  The Sagdeev potential technique has been used to investigate the existence
and the polarity of dust ion acoustic solitary structures in an unmagnetized
collisionless nonthermal dusty plasma consisting of negatively charged static
dust grains, adiabatic warm ions and nonthermal electrons when the velocity of
the wave frame is equal to the linearized velocity of the dust ion acoustic
wave for long wave length plane wave perturbation, i.e., when the velocity of
the solitary structure is equal to the acoustic speed. A compositional
parameter space has been drawn which shows the nature of existence and the
polarity of dust ion acoustic solitary structures at the acoustic speed. This
compositional parameter space clearly indicates the regions for the existence
of positive and negative potential dust ion acoustic solitary structures.
Again, this compositional parameter space shows that the present system
supports the negative potential double layer at the acoustic speed along a
particular curve in the parametric plane. However, the negative potential
double layer is unable to restrict the occurrence of all negative potential
solitary waves. As a result, in a particular region of the parameter space,
there exist negative potential solitary waves after the formation of negative
potential double layer. But the amplitudes of these supersolitons are bounded.
A finite jump between amplitudes of negative potential solitons separated by
the negative potential double layer has been observed, and consequently, the
present system supports the supersolitons at the acoustic speed in a
neighbourhood of the curve along which negative potential double layer exist.
The effects of the parameters on the amplitude of the solitary structures at
the acoustic speed have been discussed.
