Electron acoustic rogue waves in Earth’s magnetosphere

Abstract

The characteristics of electron acoustic (EA) rogue waves (RWs), triplet rogue waves (TRWs) and super rogue waves (SRWs) in a multicomponent beam plasma composed of cold electrons fluid, inertialess superthermal hot electrons and stationary ions in background embedded with an electron beam have been examined. The multiple scale perturbation method is employed to derive the nonlinear Schrödinger equation (NLSE) to analyse the EA rogue waves. From the solutions of NLSE the properties of different types of EA RWs have been analysed. The characteristics of EA RWs are strongly influenced by the electron beam and other plasma parameters. The findings of this work may be useful to examine the formation of EA nonlinear structures in the space and astrophysical plasma environments especially in Earth’s magnetospheres.

Appendix

$$\begin{aligned} T_{1}&=\omega ^{2}j^{2}-\omega ^{2}n_{\mathrm{bc}}k C_{1}+ \omega ^{2}n_{\mathrm{bc}}j C_{2}-k j^{2}C_{3}-\omega j^{2}C_{4},\\ T_{2}&=\frac{2\omega ^{2} n_{\mathrm{bc}} k^{2}}{j}-\frac{k^{2}j^{2}}{\omega }-\frac{\omega j^{2}k^{2}}{\omega ^{2}},\\ T_{3}&=3\omega ^{2}j^{2} n_{\mathrm{hc}}m_{3}-\omega ^{2} n_{\mathrm{bc}}k C_{5}+\omega ^{2}n_{\mathrm{bc}}j C_{6}-kj^{2}C_{7}, \end{aligned}$$

here

$$\begin{aligned} C_{1}&=\frac{(V_{\mathrm{g}}-V_{\mathrm{b}})(k V_{\mathrm{g}}-\omega )}{j^{2}},\\ C_{2}&=\frac{(V_{\mathrm{g}}-V_{\mathrm{b}})(2k\omega -2 k^{2} V_{\mathrm{g}})}{j^{2}}-\frac{(kV_{\mathrm{g}}-\omega )}{j^{2}},\\ C_{3}&=\frac{-V_{\mathrm{g}}(-k V_{\mathrm{g}}+\omega )}{\omega ^{2}},\\ C_{4}&=\frac{2 k^{2} V_{\mathrm{g}}^{2}-2k\omega V_{\mathrm{g}}}{\omega ^{3}}+\frac{(-kV_{\mathrm{g}}+\omega )}{\omega ^{2}},\\ C_{5}&=\frac{3k^{5}}{2 j^{4}}-\frac{3k^3P_{1}}{j^2 P_{2}}-\frac{k^4}{2 j^3 (V_{\mathrm{g}}-V_{\mathrm{b}})}+\frac{k^2 P_{3}}{P_{4} j(V_{\mathrm{g}}-V_{\mathrm{b}})},\\ C_{6}&=\frac{5k^5}{2j^4 (V_{\mathrm{g}}-V_{\mathrm{b}})}-\frac{k^3 P_{3}}{P_{4}j^2 (V_{\mathrm{g}}-V_{\mathrm{b}})}\\&\quad -\frac{4k^6}{2j^5}+\frac{2k^4 P_{1}}{j^3P_{2}}-\frac{k^4}{2j^3(V_{\mathrm{g}}-V_{\mathrm{b}})^{2}}+\frac{k^2 P_{3}}{P_{4}(V_{\mathrm{g}}-V_{\mathrm{b}})^{2}}j,\\ C_{7}&=\frac{3k^{5}}{2 j^{4}}-\frac{3k^3P_{1}}{\omega ^2 P_{2}}-\frac{k^4}{2 \omega ^3 V_{\mathrm{g}}}-\frac{k^2 P_{3}}{P_{4} \omega V_{\mathrm{g}}},\\ C_{8}&=\frac{5k^5}{2\omega ^4 V_{\mathrm{g}}}-\frac{k^3 P_{3}}{P_{4}\omega ^2 V_{\mathrm{g}}}+\frac{4k^6}{2\omega ^5}-\frac{2k^4 P_{1}}{\omega ^3 P_{2}}\\&\quad +\frac{k^4}{2\omega ^3 V_{\mathrm{g}}^{2}}-\frac{k^2 P_{3}}{P_{4} V_{\mathrm{g}}^{2}\omega },\\ P_{1}&=\frac{3k^4}{2\omega ^4}+\frac{3 n_{\mathrm{bc}}k^4}{2 j^4}+n_{\mathrm{hc}}m_{2},\\ P_{2}&=\frac{k^2}{\omega ^2}+\frac{n_{\mathrm{bc}}k^2}{j^2}-4k^2+n_{\mathrm{hc}}m_{1},\\ P_{3}&=\frac{k^2}{2\omega ^2 V_{\mathrm{g}}^2}+\frac{2k^3}{\omega ^3 V_{\mathrm{g}}}\\&\quad +n_{\mathrm{bc}}\left( \frac{k^2}{2j^2(V_{\mathrm{g}}-V_{\mathrm{b}})^{2}}-\frac{2k^3}{j^3(V_{\mathrm{g}}-V_{\mathrm{b}})}+2 n_{\mathrm{hc}}m_{2}\right) ,\\ P_{4}&=\frac{1}{V_{\mathrm{g}}^{2}}+\frac{n_{\mathrm{bc}}}{(V_{\mathrm{g}}-V_{\mathrm{b}})^{2}}+n_{\mathrm{hc}}m_{1}, \end{aligned}$$

where

$$\begin{aligned} j&=(-\omega +V_{\mathrm{b}}k)^{2}, \quad m_{1}=\frac{-\kappa +\frac{1}{2}}{\kappa -\frac{3}{2}},\\ m_{2}&=\frac{(-\kappa +\frac{1}{2})(\kappa +\frac{1}{2})}{2(\kappa -\frac{3}{2})^{2}},\,m_{3}=\frac{(-\kappa +\frac{1}{2})(\kappa +\frac{1}{2})(\kappa +\frac{3}{2})}{6(\kappa -\frac{3}{2})^{3}}. \end{aligned}$$