Electron-acoustic anti-kink, kink and periodic waves in a collisional superthermal plasma

Abstract

A qualitative analysis of the electron-acoustic wave is taken in a collisional plasma having two-temperature electrons with a fixed ion background, where hot electrons follow kappa distribution. The collision between stationary ions and cold electrons is considered. Using reduced perturbation technique, the Burgers equation for the plasma system is derived. Using traveling wave transformation, we obtain the dynamical system corresponding to the plasma system. Phase plane analysis is used in the dynamical system to study different kinds of wave features for the considered plasma system. Moreover periodic wave features and shock wave features are investigated in accordance to periodic orbits and heteroclinic orbits obtained in the phase portrait. Role of the superthermal parameter ($\kappa$), speed of the travelling wave (U) and $\alpha =n_{c_{0}}/n_{h_{0}}$ (where $n_{c_{0}}$ denotes the number density of cold electrons in equilibrium and $n_{h_{0}}$ denotes the number density of hot electrons in equilibrium) are shown on the electron-acoustic periodic waves and shock waves structures. The results hold relevance and significance in the context of space plasma.

Appendix

Here we have shown some steps for the derivation of the dynamical system (22) from the Burgers Eq. (20).

From equation (21) we get

$$\begin{aligned}{} & {} \frac{\partial \psi }{\partial \tau }= \frac{\partial \psi }{\partial \eta }. \frac{\partial \eta }{\partial \tau }=-U\frac{d\psi }{d \eta },\end{aligned}$$

(26)

$$\begin{aligned}{} & {} \frac{\partial \psi }{\partial \xi } = \frac{\partial \psi }{\partial \eta }. \frac{\partial \eta }{\partial \xi }=\frac{d\psi }{d \eta },\end{aligned}$$

(27)

$$\begin{aligned}{} & {} \quad \text {and}\quad \frac{\partial ^2\psi }{\partial \xi ^2} = \frac{d^2\psi }{d \eta ^2}. \end{aligned}$$

(28)

Now using Eqs. (26)-(28) in the Burgers Eq. (20), we get

$$\begin{aligned} -U\frac{d\psi }{d \eta }+A\psi \frac{d\psi }{d \eta }-B\frac{d^2\psi }{d \eta ^2}=0. \end{aligned}$$

(29)

Integrating the above Eq. (29) with respect to $\eta$, we get

$$\begin{aligned} -U\psi _{1}+A\frac{\psi ^{2}}{2}-B\frac{d\psi }{d \eta }=c_{1}, \end{aligned}$$

where $c_{1}$ is an integrating constant.

Using boundary conditions   $\psi \rightarrow 0$, $\frac{d\psi }{d \eta }\rightarrow 0$ as $\eta \rightarrow \infty$ or $\eta \rightarrow -\infty$, we get   $c_{1}=0$.

Then we have

$$\begin{aligned} \frac{d\psi }{d \eta }=A\frac{\psi ^{2}}{2B}-\frac{U\psi }{B}. \end{aligned}$$

(30)

Now using Eq. (30) in Eq. (29), we get

$$\begin{aligned} \frac{d^2\psi }{d \eta ^2}=\frac{A^{2}}{2B^{2}}\psi ^{3} -\frac{3UA}{2B^{2}}\psi ^{2}+\frac{U^{2}}{B^{2}}\psi . \end{aligned}$$

(31)

Now taking $\frac{d\psi }{d \eta }=z$, we get the dynamical system (22).