Nonlinear Electromagnetic Waves in a Degenerate Electron-Positron Plasma

Abstract

Using the reductive perturbation technique (RPT), the nonlinear propagation of magnetosonic solitary waves in an ultracold, degenerate (extremely dense) electron-positron (EP) plasma (containing ultracold, degenerate electron, and positron fluids) is investigated. The set of basic equations is reduced to a Korteweg-de Vries (KdV) equation for the lowest-order perturbed magnetic field and to a KdV type equation for the higher-order perturbed magnetic field. The solutions of these evolution equations are obtained. For better accuracy and searching on new features, the new solutions are analyzed numerically based on compact objects (white dwarf) parameters. It is found that including the higher-order corrections results as a reduction (increment) of the fast (slow) electromagnetic wave amplitude but the wave width is increased in both cases. The ranges where the RPT can describe adequately the total magnetic field including different conditions are discussed.

Appendix

The expressions of the coefficients appeared in Eqs. 14–16 are

$$\begin{array}{@{}rcl@{}} E & =-\frac{2D\lambda\sin\theta}{{\alpha_{2}^{2}}}\text{, }F=\frac{1} {2\alpha_{2}}-\frac{A\lambda\sin\theta}{{\alpha_{2}^{2}}}+\frac{\sin^{2}\theta }{2{\alpha_{2}^{3}}}\left( 3\lambda^{2}-\alpha_{2}-\frac{\beta}{3}\right) ,\\ G & =E\lambda+\frac{D\sin\theta}{\alpha_{2}}\text{, }H=F\lambda -\frac{\lambda\sin^{2}\theta}{{\alpha_{2}^{2}}}+\frac{A\sin\theta}{2\alpha_{2} }\text{,}\\ I & =\cos\theta\left[ \frac{D}{\lambda^{2}}-\frac{\upmu\lambda\left( 1-\upmu\right) }{\alpha_{2}}\right] \text{.} \end{array} $$

The expressions of those presented in Eqs. 17 and 18 are

$$\begin{array}{@{}rcl@{}} J & =\frac{\left( \upmu-D_{\upmu}\right) D-L\cos\theta}{\lambda} ,~K=\frac{\left[ \left( \upmu-D_{\upmu}\right) A-M\cos\theta\right] }{\lambda }-\frac{\upmu\sin\theta}{\alpha_{2}},\\ L & =\frac{\eta}{\lambda}\left[ \frac{\upmu\lambda^{3}}{\alpha_{2}} -\frac{D\nu_{2}}{\nu_{1}}\right] ,~M=-\frac{\eta}{\nu_{1}}\left[ \frac {A\nu_{2}}{\lambda}-\frac{\sin\theta\cos^{2}\theta}{\alpha_{2}}\right] . \end{array} $$

Finally, the coefficients appeared within the source term, (24), are

$$\begin{array}{@{}rcl@{}} R_{1} &=&-\frac{1}{H_{1}}\left[ \left( \sin^{2}\theta/\alpha_{2}\right) \left( \cos^{2}\theta+\lambda^{2}\left\{ -1+\left( \sin^{2}\theta /\alpha_{2}\right)\right.\right.\right.\\ &&\left.\left.\left.+\left[ 2\sin^{2}\theta/\left( 9\beta\alpha_{2}^{2}\right) \right]\right\} \right) -\left( A/\lambda\right) \sin\theta\cos^{2}\theta\right. \\ && + F\left\{ \beta\left[ \cos^{2}\theta+2\left( \lambda^{2} /\alpha_{2}\right) \sin^{2}\theta\right]-\lambda^{2}\right\}\\ &&\left.+H\left[ \left( \alpha_{2}/\lambda\right) \left( 3\lambda^{2}+\cos^{2}\theta\right) +3\left(\alpha_{1/}\alpha_{2}\right) \lambda\sin^{2}\theta\right] \right],\\ R_{2} & =&-\frac{1}{H_{1}}\left\{3\alpha_{2}\lambda\left( K\lambda +M\upmu\cos\theta\right) -\lambda^{2}\sin\theta\left[ {\upmu}^{2}-\eta^{2}\right. \right. \\ &&\left.\left.+D_{\upmu}\left( 1-\upmu\right) \right] +E\left( \beta\left\{ \cos^{2}\theta+2\left[ \left( \lambda^{2}\sin^{2}\theta\right) /\left(3\alpha_{2}\right) \right] \right\}\right.\right.\\&&\left.\left. -\lambda^{2}\right) \right\} ,\\ R_{3} & =&-\frac{1}{H_{1}}\left\{\alpha_{2}\lambda\left( \lambda K+M\upmu\cos\theta\right) -\lambda^{2}\sin\theta\left[ 2{\upmu}^{2}+D_{\upmu}\left(1-2\upmu\right) \right]\right.\\ &&\left. +G\lambda\left[ \alpha_{2}+\left( \alpha_{1}/\alpha_{2}\right) \sin^{2}\theta\right] \right. \\ && \left.+ \left[ 2/\left( 3\alpha_{2}\right) \right] E\beta\lambda^{2}\sin^{2}\theta-\lambda I\sin\theta\cos\theta\right\} ,\\ R_{4} & =&-\frac{\alpha_{2}\lambda}{H_{1}}\left( \lambda J+\upmu L\cos \theta\right), \end{array} $$