  The supercritical composition of a plasma model with cold positive ions in
the presence of a two-temperature electron population is investigated,
initially by a reductive perturbation approach, under the combined requirements
that there be neither quadratic nor cubic nonlinearities in the evolution
equation. This leads to a unique choice for the set of compositional parameters
and a modified Korteweg-de Vries equation (mKdV) with a quartic nonlinear term.
The conclusions about its one-soliton solution and integrability will also be
valid for more complicated plasma compositions. Only three polynomial
conservation laws can be obtained. The mKdV equation with quartic nonlinearity
is not completely integrable, thus precluding the existence of multi-soliton
solutions. Next, the full Sagdeev pseudopotential method has been applied and
this allows for a detailed comparison with the reductive perturbation results.
This comparison shows that the mKdV solitons have slightly larger amplitudes
and widths than those obtained from the more complete Sagdeev solution and that
only slightly superacoustic mKdV solitons have acceptable amplitudes and
widths, in the light of the full solutions.
