  The Sagdeev pseudo-potential technique and the analytic theory developed by
Das et al. [J. Plasma Phys. 78, 565 (2012)] have been used to investigate the
dust ion acoustic solitary structures at the acoustic speed in a collisionless
unmagnetized dusty plasma consisting of negatively charged static dust grains,
adiabatic warm ions, nonthermal electrons and isothermal positrons. The present
system supports both positive and negative potential solitary waves at the
acoustic speed, but the system does not support the coexistence of solitary
structures of opposite polarity at the acoustic speed. The system also supports
negative potential double layer at the acoustic speed, but does not support
positive potential double layer. Although the system supports positive
potential supersoliton at the supersonic speed, but there does not exist
supersoliton of any polarity at the acoustic speed. Solitary structures have
been investigated with the help of compositional parameter spaces and the phase
portraits of the dynamical system describing the nonlinear behaviour of the
dust ion acoustic waves at the acoustic speed. For the case, when there is no
positron in the system, there exist negative potential double layer and
negative potential supersoliton at the acoustic speed and for such case, the
mechanism of transition of supersoliton to soliton after the formation of
double layer at the acoustic speed has been discussed with the help of phase
portraits. The differences between the solitary structures at the acoustic
speed and the solitary structures at the supersonic speed have been analysed
with the help of phase portraits.
