Homotopy perturbation method for modeling electrostatic structures in collisional plasmas

Abstract

In this work, the extended Poincaré–Lighthill–Kuo (PLK) technique is devoted to reduce the fluid govern equations of complex plasma species to two coupled evolution equations (including the damping Korteweg–de Vries (dKdV) equation and damping modified Korteweg–de Vries (dmKdV) equation) for studying the dissipative soliton collisions in a collisional complex unmagnetized plasma. To find the phase shifts of both the dKdV and dmKdV solitons after collisions, the solutions of the evolution equations must be determined precisely. For this purpose, the homotopy perturbation method is introduced to find high-accuracy approximate numerical solutions to both the dKdV and dmKdV equations after few iterations (approximations). Moreover, and for the first time, we will find some approximate analytic solutions for the evolution equations in order to investigate the characteristics of the dissipative solitons after collisions. Furthermore, the comparison between the approximate numerical and analytical dissipative solutions of both the dKdV and dmKdV equations will be checked. Thereafter, a set of analytical expressions for the phase shifts of the normal and dissipative soliton collisions will be estimated. The effects of plasma configurational parameters on the phase shifts of colliding solitons will be examined.

Appendix: The values of \(X_{j}\), \(Y_{j}\), and \(Z_{j}\) of the system (7)

$$\begin{aligned} \left( \begin{array}{c} X_{j} \end{array} \right)&=\left( \begin{array}{c} 0,-{\widehat{x}}u^{(1)}n^{(1)}\text {,}-\widehat{{\dot{T}}}n^{(1)}-\widehat{x}\left( u^{(2)}n^{(1)}+u^{(1)}n^{(2)}\right) -{\widetilde{x}}u^{(1)},\\ -\widehat{{\dot{T}}}n^{(2)}-{\widehat{x}}\left( u^{(1)}n^{(3)}+u^{(2)} n^{(2)}+u^{(3)}n^{(1)}\right) -{\widetilde{x}}\left( u^{(2)}+u^{(1)} n^{(1)}\right) \end{array} \right) ,\\ \left( \begin{array} [c]{c} Y_{j} \end{array} \right)&=\left( \begin{array} [c]{c} 0,-u^{(1)}{\widehat{x}}u^{(1)}\text {,}-\widehat{{\dot{T}}}u^{(1)}-\widehat{x}u^{(2)}u^{(1)}-{\widetilde{x}}\phi ^{(1)}+\nu u^{(1)},\\ -\widehat{{\dot{T}}}u^{(2)}-{\widehat{x}}u^{(3)}u^{(1)}-u^{(2)}{\widehat{x}} u^{(2)}-u^{(1)}{\widetilde{x}}u^{(1)}-{\widetilde{x}}\phi ^{(2)}+\nu u^{(2)} \end{array} \right) ,\\ \left( \begin{array} [c]{c} Z_{j} \end{array} \right)&=\left( \begin{array} [c]{c} 0,\gamma _{2}\phi ^{(1)2}\text {,}2\gamma _{2}\phi ^{(2)}\phi ^{(1)}+\gamma _{3} \phi ^{(1)3}-{\widehat{x}}^{2}\phi ^{(1)},\\ \gamma _{2}\left( \phi ^{(2)2}+2\phi ^{(1)}\phi ^{(3)}\right) +3\gamma _{3} \phi ^{(1)2}\phi ^{(2)}+\gamma _{4}\phi ^{(1)4}-{\widehat{x}}^{2}\phi ^{(2)} \end{array} \right) ,\text { } \end{aligned}$$

where

$$\begin{aligned} \gamma _{1}&=\mu _{e}\left( 1-\beta \right) +\mu _{i}\sigma ,\\ \gamma _{2}&=\frac{1}{2}\left( \mu _{e}-\mu _{i}\sigma ^{2}\right) ,\\ \gamma _{3}&=\frac{1}{6}\left[ \mu _{e}\left( 1+3\beta \right) +\mu _{i}\sigma ^{3}\right] . \end{aligned}$$