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.
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