Effect of Landau damping on ion acoustic solitary waves in a collisionless unmagnetized plasma consisting of nonthermal and isothermal electrons

Abstract

A Korteweg–de Vries (KdV) equation including the effect of linear Landau damping of electrons is derived to study the propagation of weakly nonlinear and weakly dispersive ion acoustic waves in a collisionless unmagnetized plasma consisting of warm adiabatic ions and two species of electrons at different temperatures. It is found that the coefficient of the nonlinear term of this KdV-like evolution equation vanishes along different family of curves in different parameter planes. In this context, a modified KdV (MKdV) equation including the effect of linear Landau damping of electrons describes the nonlinear behaviour of ion acoustic waves. Again, the coefficients of the nonlinear terms of the KdV and MKdV-like evolution equations are simultaneously equal to zero along a family of curves in the parameter plane. In this situation, we have derived a further modified KdV (FMKdV) equation including the effect of linear Landau damping of electrons. The multiple time scale method has been applied to obtain the solitary wave solution of the evolution equations having the nonlinear term $\left( \phi ^{(1)}\right) ^{r}\frac{\partial \phi ^{(1)}}{\partial \xi }$, where $\phi ^{(1)}$ is the first-order perturbed electrostatic potential and $r =1,2,3$. The amplitude of the ion acoustic solitary wave decreases with time for all $r =1,2,3$.

Appendices

Appendix 1

Coefficients of Eq. (76):

$$\begin{aligned} J_{{{\mathrm{c0}}}} = \int _{-\infty }^{+\infty } \frac{\partial j_{{{\mathrm{c0}}}}}{\partial v_{||}^{2}}{{\mathrm{d}}}v_{||},\;Z_{{{\mathrm{c0}}}} = \frac{\partial f_{{{\mathrm{c0}}}}}{\partial v_{||}^{2}}\bigg |_{v_{||}=0}, \end{aligned}$$

(107)

where $J = F$, G, H, K for $j = f$, g, h, k, respectively.

Appendix 2

Coefficients of Eq. (78):

$$\begin{aligned} J_{{{\mathrm{s0}}}} = \int _{-\infty }^{+\infty } \frac{\partial j_{{{\mathrm{s0}}}}}{\partial v_{||}^{2}}{{\mathrm{d}}}v_{||},\;Z_{{{\mathrm{s0}}}} = \frac{\partial f_{{{\mathrm{s0}}}}}{\partial v_{||}^{2}}\bigg |_{v_{||}=0}, \end{aligned}$$

(108)

where $J = F$, G, H, K for $j = f$, g, h, k, respectively.

Appendix 3

Method of finding $I_{r}$ associated with Eqs. (103) and (104):

$$\begin{aligned} I_{r} = \mathcal {P} \int \limits _{-\infty }^{+\infty }\int \limits _{-\infty }^{+\infty } [{{\text{ sech } }}{} \textit{X} ]^{\frac{2}{r}} \frac{\partial [{{\text{ sech } }}{} \textit{X}' ]^{\frac{2}{r}}}{\partial X'} \frac{{{\mathrm{{d}}}}X {{\mathrm{{d}}}}X'}{X-X'} . \end{aligned}$$

(109)

Now $I_{r}$ can be written as

$$\begin{aligned} I_{r} = - \int _{-\infty }^{\infty } \frac{\partial [{{\text{ sech } }} \textit{z} ]^{\frac{2}{r}}}{\partial z} I_{1r} dz , \end{aligned}$$

(110)

where $X=z'$, $X'=z$ and

$$\begin{aligned} I_{1r} = \mathcal {P} \int _{-\infty }^{\infty } \frac{[{{\text{ sech } }} \textit{z}' ]^{\frac{2}{r}} }{z-z'} dz' . \end{aligned}$$

(111)

Using the following known result

$$\begin{aligned} \int _{-\infty }^{0} e^{is(z-z')} ds = \pi \delta (z-z') - i \mathcal {P} \frac{1}{z-z'} , \end{aligned}$$

(112)

form Eq. (112), we get

$$\begin{aligned} \mathcal {P} \frac{1}{z-z'} = \frac{1}{2i} \int _{-\infty }^{\infty } \frac{s}{|s|} e^{is(z-z')} ds . \end{aligned}$$

(113)

Using (113), Eq. (111) can be written as

$$\begin{aligned} I_{1r} = \frac{1}{2i} \int _{-\infty }^{\infty } \frac{s}{|s|} F(s) e^{isz} ds , \end{aligned}$$

(114)

where

$$\begin{aligned} F(s) = \int _{-\infty }^{\infty } [{{\text{ sech } }}{} \textit{z} ]^{\frac{2}{r}} e^{-isz} dz . \end{aligned}$$

(115)

Therefore, Eq. (110) can be written as

$$\begin{aligned} I_{r} = \int _{0}^{\infty } s [F(s)]^{2} ds . \end{aligned}$$

(116)