Two-dimensional dynamics of ion-acoustic waves in a magnetised electronegative plasma

Abstract

We consider a plasma model made of negative ions and electrons in Boltzmann distribution and cold mobile positive ions, under the influence of an external magnetic field, to study the propagation of ion-acoustic waves (IAWs). From the hydrodynamic equations of the two-dimensional (2D) electronegative plasma, the dynamics of the system is reduced via the reductive perturbation method to the Davey–Stewartson (DS) equations. Using the linear stability analysis of planar waves, an expression of the modulational instability growth rate is derived, and its response to system parameters, such as the negative ion concentration ratio, the electron-to-negative ion temperature ratio and the magnetic field is discussed. It comes out that the instability occurs for high values of the negative ion concentration ratio and in a very restrained area of small magnetic field values. The growth rate is maximum for high values of both the electron-to-negative ion temperature ratio and the magnetic field. The existence of IAWs in the model is confirmed. One-dromion and two-dromion solutions are investigated analytically using the Hirota’s bilinear method and numerically. The impact of the magnetic field is discussed in that context, and particular attention is given to dromion interactions under different scenarios. We noted a slowing effect on the propagation of the dromion solutions in the presence of the magnetic field. Elastic collision of two-dromion solutions is observed during their evolution, with an acceleration of the process in the presence of the magnetic field.

Appendices

Appendix A: Various coefficients

The coefficients appearing in various equations are defined as follows:

$$\begin{aligned}{} & {} \beta _1=\omega ^2 (k^2+a_1) -k_z^2,\nonumber \\{} & {} \beta _2=\omega ^2 (4k^2+a_1) -k_z^2, \nonumber \\{} & {} u_{x\phi }= \frac{\beta _1 }{k_x \omega },\nonumber \\{} & {} u_{z\phi }= \frac{k_z }{\omega },\nonumber \\{} & {} u_{y\phi }= -\frac{i\beta _1 \Omega }{k_x \omega ^2 }, \nonumber \\{} & {} u_{x\xi }=- \frac{2\omega }{ k^2+a_1}+ \frac{\beta _1}{k_x^2 \omega (k^2+a_1)} -\frac{\beta _2 v_{gx}}{k_x\omega ^2 (k^2+a_1)}, \nonumber \\{} & {} u_{x\eta }= - \frac{2k_z \omega }{ k_x (k^2+a_1)}+ \frac{2k_z}{k_x \omega (k^2+a_1)}\nonumber \\ {}{} & {} \qquad \quad -\frac{\beta _2 v_{gz}}{k_x\omega ^2 (k^2+a_1)}, \nonumber \\{} & {} u_{y\xi }=- \frac{2\Omega }{ k^2+a_1}+ \frac{\beta _1 \Omega }{k_x^2 \omega ^2(k^2+a_1)} -\frac{2k_z^2 \Omega v_{gx}}{k_x\omega ^3 (k^2+a_1)},\nonumber \\{} & {} u_{z\xi }=\frac{k_z v_{gx} }{\omega ^2 (k^2+a_1)}, \nonumber \\ {}{} & {} u_{z\eta }= - \frac{1}{\omega (k^2+a_1)} \left( v_{gz} \frac{k_z}{\omega }-1\right) , \end{aligned}$$

(A.1)

$$\begin{aligned}{} & {} {\alpha _\phi } = \frac{\left\{ \begin{aligned}&(4\omega ^2-\Omega ^2)(2({k^2} + {a_1})^2 \omega ^2 +k_z^2({k^2} + {a_1})\\&-2{a_2 \omega ^2})+2\beta _1 ({k^2} + {a_1})\left( 2{\omega ^2} + {\Omega ^2} \right) \end{aligned} \right\} }{2(k^2 +a_1)^2\left( \beta _2(4\omega ^2 -\Omega ^2) -4k_x^2 \omega ^2\right) },\nonumber \\{} & {} {\alpha _n} = ( {4{k^2} + {a_1}} ){\alpha _\phi } + \frac{{{a_2}}}{{{{({k^2} + {a_1})}^2}}}, \nonumber \\{} & {} {\alpha _{{u_x}}} = \frac{\omega }{k_x} ({\alpha _n} -1) - \frac{k_z }{k_x}\alpha _{{u_z}}, \nonumber \\{} & {} {\alpha _{{u_y}}}= - \frac{\Omega }{2\omega }\left( {\alpha _{{u_x}}} - \frac{\beta _1}{{k_x \omega ({k^2} + {a_1}) }}\right) ,\nonumber \\{} & {} {\alpha _{{u_z}}} = \frac{k_z }{\omega } {\alpha _\phi } +\frac{k_z }{2 \omega ({k^2} + {a_1})}. \end{aligned}$$

(A.2)

Appendix B: Equations and other coefficients

The third-order ($\varepsilon ^3$) reduced equations for $l=0$ are obtained as set of couple equations

$$\begin{aligned}{} & {} - v_{gx} \frac{{\partial \phi _0^{(2)} }}{{\partial \xi }} - v_{gz} \frac{{\partial \phi _0^{(2)} }}{{\partial \eta }} + \frac{{\partial u_{z0}^{(2)} }}{{\partial \eta }} \nonumber \\{} & {} \quad = A_{\xi }\frac{{\partial | {n_1^{(1)} } |^2 }}{{\partial \xi }}+ A_{\eta }\frac{{\partial | {n_1^{(1)} } |^2 }}{{\partial \eta }},\nonumber \\{} & {} - v_{gx} \frac{{\partial u_{z0}^{(2)} }}{{\partial \xi }} - v_{gz} \frac{{\partial u_{z0}^{(2)} }}{{\partial \eta }} + \frac{{\partial \phi _0^{(2)} }}{{\partial \eta }}\nonumber \\{} & {} \quad = B_{\xi }\frac{{\partial | {n_1^{(1)} } |^2 }}{{\partial \xi }} + B_{\eta }\frac{{\partial | {n_1^{(1)} } |^2 }}{{\partial \eta }}, \end{aligned}$$

(B.1)

where

$$\begin{aligned} A_{\xi }= & {} \frac{2v_{gx}a_2}{(k^2+a_1)^2},\\ A_{\eta }= & {} \frac{2v_{gz}a_2}{(k^2 +a_1)^2} -\frac{2k_z}{\omega (k^2+a_1)}, \\ B_{\xi }= & {} \frac{2}{(k^2+a_1)^2}\\{} & {} \times \left[ -\frac{k_z \beta _1}{k_x \omega ^2} + k_x k_z +\frac{k_z v_{gz}}{2 \omega ^3 (k^2+a_1)^2} (\beta _1+\beta _2)\right] , \end{aligned}$$

$$\begin{aligned} B_{\eta }= & {} \frac{1}{(k^2+a_1)^2}\left[ \frac{ \beta _1}{\omega ^2} -\frac{3k_z^2 }{ \omega ^2}+ 2 k_z^2 +\frac{k_z v_{gz}}{\omega ^3 } (\beta _2-\beta _1)\right] . \end{aligned}$$

The coefficients of eq. (9) are as follows:

$$\begin{aligned} \begin{aligned}&{\delta _1} = - {v_{gx}^2}{a_1}, \quad {\delta _{\xi \eta }} = - 2{v_{gx}} {v_{gz}}{a_1}, \quad {\delta _{\eta }} = 1 - {v_{gz}^2}{a_1}, \\&{\delta _2} = - v_{gx} A_{\xi },\quad {\delta _{23}} = -v_{gx} A_{\eta }- v_{gz}A_{\xi }-B_{\xi },\\&{\delta _3} =- v_{gz} A_{\eta }+B_{\eta }, \end{aligned} \end{aligned}$$

(B.2)

Appenidx C Coefficients of DSEs

The set of both eqs. (10) and (9)

$$\begin{aligned} \gamma {_{\tau }}= & {} \frac{2}{\omega }( \Omega ^4(k^2+a_1)+ k_z^2 \Omega ^2), \end{aligned}$$

(C.1)

$$\begin{aligned} \gamma {_1}= & {} \frac{1}{\gamma {_{\tau }}}\left[ \begin{array}{l} {v_{gx}} \left( \begin{array}{l} -2 k_x \omega (\omega ^2-\Omega ^2)+ k_x \omega ^2 (k^2 +a_1)u_{x\xi } + k_x \omega \Omega (k^2+a_1)u_{y\xi } \\ + k_z (\omega ^2-\Omega ^2) (k^2+a_1)u_{z\xi } \end{array} \right) \\ -\omega (\omega ^2-\Omega ^2)(k^2+a_1)u_{z\xi }-\omega ^2 (\omega ^2-\Omega ^2) \end{array}\right] , \end{aligned}$$

(C.2)

$$\begin{aligned} \gamma {_{12}}= & {} \frac{1}{\gamma {_{\tau }}}\left[ \begin{aligned}&-2 \omega (\omega ^2-\Omega ^2)(k^2+a_1) (k_z {v_{gx}} + k_x{v_{gz}})-\omega (\omega ^2-\Omega ^2)(k^2+a_1) (u_{x\eta }+u_{z\xi }) \\&+ k_x \omega ^2 (k^2+a_1) ({v_{gx}}u_{x\eta } + {v_{gz}}u_{x\xi })+ k_x \omega \Omega (k^2+a_1)({v_{gx}}u_{y\eta } + {v_{gz}}y_{x\xi })\\&+ k_z (\omega ^2-\Omega ^2)(k^2+a_1)({v_{gx}}u_{z\eta } + {v_{gz}}u_{z\xi })\end{aligned}\right] , \end{aligned}$$

(C.3)

$$\begin{aligned} \gamma {_2}= & {} \frac{1}{\gamma {_{\tau }}}\left[ \begin{array}{l} {v_{gz}}\left( \begin{array}{l}-2 k_z \omega (\omega ^2-\Omega ^2)+ k_x \omega ^2 (k^2+a_1)u_{x\eta } + k_x \omega \Omega (k^2+a_1)u_{y\eta } + k_z (\omega ^2-\Omega ^2)(k^2+a_1)u_{z\eta }\end{array}\right) \\ -\omega (\omega ^2-\Omega ^2)(k^2+a_1)u_{z\eta }-\omega ^2 (\omega ^2-\Omega ^2)\end{array}\right] ,\nonumber \\\end{aligned}$$

(C.4)

$$\begin{aligned} \gamma {_3}= & {} \frac{1}{\gamma {_{\tau }}}\left[ \begin{array}{l} - \omega (\omega ^2-\Omega ^2)\left( \begin{array}{l} (k^2+a_1) ( k_x{\alpha _{{u_x}}} +k_z \alpha _{{u_z}}) + \dfrac{1}{\omega }( \alpha _n(\beta _1+k_z^2)-2\beta _1)\\ + \dfrac{a_2}{\omega (k^2+a_1)}(2\beta _1+k_z^2) \end{array}\right) \\ + k_x \omega ^2\left( {\alpha _{{u_x}}} \bigg (\dfrac{2k_z^2 \Omega }{\omega } -\dfrac{\beta _1}{\omega }\bigg ) +\dfrac{\beta _1\alpha _{{u_z}}}{\omega } +\dfrac{2\beta _{1}^2}{k_x\omega ^2(k^2+a_1)}\right) \\ + k_x \omega \Omega \left( \begin{aligned}&{} -2{\alpha _{{u_y}}} \bigg (\frac{\beta _1 \Omega }{\omega } +\frac{k_z^2}{\omega }\bigg )+\frac{\beta _1\Omega \alpha _{{u_x}}}{\omega ^2} +\frac{2\Omega \beta _{1}^2}{k_x\omega ^3(k^2+a_1)} +\frac{k_z \beta _1\Omega \alpha _{{u_z}}}{k_x \omega ^2}\end{aligned}\right) \\ + k_z (\omega ^2-\Omega ^2)\left( \begin{aligned}&{} -{\alpha _{{u_z}}} \bigg (\frac{2\beta _1}{\omega }+ \frac{k_z^2}{\omega }\bigg )+\frac{2k_z\beta _{1}}{\omega ^2(k^2+a_1)} +\frac{k_x k_z\alpha _{{u_x}}}{\omega }\end{aligned}\right) \\ + \omega ^2 (\omega ^2-\Omega ^2)\left( {2a_2 \alpha _\phi }- \dfrac{3a_3}{(k^2+a_1)^2} \right) \end{array}\right] ,\end{aligned}$$

(C.5)

$$\begin{aligned} \gamma {_4}= & {} -\frac{(\omega ^2-\Omega ^2)}{\gamma {_{\tau }}} (2k_z \omega {v_{gz}}-1 +2a_2\omega ^2). \end{aligned}$$

(C.6)