4 Nonlinear analysis for large amplitude solitary waves

Anticipating constant profile solutions, we shall consider Eqs. (9 )-(12) in a stationary frame traveling at a constant normalized velocity $M$ (to be referred to as the Mach number), implying the transformation $\xi=x-Mt$. The space and time derivatives are thus replaced by $\partial/\partial x=d/d\xi$ and $\partial/\partial t=-Md/d\xi$, respectively, so Eqs. (9)-(12) take the form:

$$-M\dfrac{dn}{d\xi}+\frac{d(nu)}{d\xi}=0,$$ (21)

$$-M\dfrac{du}{d\xi}+u\dfrac{du}{d\xi}=\dfrac{d\phi}{d\xi}-\frac{\sigma} {n}\dfrac{dp}{d\xi},$$ (22)

$$-M\dfrac{dp}{d\xi}+u\dfrac{dp}{d\xi}+3p\dfrac{du}{d\xi}=0,$$ (23)

$$\frac{d^{2}\phi}{d\xi^{2}}=-(\beta+1)+n+\beta\left[ 1-\frac{\phi} {(\kappa-\tfrac{3}{2})}\right] ^{-\kappa+1/2}.$$ (24)

We assume that the equilibrium state is reached at both infinities ( $\xi\rightarrow\pm\infty$). Accordingly, we integrate and apply the boundary conditions $n=1$, $p=1$, $u=0$ and $\phi=0$ at $\pm\infty$. One thus obtains

$$u=M\left( 1-\frac{1}{n}\right) ,$$ (25)

$$u={M-(M}^{2}{+2\phi-3n^{2}\sigma+3\sigma)}^{1/2},$$ (26)

and

$$p=n^{3}.$$ (27)

Combining Eqs. (25)-(27), we obtain the following biquadratic equation for the cool electron density,

$${3\sigma n^{4}}-({M}^{2}{+2\phi+3\sigma)n^{2}}+{M}^{2}=0\,.$$ (28)

The solution of Eq. (28) may be written as

$${n=}\dfrac{1}{2}\left( n_{(+)}\pm n_{(-)}\right) ,$$ (29)

where

$$n_{(+)} {\equiv}\left[ \dfrac{{2\phi+}\left( {M+}\sqrt{3{\sigma} }\right) ^{2}}{3{\sigma}}\right] ^{1/2},$$ (30)

$$n_{(-)} {\equiv}\left[ \dfrac{{2\phi+\left( {M-}\sqrt{3{\sigma}}\right) ^{2}}}{3{\sigma}}\right] ^{1/2}.$$ (31)

From the boundary conditions, $n=1$ at $\phi=0$, it follows that the negative sign must be taken in Eq. (29). Furthermore, we shall assume that $M>\sqrt{3\sigma}$, i.e., that the cool electrons are supersonic, while the hot electrons are subsonic, thus we require that $M<1$.

Reality of the density variable imposes the requirement ${2\phi+\left( {M-}\sqrt{3{\sigma}}\right) ^{2}>0}$, which implies a limit on the electrostatic potential value $\vert\phi_{\max}\vert=\frac{1}{2}\left( {M-}\sqrt{3{\sigma}}\right) ^{2}$ associated with negative solitary structures (positive electric potentials, should they exist, satisfy the latter condition automatically, and are thus not limited).

Substituting the density expression (29)-(31) into Poisson's equation (24) and integrating, yields the pseudo-energy balance equation for a unit mass in a conservative force field, if one defines $\xi$ as ``time'' and $\phi$ as ``position'' variable:

$$\frac{1}{2}\left( \frac{d\phi}{d\xi}\right) ^{2}+\Psi(\phi)=0,$$ (32)

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

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

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

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

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

Figure 2: (Color online) Variation of the lower limit $M_{1}$ (lower curves) and the upper limit $M_{2}$ (upper curves) with the hot-to-cold electron density ratio $\beta$ for different values of the temperature ratio $\sigma$. 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. Curves: (a-b) $\sigma =0.01$ (solid), $0.02$ (dashed), and $0.04$ (dot-dashed), and (c) $\sigma =0.01$ (solid), $0.1$ (dashed), and $0.2$ (dot-dashed) Here, we have taken: (a) $\kappa =2$, (b) $\kappa =100$ (quasi-Maxwellian), and (c) $\kappa =1.65$.

Figure 3: (Color online) Variation of the lower limit $M_{1}$ (lower curves) and the upper limit $M_{2}$ (upper curves) with the suprathermality parameter $\kappa$ for different values of the temperature ratio $\sigma$ (upper panel), and density ratio $\beta$ (bottom panel). Solitons may exist for values of the Mach number $M$ in the region between the lower and upper curves of the same style/color. Upper panel: $\sigma =0.01$ (solid curve), $0.02$ (dashed), and $0.04$ (dot-dashed). Here, we have taken $\beta =2$. Lower panel: $\beta =1.8$ (solid), $2$ (dashed), $2.2$ (dot-dashed), and $5$ (solid circles). Here, $\sigma =0.02$.