1. INTRODUCTION
There has been a lot of study on the topic because of the fact that there lies the possibility of long-distance propagation of electron beams through the atmosphere and the potential use of intense beams for plasma heating in both scientific and industrial fields. The motion of charged particles is of more interest rather than the atomic processes involved in the creation of plasmas; therefore, we will limit our discussion to the effects on the motion of charged particles. The models used herein assume pre-formed plasmas and do not address the involved process of ionization by the beam. Mostly electron beam is injected in a plasma in the form of pulses. The ideal plasma responds immediately to a pulsed beam, providing complete neutralization of the beam space charge and current. In many cases, the electron beam can be considered as a plasma, if it satisfies the properties of a plasma-like quasi-neutrality, which happens when the ion or electron beam is space charge compensated, particularly under low-energy beam transport. In such case, one can calculate plasma density by classical formula. However, most of these beam electrons have velocity comparable to the speed of light, and therefore relativistic dynamics of the particle must be considered. In the early 80s, a scheme for accelerating electrons, employing a bunched relativistic electron beam in a cold plasma, was analyzed by Chen et al., (Reference Chen, Dawson, Huff and Katsouleas1984).
Under such circumstances, a lot of wave mode may be generated like electro-acoustic waves, electron plasma waves, dust-acoustic waves, ion-acoustic waves (IAWs), etc. In this paper, we discuss about the IAWs in such a multi-component plasma. In the physics of plasmas an IAW is a type of longitudinal oscillation of the ions as well as the electrons in plasma, similar to sound (acoustic) waves travelling in neutral gas. Since the waves propagate through positively charged ions, IAWs can interact with their electromagnetic fields, as well as by way of simple collisions. They commonly govern the evolution of mass density by way of particle density, for instance due to pressure gradients, on longer time scales, longer than the frequency corresponding to the relevant length scale. Such plasma wave modes can occur in an unmagnetized plasma or in a magnetized plasma parallel to the magnetic field. The kinetic energy of plasma particles does affect how they respond to beams. They can give rise to instabilities only if the ion and electron drift velocities are less than
$\sqrt {2kT_{\rm e} /m_{\rm e}} $
. The Maxwellian distribution is canonical distribution in the classical framework of Boltzmann–Gibbs (BG) statistical mechanics, applicable to thermal equilibrium condition. Under constrains of canonical ensemble, if we maximize the BG entropy, the Boltzmann exponential distribution of energy is derived. In the absence of any potential energy, this exponential distribution of energy gives the Maxwellian distribution of velocities. Hence, the total energy in this case is reduced to the kinetic energy.
Likewise the κ distribution is the canonical distribution in the non-extensive statistics and is applicable to both thermal equilibrium and non-equilibrium conditions. Here also we maximize the Tsallis entropy under similar constrains and a q-deformed exponential distribution of energy is obtained known as q-Maxwellian distribution velocity. Once we substitute the energies in terms of velocities, this is equivalent to a κ distribution in space and plasma physics, where κ and q-indices are related as κ = 1/(q − 1).
After a proper study of the literature available, we find that the study of propagation of a non-relativistic electron beam in a plasma in a strong magnetic field has been studied using electrostatic one-dimensional particle simulation models has been carried out by Okuda and Berchem (Reference Okuda and Berchem1987). Ion sound solitary waves with density depressions has been studied by Cairns et al., (Reference Cairns, Bingham, Dendy, Nairn, Shukla and Mamun1995). Chattopadhyay et al., (Reference Chattopadhyay, Ghosh and Paul2000) has studied the propagation of ion-acoustic solitons in relativistic plasma containing cold ions and two temperature electrons, studies were also made with warm ions by Ghosh et al., (Reference Ghosh, Paul, Das and Paul2008). Studies on similar lines were also made by Mamun (Reference Mamun1997, Reference Mamun1998); Bandopadhyay and Das (Reference Bandhopadhyay and Das2000).
In ion-acoustic solitary waves with q-non-extensive electron velocity distribution had been studied by Tribeche et al., (Reference Tribeche, Djebarni and Amour2010). They showed that for q-extensive index greater than unity, the lower limit of the acceptable Mach number is smaller than the Boltzmann counterpart, and for the range − 1 <q <1, the lower limit of the same is greater than the Boltzmann counterpart; this allows the possibility of the existence of subsonic ion-acoustic solitons. They showed the formation of rarefactive ion-acoustic solitary waves with relatively high Mach number.
Head-on collision of ion-acoustic solitary waves with q-non-extensive electron velocity distribution has been studied by Eslami et al., (Reference Eslami, Mottaghizadeh and Pakzad2011); by using KdV method, they showed that there is a phase shift in the colliding solitary waves. The importance of q-index in the formations and properties of ion-acoustic solitary waves has also been discussed. Roy et al., (Reference Roy, Saha and Chatterjee2012) used Sagdeev's potential technique and showed that the effect of ion temperature and q-parameter has significant role in large amplitude and width of solitary structures. Spatial distribution of electron number density for different values of q-parameter was studied by Lin et al., (Reference Lin, Liao and Zhu2015). Hafez et al., (Reference Hafez, Talukder and Sakthivel2016) studied the ion-acoustic solitary waves with non-extensive distributed electrons, positrons, and relativistic thermal ions in the presence of magnetic field. Such a plasma with relativistic heavy ion collision is reported in quark gluon plasma and astrophysical plasmas. In the presence of suprathermal electrons, electron-acoustic localized structure in a collisionless unmagnetized plasma containing cold and hot electrons where hot electrons have the κ distribution and ions are stationary have been investigated by Danehkar et al., (Reference Danehkar, Saini, Hellberg and Kourakis2011). The hot electrons have a non-Maxwellian suprathermal distribution. They also showed that a change screening mechanism depends strongly on excess suprathermality. They used non-linear pseudo potential technique and found only negative polarity solitary waves to exist.
In Section 2, we provide the basic formulation for the particle dynamics. Section 3 deals with the linear and non-linear analysis. The next section discusses the result and finally we conclude with some remarks on further study.
2. BASIC FORMULATION
We consider non-linear propagation of IAWs in a plasma consisting ions (positive and negative), electron beam and q-exponential distributed electrons. The q-exponential distribution is a distribution function used in probability theory that is derived from the maximization of the Tsallis entropy under constraint conditions. The q-exponential is a generalization of the exponential distribution in a similar way that Tsallis entropy is a generalization of the standard Shannon entropy. In presence of an electron beam, the usual plasma wave modes, for example, ion-acoustic, ion-cyclotron, etc., are modified because of interaction of plasma waves with electron beam component. Accordingly, relativistic form of Euler's equation in hydrodynamics is incorporated. Under these conditions, the non-linear dynamics of the IAW in this plasma governed by the following equations:
$$\displaystyle{{\partial ({\rm \gamma} _j n_j )} \over {\partial t}} + \displaystyle{{\partial ({\rm \gamma} _j n_j u_j )} \over {\partial x}} = 0,$$
$$m_p \left[ {\displaystyle{\partial \over {\partial t}} + u_p \displaystyle{\partial \over {\partial x}}} \right]({\rm \gamma} _p u_p ) = - Z_p e\displaystyle{{\partial {\rm \phi}} \over {\partial x}} - \displaystyle{1 \over {n_p}} \displaystyle{{\partial p_p} \over {\partial x}},$$
$$m_n \left[ {\displaystyle{\partial \over {\partial t}} + u_n \displaystyle{\partial \over {\partial x}}} \right]({\rm \gamma} _n u_n ) = Z_n e\displaystyle{{\partial {\rm \phi}} \over {\partial x}} - \displaystyle{1 \over {n_n}} \displaystyle{{\partial p_n} \over {\partial x}},$$
$$m_b \left[ {\displaystyle{\partial \over {\partial t}} + u_b \displaystyle{\partial \over {\partial x}}} \right]({\rm \gamma} _b u_b ) = e\displaystyle{{\partial {\rm \phi}} \over {\partial x}}.$$
The system closed by Poisson's equation
$$\displaystyle{{\partial ^2 {\rm \phi}} \over {\partial x^2}} = - 4{\rm \pi} e(n_p Z_p - n_n Z_n - n_e - n_b ).$$
The first equation is fluid continuity equation of j-species particles (j = n, p, b, respectively, stands for negative ions, positive ions, beam). Equations (2–4) are the equations of motion of positive ions, negative ions, and beam, respectively. Here m
p
, m
n
, and m
b
are mass of ions (positive, negative), and beam; u
j
is the fluid velocity of particles in this plasma with relativistic factor
${\rm \gamma} _j = 1/ \sqrt {1 - u_j^2 /c^2} \approx (1 + u_j^2 /2c^2 )$
, Z is the ion charge number, p
p
and p
n
are the ion pressure where p
p
= n
p
kT
p
and p
n
= n
n
kT
n
(k = Boltzmann constant), ϕ is the electrostatic potential, and e is the magnitude of electron charge.
The electron number density is assumed to be q-exponential distributed, as
$$n_e = [1 + (q - 1){\rm \phi} ]^{(q + 1)/2(q - 1)}, $$
where q is the exponent associated with the electron distribution. The positive and negative ions follow the standard Maxwellian distribution, respectively.
3. LINEAR AND NON-LINEAR ANALYSIS
Equations (1)–(5) may be presented in a reduced form, for convenience. For this we have scaled the time t and x by
$ {\rm \omega} _{\,pj}^{ - 1} = \left( {m_j /4{\rm \pi} n_{\,j0} Z^2 e^2} \right)^{1/2} $
and the Debye length λDe = (kT
e
/4πn
j0
Ze
2)1/2. Again we scaled the number densities of ions and beam (n
j
) by their respective unperturbed densities n
j0 and fluid velocities u
j
by the ion-sound speed c
s
= (ZkT
e
/m
j
)1/2, the ion pressure p
j
by p
j0 = n
j0
kT
j
and the electrostatic potential ϕ by kT
e
/e.
Thus, Eqs (1)–(5) can write into the normalized form:
$$\displaystyle{{\partial ({\rm \gamma} _j n_j )} \over {\partial t}} + \displaystyle{{\partial ({\rm \gamma} _j n_j u_j )} \over {\partial x}} = 0,$$
$$\left[ {\displaystyle{\partial \over {\partial t}} + u_p \displaystyle{\partial \over {\partial x}}} \right]({\rm \gamma} _p u_p ) = - {\rm \mu} _p \displaystyle{{\partial {\rm \phi}} \over {\partial x}} - A{\rm \alpha} n_p^{{\rm \alpha} - 2} \displaystyle{{\partial n_p} \over {\partial x}},$$
$$\left[ {\displaystyle{\partial \over {\partial t}} + u_n \displaystyle{\partial \over {\partial x}}} \right]({\rm \gamma} _n u_n ) = {\rm \mu} _n \displaystyle{{\partial {\rm \phi}} \over {\partial x}} - B{\rm \beta} n_n^{{\rm \beta} - 2} \displaystyle{{\partial n_n} \over {\partial x}},$$
$$\left[ {\displaystyle{\partial \over {\partial t}} + u_b \displaystyle{\partial \over {\partial x}}} \right]({\rm \gamma} _b u_b ) = \displaystyle{{\partial {\rm \phi}} \over {\partial x}},$$
$$\displaystyle{{\partial ^2 {\rm \phi}} \over {\partial x^2}} = n_e + \displaystyle{{n_b} \over {\rm \rho}} + \displaystyle{{Z_n n_n} \over {\rm \sigma}} - \displaystyle{{Z_p n_p} \over {\rm \delta}}, $$
where Aα = γ p T p /Z p T e and Bβ = γ n T n /Z n T e , μ j = m e /m j are the fractional ion to electron temperature and ratio of electron mass to ion mass, respectively. Here A, B, α, β are constants to suitably fit the pressure laws (in the general form p = Anα); ρ, σ, δ are fractions deciding relative particle concentration. Assuming space–time dependence of the field variables as exp i(kx − ωt), we get dispersion relation as
$$\eqalign{k^2 &= - \left\{ {\left( {\displaystyle{{q + 1} \over 2}} \right) + \displaystyle{{(3 - q)(q + 1)} \over 2}{\rm \phi} ^0} \right\} \cr & + \displaystyle{{{\rm \mu} _n Z_n} \over {\rm \sigma}} \displaystyle{{\left[ {2u_n^{(0)} k(E - 1) + kEu_n^{(0)} - 2{\rm \omega} (E - 1)} \right]} \over {E^2 u_n^{(0)} \left( {{\rm \omega} - ku_n^{(0)}} \right) - B{\rm \beta} \left[ {2u_n^{(0)} k(E - 1) + kEu_n^{(0)} - 2{\rm \omega} (E - 1)} \right]}} \cr & + \displaystyle{{{\rm \mu} _p Z_p} \over {\rm \delta}} \displaystyle{{\left[ {2u_p^{(0)} k(D - 1) + kDu_p^{(0)} - 2{\rm \omega} (D - 1)} \right]} \over {D^2 u_p^{(0)} \left( {{\rm \omega} - ku_p^{(0)}} \right) - A{\rm \alpha} \left[ {2u_p^{(0)} k(D - 1) + kDu_p^{(0)} - 2{\rm \omega} (D - 1)} \right]}} \cr & + \displaystyle{1 \over {\rm \rho}} \displaystyle{{\left[ {2u_b^{(0)} k(F - 1) - kFu_b^{(0)} - 2{\rm \omega} (F - 1)} \right]} \over {F^2 u_b^{(0)} \left( {{\rm \omega} - ku_b^{(0)}} \right)}},}$$
where
$$\left( {1 + \displaystyle{{u_p^{(0)^2}} \over {2c^2}}} \right) = D,$$
$$\left( {1 + \displaystyle{{u_n^{(0)^2}} \over {2c^2}}} \right) = E,$$
$$\left( {1 + \displaystyle{{u_b^{(0)^2}} \over {2c^2}}} \right) = F.$$
The derivation of the dispersion relation is given in the Appendix.
From above expression, it is clear that phase velocity is affected by ρ, σ, δ, and q.
The propagation equation of the IAWs may investigate by introducing new coordinates ξ and τ defined as:
$${\rm \xi} \hskip-1.2pt = {\rm \varepsilon}^{1/2} (x - Mt),{\rm \tau} = {\rm \varepsilon} ^{3/2} t. $$
Here, M is Mach number and ε is a small parameter measuring the weakness of dispersion and non-linearity.
Now perturbation expansion for these variables:
$$\left( \matrix{u_p^{} \hfill \cr u_n^{} \hfill \cr u_b^{} \hfill} \right) = \left( \matrix{u_p^0 \hfill \cr u_n^0 \hfill \cr u_b^0 \hfill} \right) + {\rm \varepsilon} \left( \matrix{u_p^1 \hfill \cr u_n^1 \hfill \cr u_b^1 \hfill} \right) + {\rm \varepsilon} ^2 \left( \matrix{u_p^2 \hfill \cr u_n^2 \hfill \cr u_b^2 \hfill} \right) + \cdots, $$
$$\left( \matrix{n_p^{} \hfill \cr n_n^{} \hfill \cr n_b^{} \hfill} \right) = \left( \matrix{1 \hfill \cr 1 \hfill \cr 1 \hfill} \right) + {\rm \varepsilon} \left( \matrix{n_p^0 \hfill \cr n_n^0 \hfill \cr n_b^0 \hfill} \right) + {\rm \varepsilon} ^2 \left( \matrix{n_p^1 \hfill \cr n_n^1 \hfill \cr n_b^1 \hfill} \right) + \cdots, $$
$${\rm \phi} = {\rm \phi} ^0 + {\rm \varepsilon} ^1 {\rm \phi} ^1 + {\rm \varepsilon} ^2 {\rm \phi} ^2 + {\rm \varepsilon} ^3 {\rm \phi} ^3 + \cdots. $$
By using Eq. (16) and (17–19), in the set of Eq. (7–11) collecting the terms of lowest order (ε3/2), we obtain
$$\eqalign{n_p^{(1)} & = G \cdot u_p^{(1)}, n_n^{(1)} = H \cdot u_n^{(1)}, n_b^{(1)} = I \cdot u_b^{(1)}, n_p^{(1)} \cr & = - \displaystyle{{{\rm \mu} _p} \over J} \cdot {\rm \phi} ^{(1)}, n_n^{(1)} = - \displaystyle{{{\rm \mu} _n} \over L} \cdot {\rm \phi} ^{(1)}, n_b^{(1)} = - \displaystyle{1 \over N} \cdot {\rm \phi} ^{(1)}} $$
where
$$G = \displaystyle{{\left( {u_p^1 - M} \right)u_p^0 /c_{}^2 + \left( {1 + u_p^{(0)2} /2c^2} \right)} \over {\left( {u_p^0 - M} \right)\left( {1 + u_p^{(0)2} /2c^2} \right)}},$$
$$H = \displaystyle{{\left( {u_n^1 - M} \right)u_n^0 /c_{}^2 + \left( {1 + u_n^{(0)2} /2c^2} \right)} \over {\left( {u_n^0 - M} \right)\left( {1 + u_n^{(0)2} /2c^2} \right)}},$$
$$I = \displaystyle{{\left( {u_b^1 - M} \right)u_b^0 /c_{}^2 + \left( {1 + u_b^{(0)2} /2c^2} \right)} \over {\left( {u_b^0 - M} \right)\left( {1 + u_b^{(0)2} /2c^2} \right)}},$$
$$J = \displaystyle{{\left( {u_p^0 - M} \right)\left\{ {\left( {1 + u_p^{(0)2} /2c^2} \right) + u_p^{(0)2} /c^2} \right\}} \over G} + A{\rm \alpha}, $$
$$L = \displaystyle{{\left( {u_n^0 - M} \right)\left\{ {\left( {1 + u_n^{(0)2} /2c^2} \right) + u_n^{(0)2} /c^2} \right\}} \over H} + B{\rm \beta}, $$
$$N = \displaystyle{{\left( {u_b^0 - M} \right)\left\{ {\left( {1 + u_b^{(0)2} /2c^2} \right) + u_b^{(0)2} /c^2} \right\}} \over I}.$$
Replacing the different quantities in first order in Eq. (20) into a single variable ϕ1 after manipulation, we derive the non-linear equation for propagation of ion-acoustic solitary wave in the aforesaid plasma system, as described in the following:
$$\displaystyle{{\partial {\rm \phi} ^1} \over {\partial {\rm \tau}}} + P{\rm \phi} ^1 \displaystyle{{\partial {\rm \phi} ^1} \over {\partial {\rm \xi}}} + Q\displaystyle{{\partial ^3 {\rm \phi} ^1} \over {\partial {\rm \xi} ^3}} = 0.$$
Equation (A16) is a well-known KdV equation. Coefficients P and Q are defined as:
$$\eqalign{P & = 2\left\{ {\displaystyle{{X_p Z_p} \over {\rm \delta}} - \displaystyle{{X_n Z_n} \over {\rm \sigma}} - \displaystyle{{X_b} \over {\rm \rho}} - \displaystyle{{\left( {3 - q} \right)\left( {q + 1} \right)} \over 2}} \right\}\; {\rm and}\; \cr Q & = \left\{ {\displaystyle{{Y_p Z_p} \over {\rm \delta}} - \displaystyle{{Y_n Z_n} \over {\rm \sigma}} - \displaystyle{{Y_b} \over {\rm \rho}}} \right\}.}$$
Here
$$\eqalign{X_p & = \left[ {\displaystyle{{A{\rm \alpha} ({\rm \alpha} - 1){\rm \mu} _p} \over {2J^2}} - \left( {u_p^0 - M} \right)\displaystyle{{3{\rm \mu} _p u_p^0} \over {2c^2 J^2 G^2}} - \displaystyle{{{\rm \mu} _p} \over {2J^2 G^2}} \left( {1 + \displaystyle{{3u_p^{0^2}} \over {2c^2}}} \right)} \right] \cr & \quad - A{\rm \alpha} X{\rm ^{\prime}}_p,}$$
$$\eqalign{X_n & = \left[ {\displaystyle{{B{\rm \beta} ({\rm \beta} - 1){\rm \mu} _n} \over {2L^2}} - \left( {u_n^0 - M} \right)\displaystyle{{3{\rm \mu} _n u_n^0} \over {2c^2 L^2 H^2}} - \displaystyle{{{\rm \mu} _n} \over {2L^2 H^2}} \left( {1 + \displaystyle{{3u_p^{0^2}} \over {2c^2}}} \right)} \right] \cr & \quad - B{\rm \beta} X{\rm ^{\prime}}_n,}$$
$$\eqalign{& X_b = \left[ { - \left( {u_b^0 - M} \right)\displaystyle{{3u_p^0} \over {2c^2 N^2 I^2}} - \displaystyle{1 \over {2N^2 I^2}} \left( {1 + \displaystyle{{3u_b^{0^2}} \over {2c^2}}} \right)} \right], \cr & Y_b = \left[ {\displaystyle{{(u_b^0 - M)(1 + 3u_b^{0^2} /2c^2 )Y{\rm ^{\prime}}_b} \over I} - \left( {1 + \displaystyle{{3u_b^{0^2}} \over {2c^2}}} \right)\displaystyle{1 \over N}} \right],} $$
$$\eqalign{& Y_p = \left[ {\displaystyle{{\left( {u_p^0 - M} \right)\left( {1 + 3u_p^{0^2} /2c^2} \right)Y{\rm ^{\prime}}_p} \over G} - \left( {1 + \displaystyle{{3u_p^{0^2}} \over {2c^2}}} \right)\displaystyle{{{\rm \mu} _p} \over J}} \right], \cr & Y_n = \left[\displaystyle{{\left( {u_n^0 - M} \right)\left( {1 + 3u_n^{0^2} /2c^2} \right)Y{\rm ^{\prime}}_n} \over H} - \left( {1 + \displaystyle{{3u_p^{0^2}} \over {2c^2}}} \right)\displaystyle{{{\rm \mu} _n} \over L}\right].} $$
Here the quantities X p ′, X n ′, Y p ′, and Yn ′ are
$$X_p^{\prime} = \displaystyle{\matrix{\left( { - {\rm \mu} _p^2 \left( {1 + u_p^{0^2} /2c^2} \right)/J^2 G} \right) - \left( {{\rm \mu} _p^2 \left( {u_p^0 - M} \right)/2c^2 J^2 G^2} \right) \cr - \left( {{\rm \mu} _p^2 u_p^0 \left( {2u_p^0 - M} \right)/2c^2 J^2 G} \right)} \over {\left( {u_p^0 - M} \right)\left( {1 + u_p^{0^2} /2c^2} \right)}},$$
$$X_n^{\prime} = \displaystyle{\matrix{\left( { - {\rm \mu} _n^2 \left( {1 + u_n^{0^2} /2c^2} \right)/L^2H} \right) - \left( {{\rm \mu} _n^2 \left( {u_n^0 - M} \right)/2c^2L^2H^2} \right)\cr - \left( {{\rm \mu} _n^2 u_n^0 \left( {2u_n^0 - M} \right)/2c^2L^2H} \right)} \over {\left( {u_n^0 - M} \right)\left( {1 + u_n^{0^2} /2c^2} \right)}},$$
$$\eqalign{& Y_p ^{\rm {\prime}} = \displaystyle{{\left( {2u_p^0 {\rm \mu} _p /2c^2 JG} \right) + {\rm \mu} _p \left( {\left( {1 + u_p^{0^2} /2c^2} \right)/J} \right)} \over {\left( {u_p^0 - M} \right)\left( {1 + u_p^{0^2} /2c^2} \right)}},\; \cr & Y_b^{\rm {\prime}} = \displaystyle{{\left( { - \left( {2u_n^0 /2c^2 NI} \right)} \right) - \left( {\left( {1 + u_b^{0^2} /2c^2} \right)/N} \right)} \over {(u_b^0 - M)\left( {1 + u_b^{0^2} /2c^2} \right)}},}$$
$$Y_n {\!\! ^{\prime}} = \displaystyle{{\left( { - \left( {2u_n^0 {\rm \mu} _n /2c^2 LH} \right)} \right) - {\rm \mu} _n \left( {\left( {1 + u_n^{0^2} /2c^2 /L} \right)} \right)} \over {\left( {u_n^0 - M} \right)\left( {1 + u_n^{0^2} /2c^2} \right)}}.$$
In order to derive the solution of Eq. (21), we use the transformation η = ξ − Uτ and using the boundary conditions: ϕ → 0, d 2ϕ/dξ 2 → 0, ξ → ∞, and integrating and simplifying Eq. (21), we obtain the following solution:
$${\rm \phi} = {\rm \phi} _m \sec h^2 \left( {{{\rm \eta} / {\rm \Delta}}} \right)$$
where
${\rm \phi} _m = 3M/A\;{\rm and}\,\,{\rm \Delta} = \sqrt {4B/M} $
are the amplitude and width of the solitary waves accordingly.
4. RESULTS
The numerical study of the above problem has been made by feeding the solutions with values of parameters that aptly fit the situation. It has been analytically found and graphically demonstrated that the ion-acoustic solitary structures become deeper with increasing Mach number (U) as shown in Figure 1. Figure 2 shows that the solitary wave structure for different streaming velocity for the beam. It has been found that the solitons become narrower and shorter with increasing beam-streaming velocity. However, the waves grow deeper when the positron streaming velocity increases (Fig. 3).
Fig. 1. Solitary wave structure for different U (Mach number).
Fig. 2. Solitary wave structure for different streaming velocity for the beam (
$u_b^0 $
).
Fig. 3. Solitary wave structure for different streaming velocity of positive ion (
$u_b^0 $
).
It has been found that for negative values of q, the amplitude of the rarefactive solitons and its width increases with the increase in q value (Fig. 4). This is in accordance with the findings of Tribeche et al., (Reference Tribeche, Djebarni and Amour2010). Similar findings had been reported by Danehkar et al., (Reference Danehkar, Saini, Hellberg and Kourakis2011) in which the IAWs were found behave likewise using Sagdeevs’ method. Hafez et al., (Reference Hafez, Talukder and Sakthivel2016) analyze the formulation and characteristics of compressive solitary structures with relativistic ions; however, in our work, we considered the electron to acquire relativistic speed, so we found the formulation of rarefactive solitary structures. According to Hafez, for weakly relativistic on ion-acoustic solitary wave, the depth and width of solitary profile decrease with decrease in the value of q which is similar to our findings.
Fig. 4. Solitary wave structure for different values of non-extensive parameter (q).
The results can be thus interpreted as the wave advances, the initially smooth perturbation becomes steeper and oscillations appear almost negligible; their amplitude increases with increasing distance traversed by the wave, and the width of the wave front, starting with a certain instant, remains constant. The solitary structures are rarefactive, that is, for ions there will be a fall in density of ions. This may be put forward like this that as the positive ion velocity increases compared with negative ion and electron beam velocity, the positive ions tend to come close and due to Columbic repulsion move apart thus causing a fall in density of positive ions.
5. REMARKS
Thermal electron beam has many applications in both scientific and industrial sectors. It is applied for remelting, welding, surface layer treatment, perforation, surface structuring, micro treatment, evaporation, etc. Thus, the quality and property of electron beam is a determining factor in the process. The findings of this research paper may be quite helpful in controlling such processes and finding the unwanted features that might creep in.
ACKNOWLEDGMENT
The authors would like to thank Jadavpur University, Kolkata for providing support to carry out the research work.
APPENDIX
The normalized basic equations are
$$\displaystyle{{\partial ({\rm \gamma} _j n_j )} \over {\partial t}} + \displaystyle{{\partial ({\rm \gamma} _j n_j u_j )} \over {\partial x}} = 0,$$
$$\left[ {\displaystyle{\partial \over {\partial t}} + u_p \displaystyle{\partial \over {\partial x}}} \right]({\rm \gamma} _p u_p ) = - {\rm \mu} _p \displaystyle{{\partial {\rm \phi}} \over {\partial x}} - A{\rm \alpha} n_p^{{\rm \alpha} - 2} \displaystyle{{\partial n_p} \over {\partial x}},$$
$$\left[ {\displaystyle{\partial \over {\partial t}} + u_n \displaystyle{\partial \over {\partial x}}} \right]({\rm \gamma} _n u_n ) = {\rm \mu} _n \displaystyle{{\partial {\rm \phi}} \over {\partial x}} - B{\rm \beta} n_n^{{\rm \beta} - 2} \displaystyle{{\partial n_n} \over {\partial x}},$$
$$\left[ {\displaystyle{\partial \over {\partial t}} + u_b \displaystyle{\partial \over {\partial x}}} \right]({\rm \gamma} _b u_b ) = \displaystyle{{\partial {\rm \phi}} \over {\partial x}},$$
$$\displaystyle{{\partial ^2 {\rm \phi}} \over {\partial x^2}} = n_e + \displaystyle{{n_b} \over {\rm \rho}} + \displaystyle{{Z_n n_n} \over {\rm \sigma}} - \displaystyle{{Z_p n_p} \over {\rm \delta}}.$$
The perturbation expansion of the field variables are given by
$$\left( {\matrix{ {u_p} \cr {u_n} \cr {u_b} \cr {\rm \phi} \cr}} \right) = \left( {\matrix{ {u_p^0} \cr {u_n^0} \cr {u_b^0} \cr {{\rm \phi} ^0} \cr}} \right) + {\rm \varepsilon} \left( {\matrix{ {u_p^{(1)}} \cr {u_n^{(1)}} \cr {u_b^{(1)}} \cr {{\rm \phi} ^{(1)}} \cr}} \right) + {\rm \varepsilon} ^2 \left( {\matrix{ {u_p^{(2)}} \cr {u_n^{(2)}} \cr {u_b^{(2)}} \cr {{\rm \phi} ^{(2)}} \cr}} \right) + \cdots,$$
$$\left( {\matrix{ {n_p} \cr {n_n} \cr {n_b} \cr}} \right) = \left( {\matrix{ 1 \cr 1 \cr 1 \cr}} \right) + {\rm \varepsilon} \left( {\matrix{ {n_p^{(1)}} \cr {n_n^{(1)}} \cr {n_b^{(1)}} \cr}} \right) + {\rm \varepsilon} ^2 \left( {\matrix{ {n_p^{(2)}} \cr {n_n^{(2)}} \cr {n_b^{(2)}} \cr}} \right) + \cdots.$$
The electrons have the q-extensive distributions is given by,
$$\eqalign{n_e &= [1 + (q - 1){\rm \phi} ]^{(q + 1)/2(q - 1)} \cr & = \left[ \left\{ 1 + \left(\displaystyle{q + 1 \over 2} \right){\rm \phi} ^0 + \displaystyle{(3 - q)(q + 1) \over 4}{\rm \phi}^{0^2} \right\}\right. \cr & \quad \left.+ {\rm \varepsilon} \left\{ \left(\displaystyle{q + 1 \over 2} \right){\rm \phi}^{(1)} + \displaystyle{(3 - q)(q + 1) \over 2}{\rm \phi} ^0 {\rm \phi} ^{(1)} \right\} \right. \cr & \quad \left. + {\rm \varepsilon} ^2 \left\{ \left( \displaystyle{q + 1 \over 2} \right){\rm \phi} ^{(2)} + \displaystyle{(3 - q)(q + 1) \over 2}{\rm \phi}^0 {\rm \phi}^{(2)} \right.\right. \cr & \quad + \left.\left. \displaystyle{(3 - q)(q + 1) \over 4}{\rm \phi}^{(1)^2} \right\} \right].}$$
Assuming space–time dependence of the field variables as e i(kx−ωt) we get,
$$\displaystyle{\partial \over {\partial x}} \equiv ik,$$
$$\displaystyle{\partial \over {\partial t}} \equiv - i{\rm \omega}.$$
Now,
$$n_p^{{\rm \alpha} - 2} = 1 + {\rm \varepsilon} ({\rm \alpha} - 2)n_p^{(1)} + {\rm \varepsilon} ^2 ({\rm \alpha} - 2)n_p^{(2)} + \cdots,$$
$$n_n^{{\rm \beta} - 2} = 1 + {\rm \varepsilon} ({\rm \beta} - 2)n_n^{(1)} + {\rm \varepsilon} ^2 ({\rm \beta} - 2)n_n^{(2)} + \cdots.$$
From momentum equations, we get by linearizing in powers of ε,
$$- i{\rm \omega} {\rm \gamma} _p u_p^{(1)} + iku_p^0 {\rm \gamma} _p u_p^{(1)} = - {\rm \mu} _p ik{\rm \phi} ^{(1)} - A{\rm \alpha} ikn_p^{(1)}, $$
or
$$k[{\rm \gamma} _p u_p^{(0)} u_p^{(1)} + {\rm \mu} _p {\rm \phi} ^{(1)} + A{\rm \alpha} n_p^{(1)} ] = {\rm \omega} {\rm \gamma} _p u_p^{(1)}.$$
Similarly,
$$k[{\rm \gamma} _n u_n^{(0)} u_n^{(1)} - {\rm \mu} _n {\rm \phi} ^{(1)} + B{\rm \beta} n_n^{(1)} ] = {\rm \omega} {\rm \gamma} _n u_n^{(1)},$$
and
$$k[{\rm \gamma} _b u_b^{(0)} u_b^{(1)} - {\rm \phi} ^{(1)} ] = {\rm \omega} {\rm \gamma} _n u_n^{(1)}.$$
From continuity equation of positron by linearizing ε we get,
$$\eqalign{& i{\rm \omega} \left\{ {n_p^{(1)} \left( {1 + \displaystyle{{u_p^{(0)^2}} \over {2c^2}}} \right) + \displaystyle{{2u_p^{(0)} u_p^{(1)}} \over {2c^2}}} \right\}\cr & \qquad = ik\left\{ {2\displaystyle{{u_p^{(0)^2} u_p^{(1)}} \over {2c^2}} + n_p^{(1)} u_p^{(0)} \left( {1 + \displaystyle{{u_p^{(0)^2}} \over {2c^2}}} \right) + u_p^{(1)} \left( {1 + \displaystyle{{u_p^{(0)^2}} \over {2c^2}}} \right)} \right\},}$$
or
$$n_p^{(1)} A^{\prime}\left( {k - {\rm \omega} u_p^{(0)}} \right) = \left\{ {\displaystyle{{ - 2{\rm \omega} (A{\rm ^{\prime}} - 1)} \over {2c^2 u_p^{(0)}}} + 2k(A{\rm ^{\prime}} - 1) + kA{\rm ^{\prime}}} \right\}u_p^{(1)},$$
where
$$\left( {1 + \displaystyle{{u_p^{(0)^2}} \over {2c^2}}} \right) = A{\rm ^{\prime}} = D.$$
Therefore, from Eqs (A13) and (A16) we get,
$$u_p^{(1)} = \displaystyle{{Du_p^{(0)} \left( {{\rm \omega} - ku_p^{(0)}} \right)} \over {\left[ {2u_p^{(0)} k(D - 1) + kDu_p^{(0)} - 2{\rm \omega} (D - 1)} \right]}}n_p^{(1)}$$
Putting this value on the result of momentum equation we get,
Hence,
$$- \displaystyle{{Z_p n_p} \over {\rm \delta}} \!=\! - \displaystyle{{{\rm \mu} _p Z_p} \over {\rm \delta}} \displaystyle{{\left[ {2u_p^{(0)} k(D - 1) + kDu_p^{(0)} - 2{\rm \omega} (D - 1)} \right]} \over {\matrix{D^2 u_p^{(0)} \left( {{\rm \omega} - ku_p^{(0)}} \right) \cr - A{\rm \alpha} \left[ {2u_p^{(0)} k(D - 1) + kDu_p^{(0)} - 2{\rm \omega} (D - 1)} \right]}} } {\rm \phi} ^{(1)}.$$
Similarly,
$$\displaystyle{{Z_n n_n} \over {\rm \sigma}} = - \displaystyle{{{\rm \mu} _n Z_n} \over {\rm \sigma}} \displaystyle{{\left[ {2u_n^{(0)} k(E - 1) + kEu_n^{(0)} - 2{\rm \omega} (E - 1)} \right]} \over {\matrix{E^2 u_n^{(0)} \left( {{\rm \omega} - ku_n^{(0)}} \right) \cr - B{\rm \beta} \left[ {2u_n^{(0)} k(E - 1) + kEu_n^{(0)} - 2{\rm \omega} (E - 1)} \right]}} } {\rm \phi} ^{(1)},$$
where
$$\left( {1 + \displaystyle{{u_n^{(0)^2}} \over {2c^2}}} \right) = E,$$
$$\displaystyle{{n_b} \over {\rm \rho}} = - \displaystyle{1 \over {\rm \rho}} \displaystyle{{\left[ {2u_b^{(0)} k(F - 1) - kFu_b^{(0)} - 2{\rm \omega} (F - 1)} \right]} \over {F^2 u_b^{(0)} \left( {{\rm \omega} - ku_b^{(0)}} \right)}}{\rm \phi} ^{(1)},$$
where
$$\left( {1 + \displaystyle{{u_b^{(0)^2}} \over {2c^2}}} \right) = F.$$
Putting this value in Poisson's equation, we will have
$$- k^2 {\rm \phi} ^{(1)} = n^{(1)} _e + \displaystyle{{n^{(1)} _b} \over {\rm \rho}} + \displaystyle{{Z_n n^{(1)} _n} \over {\rm \sigma}} - \displaystyle{{Z_p n^{(1)} _p} \over {\rm \delta}} $$
$$\eqalign{ - k^2 {\rm \phi} ^{(1)} &= \left\{ {\left( {\displaystyle{{q + 1} \over 2}} \right){\rm \phi} ^{(1)} + \displaystyle{{(3 - q)(q + 1)} \over 2}{\rm \phi} ^0 {\rm \phi} ^{(1)}} \right\} - \displaystyle{{{\rm \mu} _n Z_n} \over {\rm \sigma}} \displaystyle{{\left[ {2u_n^{(0)} k(E - 1) + kEu_n^{(0)} - 2{\rm \omega} (E - 1)} \right]} \over {E^2 u_n^{(0)} \left( {{\rm \omega} - ku_n^{(0)}} \right) - B{\rm \beta} \left[ {2u_n^{(0)} k(E - 1) + kEu_n^{(0)} - 2{\rm \omega} (E - 1)} \right]}}{\rm \phi} ^{(1)} \cr & - \displaystyle{{{\rm \mu} _p Z_p} \over {\rm \delta}} \displaystyle{{\left[ {2u_p^{(0)} k(D - 1) + kDu_p^{(0)} - 2{\rm \omega} (D - 1)} \right]} \over {D^2 u_p^{(0)} \left( {{\rm \omega} - ku_p^{(0)}} \right) - A{\rm \alpha} \left[ {2u_p^{(0)} k(D - 1) + kDu_p^{(0)} - 2{\rm \omega} (D - 1)} \right]}}{\rm \phi} ^{(1)} - \displaystyle{1 \over {\rm \rho}} \displaystyle{{\left[ {2u_b^{(0)} k(F - 1) - kFu_b^{(0)} - 2{\rm \omega} (F - 1)} \right]} \over {F^2 u_b^{(0)} \left( {{\rm \omega} - ku_b^{(0)}} \right)}}{\rm \phi} ^{(1)}.} $$
$$\eqalign{k^2 &= - \left\{ {\left( {\displaystyle{{q + 1} \over 2}} \right) + \displaystyle{{(3 - q)(q + 1)} \over 2}{\rm \phi} ^0} \right\} + \displaystyle{{{\rm \mu} _n Z_n} \over {\rm \sigma}} \displaystyle{{\left[ {2u_n^{(0)} k(E - 1) + kEu_n^{(0)} - 2{\rm \omega} (E - 1)} \right]} \over {E^2 u_n^{(0)} \left( {{\rm \omega} - ku_n^{(0)}} \right) - B{\rm \beta} \left[ {2u_n^{(0)} k(E - 1) + kEu_n^{(0)} - 2{\rm \omega} (E - 1)} \right]}} \cr &\quad + \displaystyle{{{\rm \mu} _p Z_p} \over {\rm \delta}} \displaystyle{{\left[ {2u_p^{(0)} k(D - 1) + kDu_p^{(0)} - 2{\rm \omega} (D - 1)} \right]} \over {D^2 u_p^{(0)} \left( {{\rm \omega} - ku_p^{(0)}} \right) - A{\rm \alpha} \left[ {2u_p^{(0)} k(D - 1) + kDu_p^{(0)} - 2{\rm \omega} (D - 1)} \right]}} + \displaystyle{1 \over {\rm \rho}} \displaystyle{{\left[ {2u_b^{(0)} k(F - 1) - kFu_b^{(0)} - 2{\rm \omega} (F - 1)} \right]} \over {F^2 u_b^{(0)} \left( {{\rm \omega} - ku_b^{(0)}} \right)}}.} $$
This is the linear dispersion relation.
