5 Nonlinear Wave Structures

To consider the nonlinear features of electrostatic wave structures, we have numerically solved Eq. (36) for various plasma parameters, in order to investigate their effects. We found that both negative and positive electric potentials arise in the ranges of allowed Mach numbers obtained for negative and positive polarity soliton existence domains in Section 4.

Figure 4: (a) The pseudopotential $\Psi (\phi )$ of negative polarity electrostatic solitons and the associated solutions: (b) electric potential pulse $\phi$, (c) density $n_{c}$ and (d) velocity $u_{c}$ of the cool electron fluid, and (e) density $n_{p}$ and (f) velocity $u_{p}$ of the positron fluid are depicted versus position $\xi$ for different values of the positron-to-cool electron density ratio $\beta$. We have taken: $\beta =0.0$ (solid curve), $0.005$ (dashed curve), and $0.01$ (dot-dashed curve). The other parameter values are: $\alpha =1$, $\sigma =\theta =0.01$, $\kappa =4.0$ and $M=1.1$.

Figure 5: (a) The pseudopotential $\Psi (\phi )$ of positive polarity electrostatic solitons and the associated solutions: (b) electric potential pulse $\phi$, (c) density $n_{c}$ and (d) velocity $u_{c}$ of the cool electron fluid, and (e) density $n_{p}$ and (f) velocity $u_{p}$ of the positron fluid are depicted versus position $\xi$ for different values of the positron-to-cool electron density ratio $\beta$. We have taken: $\beta =0.005$ (solid curve), $0.010$ (dashed curve), and $0.015$ (dot-dashed curve). The other parameter values are: $\alpha =1$, $\sigma =\theta =0.01$, $\kappa =4.0$ and $M=0.5$.

Figure 4(a) shows the variation of the pseudopotential $\Psi (\phi )$ of negative polarity solitons with the normalized negative potential $\phi$, for different values of the positron-to-cool electron density ratio $\beta$ (keeping $\alpha =1$, $\sigma =\theta =0.01$, $\kappa =4.0$ and Mach number $M=1.1$, all fixed). The electrostatic pulse $\phi$ shown in Fig. 4(b) is obtained via a numerical integration. The negative pulse amplitude decreases with increasing $\beta$. We algebraically determined the fluid density (Fig. 4c) and velocity disturbance (Fig. 4d) of the cool electrons, as well as the fluid density (Fig. 4e) and velocity disturbance (Fig. 4f) of the positrons. It is found that an increase in the positron-to-cool electron density ratio $\beta$ decreases the disturbances and amplitudes of $n_{c}$, $u_{c}$, $n_{p}$ and $u_{p}$ in the negative polarity electrostatic mode. This means that increasing the positron density reduces the negative potential solitary waves, in agreement with the previous results [52]. We also note that the profiles become less steeper but broader.

Figure 6: (a) The pseudopotential $\Psi (\phi )$ of positive polarity electrostatic solitons and the associated solutions: (b) electric potential pulse $\phi$, and (c) density $n_{p}$ and (d) velocity $u_{p}$ of the positron fluid are depicted versus position $\xi$ for different values of the positron-to-hot electron temperature ratio $\theta$. We have taken: $\theta =0.0$ (solid curve), $0.01$ (dashed curve), and $0.02$ (dot-dashed curve). The other parameter values are: $\alpha =1$, $\sigma =0.01$, $\beta =0.015$, $\kappa =4.0$ and $M=0.5$.

Similarly, Figure 5(a) depicts the variation of the pseudopotential $\Psi (\phi )$ of positive polarity solitons associated with the positrons for different values of the positron-to-cool electron density ratio $\beta$ (keeping $\alpha =1$, $\sigma =\theta =0.01$, $\kappa =4.0$ and Mach number $M=0.5$, all fixed). As seen in Fig. 5(b), that the positive pulse amplitude rises with an increase in $\beta$, in contrast to what we see in Fig. 4(b). Furthermore, an increase in $\beta$ increases the disturbances, amplitudes and steepness of $n_{c}$ $u_{c}$, $n_{p}$ and $u_{p}$ in the positive polarity electrostatic mode. This means that increasing the positron density increases the positive potential solitary waves, which agrees with the results of Ref. [52] (they used $n_{p,0}/n_{h,0}$ rather than $\beta=n_{p,0}/n_{c,0}$).

Figure 7: (a) The pseudopotential $\Psi (\phi )$ of positive polarity electrostatic solitons and the associated solutions: (b) electric potential pulse $\phi$, and (c) density $n_{p}$ and (d) velocity $u_{p}$ of the positron fluid are depicted versus position $\xi$, for different values of the spectral index $\kappa$. We have taken: $\kappa =3$ (solid curve), $4$ (dashed curve), and $6$ (dot-dashed curve). The other parameter values are: $\alpha =1$, $\sigma =\theta =0.01$, $\beta =0.015$ and $M=0.5$.

The thermal effect of the positrons through $\theta=T_{p}/T_{h}$ is shown in Fig. 6. The soliton excitation $\phi$ is slightly amplified with an increase in the temperature ratio $\theta$, which agrees with the results of Ref. [52] (they used $T_{h}/T_{p}$ rather than $\theta=T_{p}/T_{h}$). Furthermore, an increase in $\theta$ slightly increases the disturbance of $n_{c}$ and $u_{c}$ (not shown here), however, significantly increases and steepens the disturbance of $n_{p}$ and $u_{p}$ in the positive polarity electrostatic mode (Fig. 6(c) and (d)). The temperature ratio $\theta$ does not make a significant contribution to the negative polarity electrostatic solitary waves due to the small value of $\beta$.

Figure 8: The dependence of the pulse amplitude $\phi _{m}$ of positive polarity electrostatic solitons on the Mach number-to-sound-speed ratio $M/\sqrt {3{\sigma }}$ is depicted, for different values of the spectral index $\kappa$. From top to bottom: $k=2$ (solid curve); $2.5$ (dashed curve); $3$ (dot-dashed curve); $4$ (crosses); $10$ (circles); $100$ (solid squares). Here, $\alpha =1$ and $\sigma =\theta =\beta =0.01$.

Figure 7(a) shows the pseudopotential $\Psi (\phi )$ of positive polarity solitons for different values of the spectral index $\kappa$ (keeping $\alpha =1$, $\sigma =\theta =0.01$, $\beta =0.015$ and Mach number $M=0.5$). The positive polarity electrostatic pulse shown in Fig. 7(b) is found to increase for lower $\kappa$, implying an amplification of the electric potential disturbance as the suprathermality increases. It can be seen that the positron fluid density (Fig. 7(c)) and velocity disturbance (Fig. 7(d)) are increased in the positive polarity electrostatic mode, and again, for lower $\kappa$ values.

As inherently super-acoustic solitons are taken, it is important to see the effect of a varying true Mach number, so we explore the pulse amplitude $\phi _{m}$ of the positive polarity electrostatic solitons as a function of the propagation speed $M$, measured relative to the true acoustic speed, $M_{1}$. The variation of the soliton amplitude $\phi _{m}$ as a function of the true Mach number, $M/M_{1}$, is numerically obtained from Eq. (36). Noting that the lower limit ($M_1$) for positive polarity solitary structures is about $\sqrt{3{\sigma}}$, we have plotted the soliton amplitude $\phi_{\mathrm{m}}$ against the ratio $M/\sqrt {3{\sigma }}$, for a range of values of the parameter $\kappa$ in Fig. 8. It is seen that the soliton amplitude $\phi _{m}$ increases with $M/\sqrt {3{\sigma }}$ for all values of $\kappa$. Moreover, the soliton amplitude increases with growing the suprathermality (reducing $\kappa$) at a fixed true Mach number, $M/\sqrt {3{\sigma }}$, in contrast to the results obtained previously [31]. However, the maximum value of soliton amplitude is found to be for a Maxwellian distribution ( $\kappa\rightarrow\infty$) at larger true Mach numbers ( $M/\sqrt{3{\sigma}}>4$).