We next investigate the conditions for existence of solitons. First, we need to ensure that the origin at is a root and a local maximum of in Eq. (33), i.e., , and at [43,44,45], where primes denote derivatives with respect to . It is easily seen that the first two constraints are satisfied. We thus impose the condition
An upper limit for is found through the fact that the cool electron density becomes complex at , and hence the largest soliton amplitude satisfies . This yields the following equation for the upper limit in M:
For comparison, for a Maxwellian distribution (here recovered as ), the constraints reduce to
(37)  
(38) 
In the opposite limit of ultrastrong suprathermality, i.e., , the Mach number threshold approaches a nonzero limit , which is essentially the thermal speed, as noted above (recall that by assumption). The upper limit is then given by
(39) 
We have studied the existence domain of electronacoustic solitary waves for different values of the parameters. The results are depicted in Figs. 23. Solitary structures of the electrostatic potential may occur in the range , which depends on the parameters , , and . We recall that we have also assumed that cool electrons are supersonic (in the sense ) [43,44,45], and the hot electrons subsonic (), and care must be taken not to go beyond the limits of the plasma model.

The interval where solitons may exist is depicted in Fig. 2, in two opposite cases: in (a) and (c) two very low, and in (b) one very high value of . We thus see that for both a quasiMaxwellian distribution and one with a large excess suprathermal component of hot electrons, both and decrease with an increase in the relative density parameter for fixed and soliton speed . Further, the upper limit falls off more rapidly, and thus the existence domain in Mach number becomes narrower for higher values of the hottocool electron density ratio. Comparing the two frames (a) and (b) in Fig. 2, we immediately notice that suprathermality (low ) results in solitons propagating at lower Mach number values, a trend which is also seen in Fig. 2c. Another trend that is visible in Figs. 23a is that increased thermal pressure effects of the cool electrons, manifested through increasing , also lead to a narrowing of the Mach number range that can support solitons. Finally, we note that for , the upper limit found from Eq. (36) rises above the limit required by the assumptions of the model, and the latter then forms the upper limit.
Interestingly, in Figs. 23 the existence region appears to shrink down to nil, as the curves approach each other for high values. This is particularly visible in Fig. 2c, for a very low value of ( ). This is not an unexpected result, as high values of are equivalent to a reduction in cool electron relative density, which leads to our model breaking down if the inertial electrons vanish. We recall that a value is a rather abstract case, as it corresponds to a forbidden regime, since Landau damping will prevent electronacoustic oscillations from propagating. Similarly, a high value of the temperature ratio, such as , takes us outside the physically reasonable domain. Nevertheless, as it appears that the lower and upper limits in approach each other asymptotically for high values of , we have carried out calculations for increasing , up to for as an academic exercise, and can confirm that the two limits do not actually intersect.

Figure 3 shows the range of allowed Mach numbers as a function of , for various values of the temperature ratio . As discussed above, increasing towards a Maxwellian distribution ( ) broadens the Mach number range and yields higher values of Mach number. On the other hand, both upper and lower limits decrease as the limiting value is approached. The qualitative conclusion is analogous to the trend in Fig. 2: stronger excess suprathermality leads to solitons occurring in narrower ranges of . Furthermore, as illustrated in Figs. 2 and 3a, the Mach number threshold approaches the upper limit for high values of and : both increased hotelectron density and coolelectron thermal effects shrink the permitted soliton existence region.
Figure 3b depicts the range of allowed Mach numbers as a function of for various values of the density parameter (for a fixed indicative value). We note that both curves decrease with an increase in . Although it lies in the damped region, we have also depicted a high regime for comparison (solidcrosses curve).
We conclude this section with a brief comparison of our work with that of Ref. [39]. The latter did not consider existence domains at all, let alone their dependence on plasma parameters, but merely plotted some Sagdeev potentials and associated soliton potential profiles for chosen values of some of the parameters, so as to extract some trends. En passant, there is indirect mention of an upper limit in , in that it is commented that as increasing values of are considered, at some stage solitary waves cease to exist. [39]


Ashkbiz Danehkar