2. The Fluid Equations

In statistical mechanics, there exist generally two types of systems such as extensive and nonextensive systems, depending on the range of interparticle forces. The extensive systems are considered for short-range interparticle forces and need Boltzmann–Gibbs (B-G) statistics. On the other hand, nonextensive systems hold for long-range interparticle forces, need generalized B-G statistics, and also include dissipative systems having nonvanishing thermodynamic currents. Tsallis [Reference Tsallis33] introduced q-nonextensive distribution which is used to model astrophysical regions (i.e., gravitational systems and stellar polytropes) [Reference Lima, Silva and Santos34, Reference Plastino and Plastino35] where Maxwellian distribution might become inadequate. Tsallis q-entropy and the probabilistically independent postulate are known to be the basis of the nonextensive statistics. The principle of maximum q-entropy gives the power-law q-distribution which is called q-exponentials in nonextensive statistics. The nonextensivity is also known as pseudoadditivity or nonadditivity which gives both the q-entropy and the energy (for more details, see [Reference Shalini, Saini and Misra36]). The parameter q measures the degree of nonextensivity of the system. The q-nonextensive number density of ions is given as follows [Reference Roy, Saha, Chatterjee and Tribeche37]:

(1) $\begin{array}{}{n}_i=n_{i_0}{\left(1-\left(q-1\right)\frac{e\psi }{K_B{T}_i}\right)}^{\left(\right)q+1/2\left(q-1\right)\left(\right)}.\end{array}$

We consider a magnetized dusty plasma comprising of inertial negatively charged dust, nonextensive ions, and Maxwellian electrons to study the head-on collision and phase shifts of two DA shock waves under the influence of nonextensively modified polarization force. Consider charge on negatively charged dust grain be $q_d=-Z_{d}e$ , and $n_{e_0}$ , $n_{i_0}$ , and $n_{d_0}$ are the unperturbed electron, ion, and dust number densities, respectively. Z_d is the dust grain charge. The charge neutrality condition at equilibrium yields $n_{e_0}+Z_d{n}_{d_0}=n_{i_0}$ which can be expressed as μ_e + 1 = μ_i, where ${\mu }_e=n_{e_0}/Z_d{n}_{d_0}=1/\rho -1$ and ${\mu }_i=n_{i_0}/Z_d{n}_{d_0}=\rho /\rho -1$ and $\rho =n_{i_0}/n_{e_0}$ . The electrons are considered Maxwellian with density $n_e=n_{e_0}\exp \left(e\psi ,/,K_B{T}_e\right)$ . After Taylor expansion of the RHS of equation (1) for $e\psi /K_B{T}_i\ll 1$ , we have

(2) $\begin{array}{}{n}_i=n_{i_0}\left(1-h_1\frac{e\psi }{K_B{T}_i}+h_2{\left(\frac{e\psi }{K_B{T}_i}\right)}^2\right),\end{array}$

where h ₁ = q + 1/2 and h ₂ = (q + 1)(3 − q)/8. The expression for polarization force F_pol is given by [Reference Hamaguchi and Farouki15]

(3) $\begin{array}{}{F}_{pol}=\frac{q_d^2}{2}\frac{\nabla {\lambda }_D}{{\lambda }_D^2}.\end{array}$

We have now derived the polarization force (i.e., deformation of Debye sphere around the dust grains) acting on dust grains modified by nonextensive ions in a nonuniform dusty plasma comprising of nonextensive ions and Maxwellian electrons where the expression for modified Debye length λ_D for negative dust (i.e., $n_e{T}_i\ll n_i{T}_e$ ) is ${\lambda }_D\simeq \sqrt{K_B{T}_i/4\pi e^2{n}_i{h}_1}$ in the presence of nonextensive ions. The expression for nonextensively modified polarization force is obtained as

(4) $\begin{array}{}{F}_{\text{pol}}=-Z_{d}eR{h}_0{\left(\frac{n_i}{n_{i_0}}\right)}^{1/2}\nabla \psi ,\end{array}$

where $R=Z_d{e}^2/4K_B{T}_i{\lambda }_{D{i}_0}$ measures the polarization force effect, i.e., interaction of highly negatively charged dust with nonextensive ions with ${\lambda }_{D{i}_0}=\sqrt{K_B{T}_i/4\pi e^2{n}_{i_0}{h}_1}$ and $h_0=\left(h_1-\left(2h_2-\left(h_1^2,/,2\right)\left(e\psi ,/,K_B{T}_i\right)\right)\right)$ . Thus, nonextensivity of ions (via q) notably alters the polarization force F_pol and also polarization parameter R. When q ⟶ 1, the polarization force expression equation (4) agrees well with the expression for the case of Maxwellian ions and electrons derived by Asaduzzaman et al. [Reference Asaduzzaman, Mamun and Ashrafi18].

The dynamics of DA shocks waves in polarized dusty plasma is characterized by the following set of normalized fluid equations (continuity, momentum, and Poisson):

(5) $\begin{array}{}\frac{\partial n_d}{\partial t}+\nabla .\left(n_d{u}_d\right)=0,\end{array}$

(6) $\begin{array}{}\frac{\partial u_d}{\partial t}+\left(u_d.\nabla \right)u_d={\chi }_p\nabla \psi +{\eta }_l{\nabla }^2{u}_d-\Omega \left(u_d\times \hat{z}\right),\end{array}$

(7) $\begin{array}{}{\nabla }^2\psi =n_d+n_e-n_i,\end{array}$

where $\nabla =\hat{x}\left(\partial /\partial x\right)+\hat{y}\left(\partial /\partial y\right)+\hat{z}\left(\partial /\partial z\right)$ and $u_d=\hat{x}{u}_{dx}+\hat{y}{u}_{dy}+\hat{z}{u}_{dz}$ . n_d, u_d, and ψ are the number density, velocity, and potential of dust fluid, σ = T_i/T_e, and ${\chi }_p=1-R\left(h_1-\left(2h_2+\left(h_1^2,/,2\right)\right)\right)$ . Density n_j (j = d, e, i) is normalized by unperturbed density of dust $n_{d_0}$ , dust velocity u_d by $\sqrt{Z_d{K}_B{T}_i/m_d}$ , ψ by K_BT_i/e, time t by ${\omega }_{pd}^{-1}$ , where ${\omega }_{pd}={\left(4\pi {\left(Z_{d}e\right)}^2{n}_{d_0},/,m_d\right)}^{1/2}$ , and x by λ_Dd, where ${\lambda }_{Dd}={\left(K_B{T}_i,/,4\pi n_{d_0}{Z}_d{e}^2\right)}^{1/2}$ . ηl is the dust viscosity normalized by ${\omega }_{pd}{\lambda }_{Dd}^2$ . The magnetic field is assumed to be along the z-axis (i.e., $B=B_0\hat{z}$ ), and the dust cyclotron frequency $\Omega \left(=Z_{d}e{B}_0/m_d\right)$ is normalized by ω_pd.

3. Derivation of KdVB Equations and Phase Shifts

We employ an extended PLK perturbation method to investigate the collision between two DA shock waves in a magnetized dusty plasma. The perturbed quantities are described as follows:

(8) $\begin{array}{}Y=Y_0+\sum _{r=1}^{\infty }{ϵ}^{r+1}{Y}_r,\hspace{1em}\Gamma =\sum _{r=1}^{\infty }{ϵ}^{r+2}{\Gamma }_r,\end{array}$

where $Y=\left(n_{\text{d}},u_{\text{d}z},\psi \right)$ , Y ₀ = (1,0,0), and $\Gamma =\left(u_{\text{d}x},u_{\text{d}y}\right)$ . The scaling variables x and t are stretched by the new coordinate system ξ, η, and τ in multiple scale variables:

(9) $\begin{array}{}\xi =ϵ\left(l_{x}x+l_{y}y+l_{z}z-v_{0}t\right)+{ϵ}^2{M}_0\left(\eta ,\tau \right)+\dots ,\end{array}$

(10) $\begin{array}{}\eta =ϵ\left(l_{x}x+l_{y}y+l_{z}z+v_{0}t\right)+{ϵ}^2{N}_0\left(\xi ,\tau \right)+\dots ,\end{array}$

(11) $\begin{array}{}\tau ={ϵ}^{3}t,\end{array}$

where ξ and η denote the opposite side trajectories of DA shock waves and ϵ is a small parameter determining the strength of nonlinearity. The functions M ₀(η, τ) and N ₀(ξ, τ) are determined later. l_x, l_y, and l_z represent the direction cosines along the x, y, and z directions, respectively, such that $l_x^2+l_y^2+l_z^2=1$ . The value of dust viscosity is considered small and its value is set as ${\eta }_l=ϵ{\eta }_{l_0}$ , where ${\eta }_{l_0}$ is O(1). Making use of equations (8)–(11) into (5)–(7), at the lowest orders of ϵ, we obtain the following first-order equations:

(12) $\begin{array}{}{v}_0\left(-\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta }\right)n_{{\text{d}}_1}+l_z\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta }\right)u_{\text{d}{z}_1}=0,\end{array}$

(13) $\begin{array}{}-v_0\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta }\right)u_{{\text{d}}_1}+\left(1-R{h}_1\right)\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta }\right){\psi }_1=0,\end{array}$

(14) $\begin{array}{}\left(1-R{h}_1\right)l_x\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta }\right){\psi }_1-\Omega u_{\text{d}{y}_1}=0,\end{array}$

(15) $\begin{array}{}\left(1-R{h}_1\right)l_y\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta }\right){\psi }_1+\Omega u_{\text{d}{x}_1}=0,\end{array}$

(16) $\begin{array}{}{n}_{d1}=-a_1{\psi }_1,\end{array}$

where a ₁ = μ_eσ + μ_ih₁. On simplifying equations (12)–(16), the relations of different first-order physical quantities are obtained as follows:

(17) $\begin{array}{}{\psi }_1={\psi }_{1\xi }\left(\xi ,\tau \right)+{\psi }_{1\eta }\left(\eta ,\tau \right),\end{array}$

(18) $\begin{array}{}{n}_{{\text{d}}_1}=-\frac{\left(1-R{h}_1\right)l_z^2}{v_0^2}\left({\psi }_{1\xi }\left(\xi ,\tau \right)+{\psi }_{1\eta }\left(\eta ,\tau \right)\right),\end{array}$

(19) $\begin{array}{}{u}_{\text{d}{x}_1}=-\frac{\left(1-R{h}_1\right)l_y}{{\omega }_{cd}}\left(\frac{{\psi }_{1\xi }\left(\xi ,\tau \right)}{\partial \xi }+\frac{{\psi }_{1\eta }\left(\eta ,\tau \right)}{\partial \eta }\right),\end{array}$

(20) $\begin{array}{}{u}_{\text{d}{y}_1}=\frac{\left(1-R{h}_1\right)l_x}{{\omega }_{cd}}\left(\frac{{\psi }_{1\xi }\left(\xi ,\tau \right)}{\partial \xi }+\frac{{\psi }_{1\eta }\left(\eta ,\tau \right)}{\partial \eta }\right),\end{array}$

(21) $\begin{array}{}{u}_{\text{d}{z}_1}=-\frac{\left(1-R{h}_1\right)l_z}{v_0}\left({\psi }_{1\xi }\left(\xi ,\tau \right)-{\psi }_{1\eta }\left(\eta ,\tau \right)\right),\end{array}$

(22) $\begin{array}{}{n}_{{\text{d}}_1}=-a_1\left({\psi }_{1\xi }\left(\xi ,\tau \right)+{\psi }_{1\eta }\left(\eta ,\tau \right)\right).\end{array}$

From equation (17), one predicts that there are two DA shock waves in which ${\psi }_{1\xi }\left(\xi ,\tau \right)$ is propagating towards the right and ${\psi }_{1\eta }\left(\eta ,\tau \right)$ is propagating towards the left. On solving equations (17)–(22), the expression for phase velocity in a magnetized dusty plasma under the effect of nonextensively modified polarization force is obtained as $v_0=l_z\sqrt{\left(1-R{h}_1\right)/a_1}$ .

Finally, for the next higher orders of ϵ, we obtain

(23) $\begin{array}{c}-4l_z\frac{{\partial }^2{u}_{\text{d}{z}_3}}{\partial \xi \partial \eta }=\int \left(\frac{\partial {\psi }_{1\xi }\left(\xi ,\tau \right)}{\partial \tau }+A{\psi }_{1\xi }\left(\xi ,\tau \right)\frac{\partial {\psi }_{1\xi }\left(\xi ,\tau \right)}{\partial \xi }+B\frac{{\partial }^3{\psi }_{1\xi }\left(\xi ,\tau \right)}{\partial {\xi }^3}-C\frac{{\partial }^2{\psi }_{1\xi }\left(\xi ,\tau \right)}{\partial {\xi }^2}\right)\text{d}\eta +\int \left(\frac{\partial {\psi }_{1\eta }\left(\eta ,\tau \right)}{\partial \tau }-A{\psi }_{1\eta }\left(\eta ,\tau \right)\frac{\partial {\psi }_{1\eta }\left(\eta ,\tau \right)}{\partial \eta }-B\frac{{\partial }^3{\psi }_{1\eta }\left(\eta ,\tau \right)}{\partial {\eta }^3}+C\frac{{\partial }^2{\psi }_{1\eta }\left(\eta ,\tau \right)}{\partial {\eta }^2}\right)\text{d}\xi \\ +\int \int \left(2v_0\frac{\partial M_0}{\partial \eta }+D{\psi }_{1\eta }\left(\eta ,\tau \right)\right)\frac{{\partial }^2{\psi }_{1\xi }\left(\xi ,\tau \right)}{\partial {\xi }^2}\text{d}\xi \text{d}\eta \\ +\int \int \left(2v_0\frac{\partial N_0}{\partial \xi }+D{\psi }_{1\xi }\left(\xi ,\tau \right)\right)\frac{{\partial }^2{\psi }_{1\eta }\left(\eta ,\tau \right)}{\partial {\eta }^2}\text{d}\xi \text{d}\eta .\end{array}$

From equation (23), it is noticed that the first (second) term of the right side is proportional to η (ξ), and thus, the integrating function is independent of η (ξ). These two secular terms in the above equation must be removed to avoid spurious resonance. Hence, we obtain the following nonlinear equations:

(24) $\begin{array}{}\frac{\partial {\varphi }_{\xi }}{\partial \tau }+A{\varphi }_{\xi }\frac{\partial {\varphi }_{\xi }}{\partial \xi }+B\frac{{\partial }^3{\varphi }_{\xi }}{\partial {\xi }^3}-C\frac{{\partial }^2{\varphi }_{\xi }}{\partial {\xi }^2}=0,\end{array}$

(25) $\begin{array}{}\frac{\partial {\varphi }_{\eta }}{\partial \tau }-A{\varphi }_{\eta }\frac{\partial {\varphi }_{\eta }}{\partial \eta }-B\frac{{\partial }^3{\varphi }_{\eta }}{\partial {\eta }^3}+C\frac{{\partial }^2{\varphi }_{\eta }}{\partial {\eta }^2}=0.\end{array}$

In the above equations, we have considered ${\psi }_{1\xi }={\varphi }_{\xi }$ and ${\psi }_{1\xi }={\varphi }_{\eta }$ for simplicity. The nonlinear, dispersion, and dissipation coefficients, respectively, are given as

(26) $\begin{array}{}A=\frac{-3l_z^2\left(1-R{h}_1\right)}{2v_0}+\frac{v_{0}b}{a_1}+\frac{v_0\left(R\left(2h_2+\left(h_1^2,/,2\right)\right)\right)}{2\left(1-R{h}_1\right)},\end{array}$

(27) $\begin{array}{}B=\frac{\left(1-l_z^2\right)\left(1-R{h}_1\right)v_0}{2a_1{\Omega }^2}+\frac{v_0}{2a_1},\end{array}$

(28) $\begin{array}{}C=\frac{{\eta }_{l_0}}{2},\end{array}$

where $b={\mu }_e{\sigma }^2/2-{\mu }_i{h}_2$ . Equations (24) and (25) are the KdV–Burgers equations of two shock waves travelling towards each other in the reference frame of ξ and η.

3.1. Solution of KdVB Equation and Phase Shifts

The steady-state solutions of equations (24) and (25) are obtained as follows [Reference El-Shamy and Al-Asbali27]:

(29) $\begin{array}{}{\varphi }_{\xi }={\varphi }_m{\left(1-\text{tanh}\left(w\left(\xi -\left(\frac{6C^2}{25B}\right)\tau \right)\right)\right)}^2,\end{array}$

(30) $\begin{array}{}{\varphi }_{\eta }={\varphi }_m{\left(1+\text{tanh}\left(w\left(\eta +\left(\frac{6C^2}{25B}\right)\tau \right)\right)\right)}^2.\end{array}$

The electrostatic potential for weak head-on collision is given as

(31) $\begin{array}{}\Phi ={\varphi }_m{\left(1-\text{tanh}\left(w\left(\xi -\left(\frac{6C^2}{25B}\right)\tau \right)\right)\right)}^2+{\varphi }_m{\left(1+\text{tanh}\left(w\left(\eta +\left(\frac{6C^2}{25B}\right)\tau \right)\right)\right)}^2,\end{array}$

where ${\varphi }_m=3C^2/25AB$ is the maximum amplitude and w = 10B/C is the width of DA shock waves.

In the next higher order, the third and fourth terms on the right-hand side of equation (23) become secular and we obtain [Reference El-Shamy and Al-Asbali27]

(32) $\begin{array}{}2v_0\frac{\partial M_0}{\partial \eta }=-D{\varphi }_{\eta },\end{array}$

(33) $\begin{array}{}2v_0\frac{\partial N_0}{\partial \xi }=-D{\varphi }_{\xi },\end{array}$

where

(34) $\begin{array}{}D=\frac{v_{0}b}{a_1}+\frac{\left(1-R{h}_1\right)l_z^2}{2v_0}+\frac{v_0\left(R\left(2h_2+\left(h_1^2,/,2\right)\right)\right)}{2\left(1-R{h}_1\right)}.\end{array}$

Equations (32) and (33) are solved using appropriate initial and boundary conditions to determine the phase shifts as

(35) $\begin{array}{}\Delta M_0\left(\eta ,\tau \right)\approx -{ϵ}^2\frac{3{\eta }_{l0}}{50v_0}\frac{CD}{AB},\end{array}$

(36) $\begin{array}{}\Delta N_0\left(\xi ,\tau \right)\approx {ϵ}^2\frac{3{\eta }_{l0}}{50v_0}\frac{CD}{AB}.\end{array}$

These phase shifts are functions of different coefficients that are dependent on various physical parameters. Therefore, any variation in physical parameters makes a significant change in the phase shifts.

4. Numerical Analysis and Discussion

In this investigation, the head-on collision of DA shock waves in a magnetized dusty plasma consisting of nonextensive ions and Maxwellian electrons using the extended PLK perturbation method is presented. The numerical values of the typical physical parameters of planetary rings [Reference Bandyopadhyay, Prasad, Sen and Kaw20, Reference Singh, Ghai, Kaur and Saini24], $n_{i_0}=5\times {10}^7{\text{cm}}^{-3},n_{e_0}=4\times {10}^7{\text{cm}}^{-3},Z_d=3\times {10}^3,n_d={10}^7{\text{cm}}^{-3},T_e=50\text{eV},T_i=0.05\text{eV},\text{and}R=0-0.14$ , are used for analysis. The shock waves are formed as a result of balance of nonlinearity, dispersion, and dissipation. The KdVB equations have been derived in equations (24) and (25). Numerically, it is seen that the nonlinear coefficient A (equation (26)) is negative (i.e., A < 0) and the dispersion coefficient B (equation (27)) is positive (i.e., B > 0) for the chosen set of parameters. It implies that AB < 0; as a result, negative potential shock structures are formed. Since the various physical parameters such as nonextensivity of ions, polarization force, obliqueness, magnetic field strength, and other plasma parameters have a profound influence on the characteristics of DA shock waves, it is of paramount importance to carry out numerical analysis.

4.1. Influence of Nonextensivity of Ions (via q) on the Polarization Force

Figure 1 depicts the variation of the polarization force parameter R with nonextensive parameter (q). Since the parameter R is a function of nonextensivity of ions (via q), any variation in q will change the value of R. It is observed that as the nonextensive parameter q is increased, the polarization force parameter R increases. Thus, nonextensivity of ions has a profound influence on the polarization force parameter R (i.e., polarization force).

Figure 1: The variation of polarization force parameter (R) with nonextensivity of ions (via q).

4.2. The Variation of Profile of Negative Polarity DA Shock Waves

Figure 2(a) depicts the variation of the amplitude of negative polarity DA shock waves in R − q plane. It is found that the amplitude of negative polarity DA shock waves is increased with rise in both polarization force parameter (via R) and nonextensive parameter (via q). Figure 2(b) shows that the amplitude of negative polarity shock waves is also enhanced with an increase in both magnetic field strength (via Ω) and obliqueness (via l_z). It is remarked that the amplitude of DA shock waves is modified with the variation in plasma parameters due to the change in nonlinear as well as dispersive effects under the influence of polarization force in the presence of dissipation effect.

Figure 2: The variation of rarefactive DA shocks profile with respect to ξ (a) for different values of polarization parameter (R) and nonextensivity parameter (q) and (b) for different values of viscosity ${\eta }_{l_0}$ and magnetic field strength whereas other parameters ρ = 1.3, l_z = 0.8, σ = 0.25, Ω = 0.8, and η_l = 0.35 are fixed.

4.3. Evolution and Phase Shift of Two Negative Polarity DA Shock Waves

The collision process of DA shock waves in a magnetized dusty plasma for different time intervals is depicted in Figure 3. The mesial line indicated in the figure separates the before and after collision process. The upper red curves illustrate the profiles of DA shocks before collision and lower blue curves for after collision. This whole process illustrates that two opposite directional propagating negative polarity DA shock waves come closer, interact, and then depart (for more clarity, see animation of collision in the supplementary section (available here)). Figures 4(a) and 4(b) depict the variation of phase shift occurred during the collision process between two negative polarity DA shock waves for different values of dust fluid viscosity ( ${\eta }_{l_0}$ ). It is recognised that the magnitude of phase shift escalates with rise in viscosity (clearly seen in color bar). Figures 5(a) and 5(b) depict the variation of phase shift occurred during the collision process between two negative polarity DA shock waves for different values of magnetic field strength (via Ω). It is found that the magnitude of phase shift enhances as the value of Ω increases (see side color bar). Figure 6(a) illustrates the variation of phase shift with nonextensive parameter (q) for different values of polarization parameter (R) in the presence of magnetic field. It is seen that magnitude of phase shift escalates with rise in nonextensive parameter (q) and suppresses with increase in polarization force (via R). Figure 6(b) shows the same variation in the absence of magnetic field. It is found that magnitude of phase shift is more (almost four times greater) than the case in the absence of magnetic field. It is remarked that magnetic field strength has a significant influence on the profile of DA shocks. It is concluded that the phase shift of DA shock waves is significantly affected under the influence of polarization force (via R), nonextensivity parameter (via q), strength of magnetic field (via Ω), and other plasma parameters.

Figure 3: The collision of two DA shock waves at different time intervals for q = 0.33, R = 0.1, ρ = 1.3, l_z = 0.8, σ = 0.25, Ω = 0.8, and ${\eta }_l=0.2$ . (a) τ = 50, (b) τ = 40, (c) τ = 30, (d) τ = 20, (e) τ = 0, (f) τ = −50, (g) τ = −40, (h) τ = −30, and (i) τ = −20.

Figure 4: Collision process of two DA shock waves for (a) ${\eta }_{l_0}=0.1$ and (b) ${\eta }_{l_0}=0.2$ and ρ = 1.3, l_z = 0.8, σ = 0.25, Ω = 0.8, R = 0.11, and q = 0.33 are fixed.

Figure 5: Collision process of two DA shock waves for (a) Ω = 0.2 and (b) Ω = 0.3 and ρ = 1.3, l_z = 0.8, σ = 0.25, R = 0.11, ${\eta }_l=0.2$ , and q = 0.3 are fixed.

Figure 6: The variation of phase shift of two DA shock waves vs. q for different values of R (a) with magnetic field and (b) without magnetic field and ρ = 1.3, l_z = 0.8, σ = 0.25, and ${\eta }_l=0.2$ are fixed.