (22)] with that in the work of Rasheed et al.^20^. In the latter, the authors investigated the characteristic of ion-acoustic solitons in an electron-positron-ion plasma with nonrelativistic flow of degenerate electrons and positrons and classical cold ions. However, they considered the ultra-relativistic degeneracy effect at a very low temperature (in comparison with the Fermi temperature) and assumed the chemical energy to be equal to the Fermi energy. Such an assumption may be valid for a fully degenerate plasma (at zero-temperature, or Tj≪TFj) but not for plasmas with finite temperature degeneracy. We have considered this issue in the present investigation and thereby generalized and advanced the work of Rasheed et al.^20^ by considering the relativistic flow of both degenerate electrons and positrons at finite temperature and warm classical, nonrelativistic, nondegenerate ions. With this assumption, the electron and positron distributions [Eq. (22)] are significantly modified by the relativistic momentum and energy as well as the finite temperature effects on the chemical energy. The latter can no longer be approximated as the Fermi energy as in Ref.^20^. Nevertheless, the expression of the number density in Ref.^20^ can be recovered in the limit of βj≫1 and with a replacement of the chemical energy by the Fermi energy.
We note that the degeneracy parameters ξe and ξp are related by ξe=-σpξp-2/βe and the relativistic parameters βe and βp are related by βp=βeσp. While βj can have any value smaller or larger than unity, the values of ξj can be obtained by using Eq. (12) and the charge neutrality condition αe=1+αp at ϕ=0. In the case of βj<1, an expression of ξe can be obtained as
where τe=TFe/Te. However, an explicit expression of ξe for the case of βj>1 can not be obtained in a straightforward way. So, we will use some approximate results for ξj that were obtained in different contexts^24^. Moreover, the normalized chemical potential ξj can assume from large negative to large positive values. For example, in metallic plasmas, it has been shown that both the Thomas-Fermi (TF) model and the ideal free electron gas (IFEG) model predict approximately the same results for the electron chemical potential^40^. Given an electron mass density ne0∼0.5 gm cm-3, as the thermal energy reduces from 5×102 ev to 0.1 ev, the chemical potential ξe increases from -5 to more or less 20, i.e., 0≲ξe≲20 for 0.1≲T(ev)≲1.4 and -5≲ξe≲0 for 1.4≲T(ev)≲102. We, however, assume that at finite temperature, the electrons and positrons have energy states in between kBTj and kBTFj. So, negative values of ξe may not be admissible, otherwise one can have TFe<Te. In astrophysical environments, it has been found that as the particle temperature drops from 24 keV to 12 keV, the normalized electron chemical potential increases from 0.01 to 10 and reaches a steady state value^24^. In particular, in the limit of full degeneracy (TFj≫Tj), μe≈kBTFe=ħ2(3π2ne0)2/3/2m, so that one obtains νe≈(2/3)τe, cs≈2/3kBTFe/mi, λD≈2/3kBTFe/miωpi2, and the Fermi pressure law, Pj=(2/5)nj0EFj(nj/nj0)5/3, where EFj≡kBTFj, i.e., the well-known results for fully degenerate plasmas are retrieved.
In the previous section “The model”, we have described the basic fluid model for the excitation of ion-acoustic waves. Nevertheless, it is pertinent to discuss the validity domains of the general theory as well as the existence domains of small amplitude ion-acoustic waves and solitons, to be investigated in sections “Linear analysis” and “Nonlinear evolution of ion-acoustic solitons”. It is also highly demanding to identify precisely the key physical parameters and their regimes where the linear wave mode and the nonlinear excitation of ion-acoustic solitons can be looked for. Clearly, the theory is more applicable to intermediate regimes of high density and moderate temperature plasmas where (i) the particle’s thermal energy is close to the rest mass energy, i.e., βj≡kBTj/mc2∼1 (So, either βj<1 or βj>1) and (ii) the thermal and Fermi energies of electrons and positrons do not differ significantly, i.e., Tj/TFj≲1 (j=e,p stand for electrons and positrons). The case with Tj/TFj>1 is not admissible as we have assumed the electrons and positrons to have energy states in between kBTj and kBTFj at finite temperature. Also, in this case, the electron chemical potential may be negative which may violate our assumption of ηj>1.
In the fluid model, the electron/positron inertia has been neglected due to Hjm≪mi. This assumption is justified if the particle number density is well below the critical density 3.6×1039cm-3 and the particle temperature is not significantly high, i.e., Tj≲1010 K. Also, we have safely neglected the time variation of the relativistic Fermi pressure, because even in plasmas with relativistic flow of electrons and positrons, the phase velocity of ion-acoustic waves should remain well below the speed of light in vacuum c. This will be clarified in section “Linear analysis”.
Relativistic, multi-component, astrophysical plasmas can occur in a wide variety of high-energy-emitting objects like white dwarfs, neutron stars, black holes, and active galactic nuclei (AGNs). In these environments, the particle distribution strongly depends on the various physical processes including the pair creation and annihilation. The latter can largely occur for plasmas in local thermal and chemical equilibrium. It has been shown that such annihilation rate can be of the order of 1015s-1 near Tj=104KeV≈1.16×1011 K. However, at these situations it significantly drops with decreasing values of Tj and freezes out at Tj=16KeV≈1.85×108 K^24^. In the core of white dwarfs with central temperature Tj∼107-1010K and particle number density ∼1028-1035cm-3, the ion plasma frequency (∼1017s-1 for nj0∼1028cm-3) can still be much higher than the annihilation rate (∼1015s-1), i.e., the electron-positron annihilation time in high-density regimes can remain longer than the ion plasma period. So, for the excitation of ion-acoustic waves in high density plasmas, the electron-positron annihilation can be safely neglected.
On the other hand, in astrophysical environments, when the temperatures of electrons and positrons drop below 109 K but still higher than 107 K, the electrons and positrons may not be in thermal and chemical equilibrium. However, since they can even strongly scatter with the plasma, their distributions can still be the Fermi-Dirac as described in Eq. (13). Such a deviation from the chemical equilibrium implies that the electron and positron degeneracy parameters evolve separately and hence the appearance of different ξe and ξp. It has been found that as the temperature reduces below 109 K, the degeneracy parameter ξj≡μj/kBTj increases from 10-2, but reaches a steady state value 10 at a smaller value of Tj. So, the values of ξj>1 are reasonably good as we have assumed ηj>1 for the expansion of the normalized densities nj [Eq. (22)]. Also, at this nonequilibrium state, the positron to electron density ratio δ drops below the unity^24^. This is also justified from the charge neutrality condition at equilibrium αe=1+αp. Furthermore, in most astrophysical plasma environments^38^, the electron and positron temperatures do not differ significantly and the ion temperature is typically low compared to that of electrons or positrons, i.e., σp∼1 and σi<1.
It is to be mentioned that although the conditions Tj/TFj≲1 and βj∼1 may be fulfilled in the laser fusion experiments, e.g., at the National Ignition Facility (NIF) with number density ∼1025 cm-3^7^, as well as in electrical explosion of metal wires with mass density^40^ 1023 cm-3, the electron-positron annihilation rate in these environments can no longer be negligible compared to the ion plasma oscillation frequency. Thus, the present plasma model can be more relevant in the environments of white dwarfs. Before we specify the parameter regimes, the plasma parameters responsible for the description of ion-acoustic waves can be identified as βj ξj, σi, σp, and δ. Since each pair of (βe,βp) and (ξe,ξp) are connected to each other by a couple of relations as mentioned in section “The model”, the key parameters are βe, ξe, σi, σp, and δ. Thus, the parameter regimes for the existence of ion-acoustic wave mode and ion-acoustic solitons may be classified in two cases as
In the following sections “Linear analysis” and “Nonlinear evolution of ion-acoustic solitons”, we will study the linear and nonlinear theory of ion-acoustic waves in relativistic degenerate e-p-i plasmas. Specifically, we will focus on the two parameter regimes as defined before to establish the existence of ion-acoustic wave modes and ion-acoustic solitons. The properties of these solitons will also be studied with the variation of parameters.
We consider the propagation of electrostatic waves in relativistic degenerate e-p-i plasmas in the limit of small amplitude perturbations for which any nonlinear effects can be neglected and look for the existence and the characteristics of the ion-acoustic mode through a linear dispersion relation. In order to obtain this dispersion relation for IAWs, we linearize Eqs. (6)–(10) by considering the dependent variables as a sum of their equilibrium and perturbation parts, i.e., nj=1+nj1, vi=0+vi1, ϕ=0+ϕ1 etc. Next, we assume the perturbed (with suffix 1) quantities to vary as plane waves with the wave number k (normalized by λD-1) and the wave frequency ω (normalized by ωpi) of the form ∼exp(ikx-iωt). Thus, we obtain the following dispersion relation for IAWs.
where Λ=νeαea0e+αpa0p/σp and a0j (for j=e,p) is given by
From the expression of ω2/k2 [Eq. (27)] some important consequences are to be noted as follows.
In what follows, we study the dispersion characteristics of IAWs by numerically solving Eq. (27) for ω for different values of βe and ξe that fall within the parameter regimes defined in Case I and Case II. The dispersion curves for the IAW mode are shown in Fig. 1. While the parameter βe characterizes the measure of the thermal energy relative to the rest mass energy, ξe measures the degree of degeneracy of electrons and hence that of positrons. It is found that with a small increase of the value of βe, the IAW frequency also increases (See the dotted and dash-dotted lines). However, the frequency gets significantly reduced with an increasing value of ξe (See the solid and dashed lines). Physically, at higher thermal energies of electrons and positrons beyond the rest mass energy, more number of wave crests may pass a particular point (due to frequency increase) in a given interval of time and so the ion-acoustic wave of constant amplitude may transmit more energy per unit time. However, the transmission of the wave energy may be reduced when electrons and positrons approach the Fermi level with an increasing value of ξe (See the solid and dashed lines). From Fig. 1 it is also noted that as the values of βe are increased or those of ξe are decreased, the domain of ω in terms of k(≲1) for the existence of IAW mode reduces. In these situations, the IAWs can propagate with longer wavelengths and hence with higher energies. On the other hand, the effects of increasing values of σp and σi are to enhance a bit the wave frequency. However, such effect becomes significant in the regime of higher values of k>1. The latter may not be relevant to the study of low-frequency IAWs with longer wavelengths, because otherwise, the ion-acoustic wavelength may become smaller than the Debye screening length for which the plasma collective behaviors may disappear.
Figure 1 The dispersion relation [Eq. (27)] is plotted to show the variation of the wave frequency (ω) against the wave number (k) for different values of βe and ξe as in the legend. The fixed parameter values are δ=0.7, σp=0.8, and σi=0.5.
In this section, we will relax the extreme condition for small amplitude perturbations for which the linear theory is no longer valid and look for how the perturbations develop into the excitation of ion-acoustic solitary waves as the nonlinear effects intervene the dynamics of relativistic degenerate e-p-i plasmas. Specifically, we will derive evolution equations for ion-acoustic solitons and study their properties in different parameter regimes that are defined in Case I and Case II. In section “Linear analysis”, we have seen that the dispersion properties of IAWs are distinct in these two cases. Also, in section “The model”, we have noted that the nonlinear contributions in the electron and positron number densities [Eq. (12)] are significantly different for βj<1 and βj>1, j=e,p. So, we will consider these two cases separately in sections “Case I, βe < 1” and “Case II, βe > 1”. We will employ the standard reductive perturbation technique to derive the evolution equation for small amplitude ion-acoustic solitons, namely the KdV equation, and consider the critical parameter regimes where the KdV equation fails, but some other nonlinear equations like mKdV and Gardner equations describe the evolution of ion-acoustic solitons.
We consider the nonlinear propagation of small-amplitude ion-acoustic perturbations and look for the evolution equation of small-amplitude ion-acoustic solitons in relativistic degenerate plasmas at finite temperature with βj<1, i.e., when the electron/positron thermal energy is slightly below their rest mass energy. In the weekly nonlinear theory, such an evolution equation of the KdV type can readily be obtained by using the standard reductive perturbation technique. To this end, we define the stretched coordinates using the Galilean transformation as
where ε (0<ε<1) is a small expansion parameter measuring the weakness of the wave amplitudes and λ is the phase velocity of the IAW normalized by cs. The new coordinates ξ and τ are, respectively, normalized by λD and ωpi-1. In a general manner, one can define the stretched coordinates using the Lorentz transformation (instead of the Galilean transformation) for the relativistic fluid model as
where γL=1/1-λ2 stands for another Lorentz factor. However, this is not necessary as the basic equations are reduced into the forms which do not involve any relativistic Lorentz factor. Also, the Lorentz transformation defined above would not change any qualitative features of the ion-acoustic wave dynamics. In fact, the factor γL may contribute to the dispersion and nonlinear coefficients of the evolution equation explicitly with its different powers, which only change their magnitudes a bit when the ion-acoustic phase velocity (λ≡ω/k) is well below the acoustic speed cs. Furthermore, defining the multiple scales [like Eq. (31)] is justified, since for a small wave number (or long wavelengths) k∼O(ε1/2), the phase factor of a plane wave kx-ωt can be expressed by using the cold (σi=0, for simplicity) plasma dispersion relation ω=k/Λ+k2 [Eq. (27)] as
and the phase velocity as λ=1/Λ. The latter will be verified later.
In what follows, the dependent variables are expanded in powers of ε as
Next, we apply the new coordinate transformations [Eq. (31)] and substitute the expansions from Eq. (34) into Eqs. (6)–(10) and Eq. (22), and then equate the coefficients of different powers of ε from the resulting equations. The lowest order of ε yields the following expressions for the first order perturbations.
Eliminating the first-order perturbations successively and looking for their nonzero solutions, from Eq. (35) we obtain the following relation for the phase velocity λ.
As expected, this expression of λ exactly agrees with that of ω/k, to be obtained from the dispersion equation (27) in the limit of long wavelength perturbations, i.e., k→0. It also justifies the consideration of λ=1/Λ in the coordinate transformation (31).
To the next higher order of ε, we have
where a1j is given by
Finally, eliminating all the second order quantities from Eqs. (37)–(40) and using the results of lowest order of ε we obtain the following KdV equation for the first-order electrostatic potential ψ≡ϕ1.
where the dispersion (A1B) and the nonlinear (B) coefficients are given by
Evidently both the dispersion (which causes wave broadening) and nonlinear (responsible for wave steepening) coefficients of the KdV equation (42) are significantly modified by the contributions of the positron species, the thermal ions, as well as the relativistic degeneracy of both electrons and positrons.
Before we analyze the characteristics of A and B, let us first obtain a traveling wave solution of Eq. (42). To this end, we apply the transformation ζ=ξ-Uτ≡ε1/2x-λ+εUt, so that U is the constant speed of ion-acoustic solitons that represents a small increment of the linear phase speed of IAWs λ, and the boundary conditions, namely ψ, dψ/dξ, and d2ψ/dξ2→0 as ξ→±∞. Thus, we obtain
where ψm and w, respectively, denote the maximum amplitude and the width of ion-acoustic solitons, given by,
and they are such that the relation ψmw2=12/A1 holds. It is to be noted that a small amplitude soliton solution, similar to that obtained in Ref.^20^ by the Sagdeev pseudopotential approach, can be recovered in the limiting case of βj≫1, and with the substitutions μj=kBTFj and σi=0.
Furthermore, the soliton energy (or soliton photon number) is given by
Since the soliton speed U is directly proportional to the amplitude ψm, but inversely to the width w, faster (slower) solitons may be taller (shorter) and narrower (wider). Furthermore, since the soliton energy E is directly proportional to the amplitude and width, ion-acoustic solitons with higher amplitudes (and/or widths) would evolve with higher energies in relativistic plasmas. From the expression of B in Eq. (43), it is evident that B is always positive and also non-zero by means of Eq. (27). So, the KdV equation (42) admits compressive or rarefactive soliton solutions according to when A1>0 or A1<0. However, when A1=0, the KdV equation fails to describe the nonlinear evolution of ion-acoustic solitons. In that case, one has to look for some higher order correction terms in the perturbation expansions, to be investigated later.
We consider the parameter regimes as in Case I which involves βe<1 and numerically investigate the properties of A1 to identify different parameter domains for which the conditions A1>0, A1<0, and A1=0 may be fulfilled. Figure 2 displays the contour plot of A1=0 in the βe-ξe plane for different values of σi, σp, and δ.
Figure 2 Contour plot of A1=0 in the βe-ξe plane for different values of σi, σp, and δ as in the legend. The region above (below) the line A1=0 corresponds to the existence regime of compressive (rarefactive) KdV solitons. For the parameter values lying on the curves of A1=0 and/or close to the curves where A1∼O(ε), the KdV equation may not be valid for the evolution of small amplitude ion-acoustic solitons.
While different points on the curves correspond to different parameter values at which A1=0, the regions above and below the curves, respectively, represent the parameter regimes where A1>0 and A1<0. In the former, the hump shaped (compressive with positive potential) solitons may exist, whereas in the latter, one can find dip shaped (rarefactive with negative potential) ion-acoustic solitons. Here, we call a line of A1=0 as the “critical line”, a point Pc≡(βec,ξec) on the critical line as the “critical point”, and any point P≡(βe,ξe) lying in the βe-ξe plane but close to the critical line [i.e., in the region where A1→0 or A1∼O(ε)] as the “close to the critical point Pc”. We note that the KdV theory may not be valid for the parameter regimes at the critical points or close to the critical points. We will treat these particular cases in sections “mKdV solitons” and “Gardner solitons” separately. From Fig. 2 it is found that as the value of the positron to electron temperature ratio (σp) is reduced (See the solid and dashed lines) or that of the positron to electron density ratio (δ) is enhanced (See the dashed and dash-dotted lines), the parameter region of rarefactive solitons corresponding to A1<0 expands, while that of compressive solitons, i.e., A1>0 shrinks. The influence of the ion temperature (σi) on the existence regions of ion-acoustic solitons is not markedly pronounced. However, an enhancement of σi expands a bit the parameter region for the compressive solitons, but reduces that of the rarefactive one (See the dotted and dash-dotted lines). Thus, from Fig. 2 it may be concluded that in contrast to typical electron-ion plasmas, within the specific domains of values of ξe and βe (and so of ξp and βp) and for a fixed positron to electron temperature ratio, higher the concentration of the positron species (δ) or lower the ion to electron temperature ratio (σi) in relativistic degenerate e-p-i plasmas, the more likely is the existence of rarefactive ion-acoustic solitons than the compressive solitons.
Having obtained the parameter regimes for the existence of ion-acoustic solitons away from the critical points, we plot the profiles of both the compressive and rarefactive solitons for different values of σi, σp, and δ as shown in Fig. 3. As an illustration, for the compressive and rarefactive solitons, we consider, respectively, the points P≡(0.4,5) and (0.1, 1.8) in the βeξe parameter space [Fig. 2], which neither lie on the critical lines nor close to them, since at these points the KdV theory may not be valid. For example, at a critical point Pc≃(0.4,1.9617) (or close to it) of the dash-dotted line of Fig. 2, the amplitude of the soliton [Eq. (45)] becomes extensively larger than unity, leading to the failure of the weekly nonlinear theory of small-amplitude perturbations. In such situations, the mKdV and Gardner equations may precisely describe the evolution of ion-acoustic solitons which will be studied shortly in sections “mKdV solitons” and “Gardner solitons”.
Figure 3 The profiles of the compressive [subplot (a)] and rarefactive [subplot (b)] ion-acoustic solitons are shown for different values of σi, σp, and δ as in the legends. The parameter values for the subplots (a) and (b), respectively, are (βe,ξe)=(0.4,5) and (βe,ξe)=(0.1,1.8). Here, U=0.01 and the values of βe and ξe are taken from the existence region (Fig. 2) in such a way that A1≠0 and A1≁o(ε).
From Fig. 3, it is found that both the widths and amplitudes of the compressive [Subplot (a)] and rarefactive [Subplot (b)] solitons decrease with increasing values of δ and the reduction is significant for rarefactive solitons (See the dashed and dash-dotted lines). Such a reduction of the amplitude and width can eventually lead to a significant decay of the soliton energy [Eq. (46)]. Although the influence of ion to electron temperature ratio σi on the profiles of rarefactive solitons is relatively small, both the amplitude and width of the compressive solitons decrease with increasing values of σi (See the dotted and dash-dotted lines). On the other hand, in contrast to the compressive solitons in which both the amplitude and width decrease, the influence of increasing the positron to electron temperature ratio σp is to increase both the amplitude and width of the rarefactive solitons. Thus, it may be concluded that the positron species (which favors the existence of rarefactive solitons, cf. Fig. 2) with thermal energies close to the electrons and with higher concentration in plasmas can reduce the soliton energy significantly. Also, the relative influence of the plasma parameters σp and δ on the profiles of the compressive and rarefactive solitons are not only different but their qualitative features also differ significantly.
In what follows, we study the influence of the relativity and degeneracy parameters βe and ξe on the profiles of the soliton amplitude and width for a set of fixed parameter values, namely σi=0.5, σp=0.8, δ=0.7, and U=0.01. We also examine the critical values of βe and ξe below or above which the polarity of solitons changes. The results are displayed in Fig. 4. The subplots (a) and (b) show the amplitudes ϕm and the subplots (c) and (d) the widths w. From subplots (a) and (b), it is seen that keeping any one of ξe and βe fixed and varying the other, there exists a critical value of ξe or βe below (or above) which the rarefactive (or compressive) ion-acoustic solitons exist. Such a critical value of ξe (or βe) is upshifted even with a small reduction of βe (or ξe). Furthermore, close to (or at) the critical values of βe and ξe [e.g., for βec∼0.34, corresponding to the dashed line of subplot (a) and ξec=1.97, corresponding to the solid line in subplot (b)], a significant increase in magnitude of the soliton amplitude is seen. Also, some subintervals of βe and ξe exist in each of which the amplitudes for rarefactive solitons are close to zero and so is the soliton energy. Such intervals corresponding to the solid lines in subplots (a) and (b), respectively, are 0<βe≲0.3 and 0<ξe≲1.7. In contrast, the amplitude of compressive solitons initially decreases but reaches a steady state as βe approaches the unity. Thus, when the degeneracy parameter is fixed at ξe=1.8 (and other parameters as above) and the relativity parameter βe varies in 0<βe≲1, the rarefactive solitons exist in 0.3≲βe≲0.6 and the compressive solitons exist in 0.7≲βe≲1 [See the solid line in subplot (a)]. On the other hand, if the relativity parameter is fixed at βe=0.7 and other parameters as above, the rarefactive solitons exist in 1.5≲ξe≲1.7 and the compressive solitons exist in 1.8≲ξe≲3 [See the dashed line in subplot (b)]. Similar domains can be obtained for some other fixed values of ξe and βe. These parameter regimes are in agreement with our previous prediction (cf. Fig. 2). From subplot (a), it can be inferred that if the value of ξe is further reduced from ξe=1.8 to ξe=1.6 (keeping the other parameters fixed as above or as in the figure caption), only the rarefactive solitons exist in 0<βe<1. However, no such domain of ξe can be found from subplot (b) for which only the rarefactive or compressive solitons can exist. It is also evident from subplots (a) and (b) that while the amplitude of compressive solitons decreases and reaches a steady state with increasing values of either ξe or βe above their critical values, the same (in magnitude) for the rarefactive solitons increases with increasing values of either ξe or βe below their critical values. These are also in agreement with what we have predicted before from Fig. 2. On the other hand, subplots (c) and (d) of Fig. 4 show the variations of the soliton width with respect to the parameters βe and ξe. In both the cases it is seen that the soliton width decreases but reaches a steady state with increasing values of both βe and ξe.
Figure 4 The variation of the amplitude [subplots (a) and (b)] and the width [subplots (c) and (d)] of the KdV soliton [Eq. (44)] are shown for different values of βe (<1) and ξe as in the legends. The fixed parameter values are δ=0.7, σi=0.5, σp=0.8, and U=0.01.
From the characteristics of the soliton amplitude and width as in Fig. 4, it may be assessed that given a set of fixed parameter values of σi, σp, δ, and U, there exist two subintervals of both βe and ξe, namely 0<βe<βec, βec<βe<1 and 0<ξe<ξec, ξec<ξe<3. In 0<βe<βec and 0<ξe<ξec, the energy of rarefactive solitons is initially very low. However, it starts increasing with increasing values of βe and ξe below their critical values βec and ξec. Such solitons with growing amplitude and hence with increasing energy can evolve in relativistic degenerate plasmas but may be unstable. On the other hand, the energy of compressive solitons (although initially high) tends to decrease in βec<βe<1 and ξec<ξe<3, but reaches a steady state at higher values of ξe and βe<1. Such solitons can evolve with a permanent profile for a longer time in the parameter space and thus can be stable. Thus, in astrophysical environments where both the chemical energy and the thermal energy of electrons and positrons are close to their rest mass energy, the ion-acoustic compressive solitons may be stable, while rarefactive solitons may become unstable with their increasing amplitudes.
In section “KdV solitons”, we have noted that the KdV equation is not valid for parameter values close to or at the critical points (P=Pc) on the curve A1=0. In this situation one needs to deal with a different set of stretched coordinates and a different expansion scheme for the evolution of small amplitude ion-acoustic perturbations. In this section, we, however, consider the case when the parameter values exactly lie on the line A1=0 and study the properties of ion-acoustic solitons at these critical parameter values. The other case (“close to the critical points”) will be studied in section “Gardner solitons”. To derive the mKdV equation for the evolution of ion-acoustic solitons at the critical points, we modify the stretched coordinates [corresponding to the higher order smallness of k, i.e., k∼O(ε)] as
However, we retain the same perturbation expansion scheme for the dependent variables and follow the same reductive perturbation technique as for the KdV equation (42). So, in the lowest order of ε, we obtain the same expressions for nj1 (j=e,p,i), vi1 and λ, as given in Eqs. (35) and (36). From the next order of ε, the second order perturbed quantities yield
For the third order perturbed quantities, we obtain
where
Finally, eliminating the third-order quantities from Eqs. (51)–(55) by using Eqs. (48)–(50), we obtain the following mKdV equation for the evolution of ion-acoustic solitons at the critical points P=Pc.
where ψ≡ϕ1 and
From Eq. (57), we note that, not only the nonlinear coefficient of the mKdV equation is modified to A2B, but the nonlinearity is also of higher order (compared to that of the KdV equation) of the first order electrostatic perturbation. This is expected as we have redefined the new space and time scales slower than the previous ones [Eq. (31)] and accordingly the nonlinear effects appear in the higher order of perturbations.
In what follows, a stationary soliton solution of Eq. (57) (different from the KdV soliton) is given by
where ψm (=±6U/A2B) and w(=B/U) are, respectively, the amplitude and width of the ion-acoustic mKdV soliton. We note that since B is always positive, for real soliton solution, A2 must be positive. Also, since ψm can be both positive and negative, the coexistence of both the compressive and rarefactive solitons is possible at the critical points Pc≡(βec,ξec) having the same amplitude (in magnitude) and the same width. Furthermore, if at some points P (other than those satisfy A1=0), A2→0, then ψm→±∞, implying that the mKdV equation (57) may no longer be valid. So, in a situation when A1≈0 and A2≈0, no finite soliton solution will exist and one thus has to look for another evolution equation with further higher order corrections. However, this is not of interest to the present study. In fact, we find that for a wide range of critical values of the parameters (at which A1=0), A2 remains positive and finite (See Table 1). The latter also ensures the existence of both compressive and rarefactive ion-acoustic mKdV solitons. The typical profiles of the mKdV soliton at different critical points Pc are shown in Fig. 5. Since the polarity of the soliton changes only due to the sign change in ψm, we plot only the profiles for |ψ| against ζ. From Fig. 5 it is found that although the amplitude can be slightly modified, the mKdV solitons can be wider at a critical point with a higher value of βec, but a lower value of ξec as in the legend. At these critical values, the soliton energy will also be higher. However, since the amplitude does not change significantly, the soliton can evolve with a stable profile. We recall that for values of the parameters near the critical points Pc of A1=0, the KdV and mKdV equations do not give any finite soliton solution. In such a situation we need to look for another evolution equation, namely the Gardner equation which we will derive in section “Gardner solitons”.
Figure 5 The profiles of the mKdV soliton [Eq. (59)] are shown at different critical points (lying on the line A1=0) as in the legend. The fixed parameter values of δ, σi, and σp are as in Fig. 4.
In this section, we study the evolution of ion-acoustic solitons in the parameter space of A1≈0. In the latter, both the KdV amd mKdV equations fail to describe the evolution of ion-acoustic solitons. So, in order to explore the finite amplitude solitons beyond the KdV and mKdV limits, we derive the standard Gardner equation. To this end, we assume that around the critical points P=Pc of A1=0, A1≃sε, where s=1 for A1>0 and s=-1 for A1<0. Since A1 appears only in the perturbation equation of the Poisson equation, [cf. Eq. (50)], the second order perturbed quantities give
i.e., they appear in the third order of ε. This should be included in the third order correction equation of the Poisson equation. As before, the first order quantities will remain the same as for the KdV and mKdV equations. Also, the second order perturbations will give the same results as for the mKdV equation. So, for the third order perturbed quantities, we obtain from Eqs. (6)–(10) and (22) the
Finally, eliminating the third order perturbations by using the second order perturbed quantities as in section “mKdV solitons”, from Eqs. (61)–(65) we obtain the following Gardner equation.
where, as before, ψ=ϕ1.
From Eq. (66), we note that in comparison with the mKdV equation (57), an additional nonlinearity proportional to s [similar to the KdV equation (42)] appears. Accordingly, Eq. (66) is often called the KdV-mKdV equation. The additional term (proportional to s) appears in Eq. (66) due to the smallness of A1: A1∼O(ε), i.e., A1≠0. So, the Gardner equation is valid for the parametric values close to the critical points of the curve A1=0 (Fig. 2). In particular, for A2→0, the Gardner equation reduces to the KdV equation (42) with the nonlinear coefficient A1≁O(ε) and with the same solution [Eq. (44)] for the finite amplitude ion-acoustic solitons in relativistic degenerate plasmas.
A stationary soliton solution of Eq. (66) can also be obtained by using the transformation ζ=ξ-Uτ as follows. Under this transformation, Eq. (66) reduces to
where the pseudo-potential V(ψ) (with U>0 and B>0) is given by
For the existence of soliton solutions, it is necessary for V(ψ) to satisfy the following conditions for some ψ=ψm ≠0.
It is straightforward to show that the first two conditions are eventually satisfied. The third condition gives
where ψ0=-s/A2 and V0=s2B/6A2. Thus, Eq. (67) reduces to
where r=A2/6. The soliton solution of Eq. (73) can then be obtained as
where w=2/-rψm1ψm2=2B/U. The profiles of the Gardner soliton [Eq. (74)] are shown in Fig. 6 at different points P≡(βe,ξe) that are close to the critical point Pc of the curve A1=0. We choose the fixed values as σi=0.5, σp=0.8, and δ=0.7 and consider a pair of different sets of values of βe and ξe for which both the compressive and rarefactive solitons can coexist. It is found that while the width remains almost unchanged, the amplitudes of both the compressive and rarefactive solitons increase (and hence solitons can evolve with increasing energies) with increasing values of ξe [subplot (a)] and βe [subplot (b)].
Figure 6 The profiles of the Gardner solitons [Eq. 74] are shown at different points (βe,ξe) that are close to the critical points, i.e., when A1∼O(ε). The fixed parameter values are δ=0.7, σi=0.5, and σp=0.8.
We move to the Case II with βe>1, i.e., when the thermal energy of electrons/positrons is slightly larger than their rest mass energy. Going back to the coefficients of the KdV equation (42), we find that apart from B>0, A1 remains positive and finite for βe>1. Consequently, the KdV equation (42) remains valid in this case and so is its solution (44) with A1>0. It follows that the parameter regimes in Case II only support the existence of compressive ion-acoustic solitons in relativistic degenerate plasmas at finite temperature. The typical potential profiles of the compressive solitons for βe>1 are shown in Fig. 7 for different values of σi, σp and δ. It is found that similar to the case of βe<1 (Fig. 3a) both the amplitude and width of the soliton profiles decrease with an increasing value of each of the parameters σi and σp. However, these amplitudes and widths are found to be increased with increasing values of δ. Similar characteristics with increasing amplitudes and widths of solitons were observed in Maxwellian e-p-i plasmas^4^. To study, in more details, the influences of the relativistic and degeneracy parameters βe (>1) and ξe on the profiles of the soliton amplitude and width, we plot ϕm and w, given in section “KdV solitons”, against βe and ξe. The results are shown in Fig. 8. Here, we consider the other parameter values fixed at σi=0.5, σp=0.8, and δ=0.7. It is found that the characteristics of the soliton amplitude and width significantly differ from those in the case of βe<1 [See Fig. 4]. From subplot (a), it is seen that although the amplitude grows initially, it reaches a steady state at higher values of βe, which may be a requisite condition for solitons to be stable with their finite energy even in strongly relativistic regime with βe>1. On the other hand, the amplitude can grow with increasing values of ξe within the domain, which may eventually lead to an instability due to increasing soliton energy. From subplots (c) and (d), one can observe that even though the width increases initially, it approaches a constant value with higher values of both βe and ξe. Thus, it may be concluded that although the soliton amplitude increases with ξe, the amplitude remains finite and small even at large ξe∼20. The latter can be achieved at small electron (or positron) thermal energy (∼0.1 ev)^40^. So, even in the case of strong relativistic degeneracy, ion-acoustic solitons having finite energy can be stable. However, the detailed discussion about the stability of ion-acoustic solitons is beyond the scope of the present investigation.
Figure 7 The profiles of the compressive KdV solitons [The case of βe>1] are shown for different values of σi, σp, and δ. The other parameter values are U=0.01, βe=2, and ξe=3.
Figure 8 The variation of the soliton amplitude [Subplots (a) and (b)] and width [Subplots (c) and (d)] of the KdV compressive soliton (The case of βe>1) are shown for different values of βe (>1) and ξe as in the legends. The fixed parameter values are δ=0.7, σi=0.5, σp=0.8, and U=0.01.
We have investigated the linear and nonlinear properties of ion-acoustic solitary waves in a multi-component relativistic degenerate plasma consisting of relativistic inertialess unmagnetized degenerate electrons and positrons at finite temperature and nonrelativistic classical thermal ions. Specifically, we have focused on the intermediate regimes relevant for astrophysical plasmas, e.g., in the core of white dwarfs, where the particle Fermi energy and the thermal energy do not differ significantly, i.e., TFj>Tj and the particle thermal energy is also close to the rest mass energy, i.e., βj≡kBTj/mc2∼1. Depending on whether the ratio βj is smaller or larger than unity and the other parameter restrictions applicable for the validity of the fluid model, we have mainly classified two parameter regimes, namely Case I and Case II as in section “Physical regimes of ion-acoustic waves”, that are relevant in astrophysical environments (e.g., in the core of white dwarf stars). The existence of ion-acoustic linear wave modes as well as the nonlinear evolution of ion-acoustic solitons are then studied in these two cases. The main theoretical results, so obtained in the linear and nonlinear regimes, are summarized as
To conclude, the relativistic high-density degenerate plasmas deviating from the thermodynamic equilibrium can appear not only in the context of laser produced plasmas or beam driven plasmas, but also in compact astrophysical objects like white dwarf stars, neutron stars. Such plasmas can support the propagation of low-frequency ion-acoustic waves and hence ion-acoustic solitons as localized bursts of different radiation spectra emanating from these compact objects. So, the present results should be useful for understanding the localization of ion-acoustic solitary waves in these astrophysical environments. Since we have considered the intermediate regime where βj∼1, the expressions for the electron and positron number densities [Eqs. (12)] may not be applicable for the extreme cases, namely nonrelativistic (βj≪1) and ultra-relativistic (βj≫1) fluid flows.
It is believed that most compact astrophysical objects are immersed in a strong magnetic field. So, a possible extension of the present study could be to a magnetized relativistic multi-component degenerate plasma at finite temperature. The importance of the quantum effects like the particle dispersion and the particle spin may be examined and included, if necessary, in the extended model. These improvements, however, requires a further study, so that the model could fit with some laboratory experiments, to be designed, or some real astrophysical observations.
One of us, RD, acknowledges support from the University Grants Commission (UGC), Government of India, for a Junior Research Fellowship (JRF) with Ref. No. 1161/(CSIR-UGC NET DEC. 2018) and F. no. 16-6 (DEC. 2018)/2019 (NET/CSIR). This work was initiated and major parts were completed when Gadadhar Banerjee was on leave from the Department of Mathematics, University of Engineering and Management (UEM), Kolkata-700 160, India, to work in the Department of Mathematics, Visva-Bharati University, India, under the Dr. D. S. Kothari Post Doctoral Fellowship Scheme of the University Grants Commission (UGC), Govt. of India with Ref. No. F.4-2/2006(BSR)/MA/18-19/0096).
R. D. and G. B. investigated, performed the numerical analysis, analyzed the results, and wrote the original draft (equal). A. P. M. conceptualized, supervised, validated and analyzed the results, and edited the manuscript. C. B. investigated and contributed to analytical results. All authors reviewed and approved the manuscript.
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
The authors declare no competing interests.
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.