Dust acoustic kinetic Alfvén wave solitons and periodic waves in a polarized dusty plasma

Abstract

This investigation presents the influence of polarization force on dust acoustic kinetic Alfvén solitary waves (DAKASWs) and phase portrait analysis in a magnetized dusty plasma composed of dust and ions as well as electrons obeying Vasyliunas–Cairns distribution. Two KdV equations with polarization force in the fluid model equations have been derived by employing extended Poincaré–Lighthill–Kuo method. Multi-soliton solution are also determined by using Hirota Bilinear method. Only negative potential (rarefactive) DAKASWs are observed. Analysis of variation in the polarization force, propagation angle, and plasma beta on the characteristics of DAKASWs and head-on collision of multi-solitons has been carried out. Galilean transformation is used to transform the KdV equation into planar dynamical systems. The dynamical system has been described in the form of phase portrait and small-amplitude Sagdeev’s pseudopotential curve. Further, under the influence of different plasma parameters, periodic waves in homoclinic and periodic orbits in phase portraits are investigated. The findings of this investigation may be useful to understand the insight of physics of nonlinear phenomena for studying dynamics of DAKASWs in Jupiter’s magnetosphere as well as astrophysical plasma environments.

Appendix A

$$\begin{aligned}&a_1 T_1 \psi _1 +X_1 v_{dx1}+ Z_1 v_{dz1} =0 \end{aligned}$$

(42)

$$\begin{aligned}&v_{dx1}=\beta _d X_1 T_1 \phi _1 \end{aligned}$$

(43)

$$\begin{aligned}&T_1 v_{dz1}= (1-\mathbb {R}H_1) \beta _d Z_1 \psi _1 \end{aligned}$$

(44)

$$\begin{aligned}&{X_1}^2 {Z_1}^2 \phi _1=-\frac{1}{\beta _d}\left( {T_1}^2 a_1 \psi _1+ T_1 Z_1 v_{dz1}\right) \end{aligned}$$

(45)

Operating $T_1$ on Eq. (42) and using Eqs. (43) and (44), we get,

$$\begin{aligned} a_1 {T_1}^2 \psi _1 + T_1 Z_1 v_{dz1}=0 \end{aligned}$$

(46)

$$\begin{aligned}&{X_1}^2 {T_1}^2 \phi _1=-\frac{1}{\beta _d}\left( a_1 {T_1}^2 \psi _1+ (1-\mathbb {R}H_1)\beta _d {Z_1}^2\psi _1 \right) \end{aligned}$$

(47)

Putting in Eq. (45), we get

$$\begin{aligned} {X_1}^2 {Z_1}^2 \phi _1= -\frac{1}{\beta _d}\left( {T_1}^2 a_1 \psi _1+(1- \mathbb {R}H_1)\beta _d {Z_1}^2 \psi _1 \right) \end{aligned}$$

(48)

Multiplying Eq. (47) with ${Z_1}^2$ and Eq. (47) with ${T_1}^2$ and adding we get,

$$\begin{aligned} \left( a_1 {Z_1}^2 {T_1}^2 +\beta _d {Z_1}^4\right) \psi _1= \left( a_1 {T_1}^4 + (1- \mathbb {R}H_1)\beta _d {T_1}^2 {Z_1}^2 \right) \psi _1 \end{aligned}$$

(49)

substituting Eq. (26) in Eq. (44), we obtain

$$\begin{aligned} v_{\mathrm{d}z1}= -(1-\mathbb {R}H_1) \beta _d \left( \psi _{1\xi }-\psi _{1\eta } \right) . \end{aligned}$$

(50)

By using Eqs. (42) and (50), the value of $v_{\mathrm{d}x1}$ can be given as

$$\begin{aligned} v_{\mathrm{d}x1}= \frac{1}{l_x}\left( \lambda a_1+\frac{\beta _d (1-\mathbb {R}H_1) {l_z}^2}{\lambda }\right) (\psi _{1\xi }-\psi _{1\eta }) \end{aligned}$$

(51)

also, substituting Eq. (26) in the lowest order of Eq. (13),

$$\begin{aligned} n_{d1}= a_1 (\psi _{1\xi }+\psi _{1\eta }) \end{aligned}$$

(52)

Lastly, substituting Eq. (26) in Eq. (48) we get,

$$\begin{aligned} \left( \partial _{ \xi }+\partial _{\eta }\right) ^2 \phi _1= - \frac{1}{\beta _d {l_x}^2} \left( a_1 +(1-\mathbb {R}H_1)\beta _d\right) (\psi _{1\xi }+\psi _{1\eta }) \end{aligned}$$

(53)

Higher-order equations are obtained as follows:

$$\begin{aligned}&T_1 (a_1 \psi _2 + a_2 {\psi _1}^2) + a_1 \frac{\partial \psi _1}{\partial \tau }+ T_2 a_1 \psi _1 + X_1 a_1 \psi _1 v_{\mathrm{d}x1} + X_1 v_{\mathrm{d}x2} \nonumber \\&\quad + Z_1 v_{\mathrm{d}z2} + X_1 a_1 v_{\mathrm{d}z1} \psi _1 + X_2 v_{\mathrm{d}x1}+ Z_2 v_{\mathrm{d}z1} =0 \end{aligned}$$

(54)

$$\begin{aligned}&v_{\mathrm{d}x2}= \beta _d (X_1 T_1 \phi _2+X_1 T_2 \phi _1 +X_2 T_1 \phi _1 +X_1 \frac{\partial \phi _1}{\partial \tau })=0 \end{aligned}$$

(55)

$$\begin{aligned}& T_1 v_{\mathrm{d}z2}+v_{\mathrm{d}x1}X_1 v_{\mathrm{d}z1}+v_{\mathrm{d}z1}Z_1 v_{\mathrm{d}z1} +\frac{\partial v_{\mathrm{d}z1}}{\partial \tau }+T_2 v_{\mathrm{d}z1}= \beta _d (1-\mathbb {R}H_1) Z_1 \psi _2\\&\quad + \beta _d Z_2 \psi + \beta _d \mathbb {R}(2 H_2 \psi _1-\frac{{H_1}^2 \psi }{2})Z_1 \psi _1 \end{aligned}$$

(56)

$$ \begin{aligned} (X_{1} )^{2} (Z_{1} )^{2} (\phi _{2} - \psi _{1} ) & = - \frac{1}{{\beta _{d} }}\left[ {T_{1} ^{2} (a_{1} \psi _{2} + a_{2} \psi _{1} ^{2} ) + Z_{1} T_{2} v_{{{\text{d}}z1}} + \frac{{\partial v_{{{\text{d}}z1}} }}{{\partial \tau }} + Z_{2} T_{1} v_{{{\text{d}}z1}} } \right. \\ & \quad \left. { + 2\lambda \frac{{\partial \psi _{1} a_{1} }}{{\partial \tau }} + 2T_{1} T_{2} \psi _{1} a_{1} + Z_{1} T_{1} v_{{{\text{d}}z2}} + T_{1} Z_{1} a_{1} \psi _{1} v_{{{\text{d}}z1}} } \right] \\ \end{aligned} $$

(57)

The unknown coefficients appearing in Eqs. (31)–(36) are illustrated as follows:

In Eqs. (31) and (32),

$$\begin{aligned}&A_1=1+ \exp {(k_1 Q^{-1/3}\xi -{k_1}^3 \tau )},\\&A_2=1+ \exp {(k_1 Q^{-1/3}\xi -{k_1}^3 \tau )}. \end{aligned}$$

In Eqs. (33) and (34),

$$\begin{aligned}&A_3=1+\exp {{\Omega _1}+\exp {\Omega _2}+U_{12}\exp {(\Omega _1+\Omega _2)}},\\&A_4=1+\exp {\varsigma _1} + \exp {\varsigma _2} + U_{12}\exp {(\varsigma _1+\varsigma _2)}. \end{aligned}$$

where $\Omega _i= k_i Q^{-1/3}\xi -{k_i}^3 \tau$, $\varsigma _i= k_i Q^{-1/3}\eta -{k_i}^3 \tau$ with $i=1$ and 2 and $U_{12}=(k_2-k_1)^2/(k_1+k_2)^2$.

In Eqs. (35) and (36),

$$\begin{aligned} A_5\,=\, & {} 1+ \exp {\Omega _1}+\exp {\Omega _2}+ \exp {\Omega _3} + U_{12}\exp {(\Omega _1+\Omega _2)}+U_{23}\exp {(\Omega _2+\Omega _3)} \\&+ U_{13}\exp {(\Omega _1+\Omega _3)}+U_{123}\exp {(\Omega _1+\Omega _2+\Omega _3),}\\ A_6\,=\, & {} 1+\exp {\varsigma _1}+\exp {\varsigma _2}+ \exp {\varsigma _3} + U_{12}\exp {(\varsigma _1+\varsigma _2)}+ U_{23}\exp {(\varsigma _2+\varsigma _3)} \\&+ U_{13}\exp {(\varsigma _1+\varsigma _3)}+ U_{123}\exp {(\varsigma _1+\varsigma _2+\varsigma _3)}. \end{aligned}$$

where $\Omega _i= k_i Q^{-1/3}\xi -{k_i}^3 \tau$, $\varsigma _i= k_i Q^{-1/3}\eta -{k_i}^3 \tau$ with $i=1$, 2, 3 and $U_{12}=(k_2-k_1)^2/(k_1+k_2)^2,$ $U_{23}=(k_2-k_3)^2/(k_2+k_3)^2,$ $U_{13}=(k_1-k_3)^2/(k_1+k_3)^2$and $U_{123}=U_{12}U_{23}U_{13}$.