Effect of adiabatically trapped-suprathermal electrons on ion-acoustic solitons in electron-ion plasma

Abstract

In this paper, the problem of nonlinear ion-acoustic (IA) solitary waves in an electron-ion plasma is analyzed, assuming electrons obey the $\kappa$-Gurevich distribution. The density of adiabatically trapped suprathermal electrons is derived from a physically relevant distribution describing such electrons. As an application, the modified Korteweg-de Vries (mK-dV) equation considering the $\kappa$-Gurevich electron density is derived. The study has revealed that the main properties (phase velocity, amplitude, and width) of small IA waves are significantly influenced by trapped suprathermal electrons. We have found that as electron suprathermality increases in plasma (i.e., as electrons move far away from their Maxwellian trapping), both amplitude and width of IA soliton decreases. Our study revealed that the IA soliton energy decreases when electrons move far from their Maxwellian trapping. Studying solitary ion acoustic waves may allow us to gain a deeper understanding of space where fast superthermal electrons are present along with ions (e.g. Earth’s auroral region, Jupiter magnetosphere).

Appendix

According to Misra and Wang (2015), equation Eq. (38) can be expressed as fellow,

$$ \frac{\partial \Phi}{\partial \tau}+iA\sqrt{\Phi} \frac{\partial \Phi}{\partial \xi}+B \frac{\partial ^{3}\Phi}{\partial \xi ^{3}}=0. $$

(46)

Applying the transformation $\zeta =\xi -U\tau$ in Eq. (46), we get

$$ \frac{d}{d\zeta}\left (B \ddot{\Phi}-U\Phi +i\frac{2}{3}A\Phi ^{3/2} \right )=0, $$

(47)

where $U$ is the solitary wave velocity and the dot denotes differentiation with respect to $\zeta$.

Integrating Eq. (47) and using the boundary conditions $\Phi ,~\ddot{\Phi}\rightarrow 0$ as $\xi \rightarrow \pm \infty$ we get

$$ B \ddot{\Phi}-U\Phi +i\frac{2}{3}A\Phi ^{3/2}=0. $$

(48)

Multiplying Eq. (48) by $2\dot{\Phi}$ and integrating with respect to $\zeta$, we obtain

$$ B \dot{\Phi}^{2}-U\Phi ^{2}+i\frac{8}{15}A\Phi ^{5/2}=0, $$

(49)

where we have used the boundary conditions $\Phi ,\ \dot{\Phi}\rightarrow 0$. From Eq. (49) we have

$$ \dot{\Phi}=\pm \Phi \sqrt{\frac{U}{B}-i\frac{8A}{15B}\sqrt{\Phi}}, $$

(50)

$$ \mbox{or},\ \int \frac{d\Phi}{\Phi \sqrt{{U}/{B}-i\left ({8A}/{15B}\right )\sqrt{\Phi}}}= \pm \int d\zeta , $$

(51)

which gives ($a=U/B$ and $b=i8A/15B$)

$$ -\frac{4}{\sqrt{a}}\tanh ^{-1}\sqrt{\frac{a-b\sqrt{\Phi}}{a}}=\pm \zeta , $$

(52)

$$ \sqrt{\frac{a-b\sqrt{\Phi}}{a}}=\mp \tanh \left (\frac{\sqrt{a}}{4} \zeta \right ). $$

(53)

Thus, we obtain the soliton solution as

$$ 1-\tanh ^{2}\left (\frac{\sqrt{a}}{4}\zeta \right )= \frac{b}{a}\sqrt{ \Phi}, $$

(54)

$$ \mbox{or},\ \Phi =\left (\frac{15U}{8A}\right )^{2}\text{sech}^{4}\left ( \sqrt{\frac{U}{16B}}\zeta \right ). $$

(55)