Anticipating constant profile solutions, we shall consider Eqs. (9
)(12) in a stationary frame traveling at a constant normalized
velocity (to be referred to as the Mach number), implying the
transformation . The space and time derivatives are thus replaced by
and
, respectively,
so Eqs. (9)(12) take the form:

(21) 

(22) 

(23) 

(24) 
We assume that the equilibrium state is reached at both infinities
(
). Accordingly, we integrate and apply the boundary
conditions , , and at . One thus obtains

(25) 

(26) 
and

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

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

(29) 
where
From the boundary conditions, at , it follows that the negative
sign must be taken in Eq. (29). Furthermore, we shall assume that
, i.e., that the cool electrons are supersonic, while the
hot electrons are subsonic, thus we require that .
Reality of the density variable imposes the requirement
, which implies a limit
on the electrostatic potential value
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 pseudoenergy balance equation for a unit mass in a conservative
force field, if one defines as ``time''
and as ``position'' variable:

(32) 
where the Sagdeev pseudopotential
is given by
Figure 2:
(Color online) Variation of the lower limit (lower curves)
and the upper limit (upper curves) with the hottocold electron
density ratio for different values of the temperature ratio .
Solitons may exist for values of the Mach number in the region between the
lower and the upper curve(s) of the same style/color. Curves: (ab)
(solid), (dashed), and (dotdashed), and (c)
(solid), (dashed), and (dotdashed) Here, we have
taken: (a) , (b)
(quasiMaxwellian), and (c)
.

Figure 3:
(Color online) Variation of the lower limit (lower curves)
and the upper limit (upper curves) with the suprathermality parameter
for different values of the temperature ratio (upper panel),
and density ratio (bottom panel). Solitons may exist for values of the
Mach number in the region between the lower and upper curves of the same
style/color. Upper panel:
(solid curve), (dashed), and
(dotdashed). Here, we have taken . Lower panel:
(solid), (dashed), (dotdashed), and (solid circles).
Here,
.

Ashkbiz Danehkar
20180328