4 Nonlinear Analysis

To obtain nonlinear wave solutions, we consider all fluid variables in a stationary frame traveling at a constant normalized velocity $M$ (to be referred to as the Mach number), which implies the transformation $\xi=x-Mt$. This replaces the space and time derivatives with $\partial/\partial x=d/d\xi$ and $\partial/\partial t=-Md/d\xi$, respectively. Now equations (10) to (16) take the following form:

$$-M\dfrac{dn}{d\xi}+\frac{d(nu)}{d\xi}=0,$$ (19)

$$-M\dfrac{du}{d\xi}+u\dfrac{du}{d\xi}=\dfrac{d\phi}{d\xi}-\frac{\sigma} {n}\dfrac{dp}{d\xi},$$ (20)

$$-M\dfrac{dp}{d\xi}+u\dfrac{dp}{d\xi}+3p\dfrac{du}{d\xi}=0,$$ (21)

$$-M\dfrac{dn_{p}}{d\xi}+\frac{d(n_{p}u_{p})}{d\xi}=0,$$ (22)

$$-M\dfrac{du_{p}}{d\xi}+u_{p}\dfrac{du_{p}}{d\xi}=-\dfrac{d\phi}{d\xi} -\frac{\theta}{n_{p}}\dfrac{dp_{p}}{d\xi},$$ (23)

$$-M\dfrac{dp_{p}}{d\xi}+u_{p}\dfrac{dp_{p}}{d\xi}+3p_{p}\dfrac{du_{p}}{d\xi }=0,$$ (24)

$$\dfrac{d^{2}\phi}{d\xi^{2}}= -\left( 1+\alpha-\beta\right) +n-\beta n_{p}$$

$$+\alpha\left( 1-\frac{\phi}{\kappa-\tfrac{3}{2}}\right)^{-\kappa+1/2},$$ (25)

The equilibrium state is assumed to be reached at both infinities ( $\xi\rightarrow\pm\infty$). Accordingly, we integrate Eqs. (19)-(24), apply the boundary conditions $n=1$, $p=1$, $u=0$, $n_{p}=1$, $p_{p}=1$, $u_{b}=0$ and $\phi=0$ at infinities, and obtain

$$u=M[1-(1/n)],$$ (26)

$$u={M-(M}^{2}{+2\phi-3n^{2}\sigma+3\sigma})^{1/2},$$ (27)

$$u_{p}=M[1-(1/n_{p})],$$ (28)

$$u_{p}=M-(M^{2}-2\phi-3n_{p}^{2}\theta+3\theta)^{1/2},$$ (29)

$$\begin{align*}\begin{array}[c]{cc}<tex2html_comment_mark> p=n^{3},\text{ \ } & p_{p}=n_{p}^{3}. \end{array}\end{align*}$$ (30)

Combining Eqs. (26)-(30), one obtains the following biquadratic equations for the cool electron density and the positron density, respectively,

$$3\sigma n^{4}-({M}^{2}{+2\phi+3\sigma)n^{2}}+{M}^{2}=0,$$ (31)

$$3\theta n_{p}^{4}-(M^{2}-2\phi+3\theta)n_{p}^{2}+M^{2}=0.$$ (32)

Eqs. (31) and (32) are respectively solved as follows:

$$n =\frac{1}{2\sqrt{3\sigma}}\left[ {2\phi+}({M+\sqrt{3{\sigma}}} )^{2}\right] ^{1/2}$$

$$\pm\frac{1}{2\sqrt{3\sigma}}{\left[ {2\phi+({M-\sqrt{3{\sigma}}}} )^{2}\right] }^{1/2},$$ (33)

$${n}_{p} {=}\frac{1}{2\sqrt{3\theta}}\left[ -{2\phi}+(M+\sqrt{3\theta })^{2}\right] ^{1/2}$$

$$\pm\frac{1}{2\sqrt{3\theta}}\left[ -{2\phi}+(M-\sqrt{3\theta})^{2}\right] ^{1/2}.$$ (34)

Eq. (33) agrees with Eq. (29) derived in Ref. [31]. From the boundary conditions, $n_{c}=n_{p}=1$ at $\phi=0$, it follows that the negative sign must be taken in equations (33) and (34). Moreover, the cool electrons and positrons are assumed to be supersonic for $M>\sqrt{3\sigma}$ and $M>\sqrt{3\theta}$, respectively, while the hot electrons are subsonic for $M<1$.

Reality of the cool electron density variable imposes the requirement ${2\phi+({M-}\sqrt{3{\sigma}})^{2}>0}$ that implies a lower boundary on the electrostatic potential value $\phi>\phi_{{\rm max}(-)}=-\frac{1}{2}({M-}\sqrt{3{\sigma}})^{2}$ associated with negative polarity solitary structures. However, reality of the positron density variable imposes ${-2\phi}+(M-\sqrt{3\theta})^{2}{>0}$, implying a higher boundary on the electrostatic potential value $\phi< \phi_{{\rm max}(+)}=\frac{1}{2}({M-}\sqrt{3\theta})^{2}$ associated with positive polarity solitary structures.

Substituting Eqs. (33)-(34) into the Poisson's equation (16), multiplying the resulting equation by $d\phi/d\xi$, integrating and taking into account the conditions at infinities ( $d\phi/d\xi\rightarrow 0$) yield a pseudo-energy balance equation:

$$\frac{1}{2}\left( \frac{d\phi}{d\xi}\right) ^{2}+\Psi(\phi)=0,$$ (35)

where the Sagdeev pseudopotential $\Psi (\phi )$ is given by

$$\Psi(\phi)= \alpha\left[ 1-\left( 1+\frac{\phi}{-\kappa+\tfrac{3}{2} }\right) ^{-\kappa+3/2}\right] +(1+\alpha-\beta)\phi$$

$$+\frac{1}{6\sqrt{3{\sigma}}}\left[ ({M+}\sqrt{3{\sigma}})^{3}-{{({M-} \sqrt{3{\sigma}})^{3}}}\right.$$

$$\left. -({2\phi+}[{M+}\sqrt{3{\sigma}}]^{2})^{3/2}\right.$$

$$\left. +{({2\phi+[{M-}\sqrt{3{\sigma}}]^{2}}})^{3/2}\right]$$

$$-\frac{\beta}{6\sqrt{3\theta}}\left[ (M_{\text{ }}{+}\sqrt{3\theta} )^{3}-{{(M{-}\sqrt{3\theta})^{3}}}\right.$$

$$\left. -(-{2\phi+}[M{+}\sqrt{3\theta}]^{2})^{3/2}\right.$$

$$\left. +{(-{2\phi+[M{-}\sqrt{3\theta}]^{2}}})^{3/2}\right] .$$ (36)

In the absence of the positrons ( $\beta\rightarrow0$), we exactly recover the pseudopotential equation derived for electron-acoustic waves with suprathermal electrons [31].

For the existence of solitons, we require that the origin at $\phi=0$ is a root and a local maximum of $\Psi$ in Eq. (36), i.e., $\Psi(\phi)=0$, $\Psi^{\prime}(\phi)=0$ and $\Psi^{\prime\prime}(\phi)<0$ at $\phi=0$, where primes denote derivatives with respect to $\phi$. It is easily seen that the first two constraints are satisfied. We thus impose the condition $F_{1}(M)=-\Psi^{\prime\prime}(\phi)\vert _{\phi=0}>0$, and we get

$$F_{1}(M)=\frac{\alpha(\kappa-\frac{1}{2})}{\kappa-\tfrac{3}{2}}-\frac {1}{(M^{2}-3\sigma)}-\frac{\beta}{(M^{2}-3\theta)}.$$ (37)

Eq. (37) provides the minimum value for the Mach number, $M_{1}(\kappa,\alpha,\sigma,\beta,\theta)$. In the limit $\beta\rightarrow0$ (without the positrons), equation (37) takes the form of Eq. (34) in Ref. [31].

An upper limit for $M$ is determined from the fact that the cool electron density becomes complex at negative potentials lower than $\phi_{{\rm max}(-)}=-\frac{1}{2}( {M-} \sqrt{3{\sigma}}) ^{2}$ for negative polarity waves, and the cool positron density at positive potentials higher than $\phi _{{\rm max}(+)}=\frac{1}{2}( M{-}\sqrt{3{\theta}}) ^{2}$ for positive polarity waves. Thus, the largest negative soliton amplitude satisfies $F_{2}(M)=\Psi(\phi)\vert _{\phi=\phi_{{\rm max}(-)}}>0$, whereas the largest positive soliton amplitude fulfills $F_{2}(M)=\Psi(\phi)\vert _{\phi=\phi_{{\rm max}(+)}}>0$. These yield the following equation for the upper limit in $M$ for negative polarity electrostatic soliton existence associated with cool electrons,

$$F_{2}^{(-)}(M) =-\tfrac{1}{2}(1+\alpha-\beta)({M-}\sqrt{3{\sigma}} )^{2}+M^{2}+\sigma$$

$$+\alpha\left[ 1-\left( 1+\frac{[{M-}\sqrt{3{\sigma}}]^{2}}{2\kappa -3}\right) ^{-\kappa+3/2}\right]$$

$$-\frac{\beta}{6\sqrt{3\theta}}\left( {[}({{M-\sqrt{3\theta})^{2}}-} ({{{M-}\sqrt{3{\sigma}})^{2}}}]^{3/2}\right.$$

$$\left. -[(M{+}\sqrt{3\theta})^{2}-{({M-}\sqrt{3{\sigma}}){^{2}}} ]^{3/2}\right)$$

$$-\beta M^{2}{-}\beta\theta-\tfrac{4}{3}M^{3/2}\left( 3\sigma\right) ^{1/4},$$ (38)

and the following equation for positive polarity electrostatic soliton existence associated with positrons,

$$F_{2}^{(+)}(M) =\tfrac{1}{2}(1+\alpha-\beta)(M{-}\sqrt{3{\theta}} )^{2}+M^{2}+\sigma$$

$$+\alpha\left( 1-\left[ 1-\frac{(M{-}\sqrt{3{\theta}})^{2}}{2\kappa -3}\right] ^{-\kappa+3/2}\right)$$

$$+\frac{1}{6\sqrt{3{\sigma}}}\left( {[({{{M-}\sqrt{3{\sigma}})^{2}}+} (M{-}\sqrt{3{\theta}}){^{2}}}]^{3/2}\right.$$

$$\left. -[({M+}\sqrt{3{\sigma}})^{2}+(M{-}\sqrt{3{\theta}})^{2} ]^{3/2}\right)$$

$$-\beta M^{2}-\beta\theta+\tfrac{4}{3}\beta M^{3/2}\left( 3\theta\right) ^{1/4}.$$ (39)

Solving equations (38) and (39) provide the upper limit $M_{2}(\kappa,\alpha,\sigma,\beta,\theta)$ for acceptable values of the Mach number for negative and positive polarity solitons to exist. The cool electrons can generally support a negative supersonic electrostatic wave, while the positrons may provide the inertia to support a positive polarity electrostatic wave. Hence, the upper limit of negative polarity electrostatic solitons can be determined from Eq. (38), while the upper limit of positive polarity electrostatic solitons may be obtained from Eq. (39). In the absence of the positrons, Eq. (38) yields exactly Eq. (36) in Ref. [31]. Taking a Maxwellian distribution ( $\kappa\rightarrow\infty$) and without the positrons ( $\beta\rightarrow0$), equations (37) and (38) take the form of Eqs. (37) and (38) in Ref. [31].

Figure 2 shows the range of allowed Mach numbers for negative polarity electrostatic solitary waves with different parameters: the positron-to-hot electron temperature ratio, $\theta$, and the positron-to-cool electron density ratio, $\beta$. The lower limit ($M_{1}$) and the upper limit ($M_{2}$) of Mach numbers are obtained from numerically solving equations (37) and (38), respectively. We see that there is a small difference between the model including the positrons and the model without the positrons ( $\beta\rightarrow0$). As the positron is assumed to have a very small fraction of the total charge ( $\beta\lesssim0.06$) and a cool temperature ( $\theta\ll0.1$), they cannot have a significant role in the dynamics of electron-acoustic waves in the model adopted here. Hence, the existence domain of electron-acoustic (negative polarity electrostatic) solitary waves are not largely affected by the cool positrons.

Figure 2: Variation of the lower limit $M_{1}$ (lower curves; panel b) and the upper limit $M_{2}$ (upper curves; panel a) of the negative polarity electrostatic solitons with the positron-to-cool electron density ratio $\beta$ for different values of the positron-to-hot electron temperature ratio $\theta$. Solitons may exist for values of the Mach number $M$ in the region between the lower and the upper curve(s) of the same style/color. (a-b) Curves: $\theta =0.0$ (solid), $0.01$ (dashed), and $0.02$ (dot-dashed). Here, $\kappa =2$, $\alpha =1$ and $\sigma =0.01$.

The soliton existence regions for positive polarity electrostatic solitary waves are shown in Fig. 3 for different parameters. Solitary structures of the electrostatic potential may occur in the range $M_{1}<M<M_{2}$, which depends on the parameters $\theta$, $\beta$, and $\kappa$. Moreover, we assume that the cool electrons and positrons are supersonic ( ${M>}\sqrt{3{\sigma}}$ and ${M>}\sqrt{3{\theta}}$, respectively), while the hot electrons are subsonic (${M<1}$). We used Eq. (37) to obtain the lower limit for negative polarity solitons. This equation may also have another solution, which could yield the lower Mach number limit for positive polarity solitary structures. However, we noticed that Mach numbers of positive polarity solitons cannot be constrained by Eq. (37) due to the small values of the density ratio $\beta$. Therefore, the lower limit ($M_{1}$) is found to be at about $\sqrt{3{\sigma}}$. The positive potential solitons numerically derived from Eq. (36) cannot also produce any solutions for Mach numbers less than $\sqrt{3{\sigma}}$ in the adopted parameter ranges of the positrons.

Figure 3: Variation of the upper limit $M_{2}$ of the positive polarity electrostatic solitons with the spectral index $\kappa$ for different values of (a) the positron-to-hot electron temperature ratio $\theta$ and (b) the positron-to-cool electron density ratio $\beta$. Upper panel: $\theta =0.0$ (solid), $0.001$ (dashed), and $0.01$ (dot-dashed). Here, $\alpha =1$ and $\beta =\sigma =0.01$. Lower panel: $\beta =0.005$ (solid), $0.01$ (dashed), and $0.015$ (dot-dashed). Here, $\alpha =1$ and $\sigma =\theta =0.01$.

As seen in Fig. 3, the upper limit ($M_{2}$) of positive polarity solitons is slightly increased with an increase in the positron-to-hot electron temperature ratio $\theta$ and a decrease in the positron-to-cool electron density ratio $\beta$. However, the effect is not significant, and also dissimilar to how the hot-to-cool electron density ratio ($\alpha$) affects electron-acoustic waves [31]. This negligible effect is mostly attributed to the small fraction of positrons and their cool temperatures in the e-p plasma system.

Figure 3 also depicts the upper limit ($M_{2}$) of allowed Mach numbers as a function of $\kappa$, for various values of $\theta$ and $\beta$. As seen, increasing $\kappa$ toward a Maxwellian distribution ( $\kappa\rightarrow\infty$) increases the upper limit ($M_{2}$) and broadens the Mach number range. It can be seen that positive polarity solitons are generated in narrower ranges of Mach numbers as hot electron suprathermality becomes stronger. This conclusion is similar to what found in electron-acoustic solitary waves with suprathermal electrons [31].