  The localized deposition of the energy of a laser pulse, as it ablates a
solid target, introduces high thermal pressure gradients in the plasma. The
thermal expansion of this laser-heated plasma into the ambient medium (ionized
residual gas) triggers the formation of non-linear structures in the
collision-less plasma. Here an electron-proton plasma is modelled with a
particle-in-cell (PIC) simulation to reproduce aspects of this plasma
expansion. A jump is introduced in the thermal pressure of the plasma, across
which the otherwise spatially uniform temperature and density change by the
factor 100. The electrons from the hot plasma expand into the cool one and the
charge imbalance drags a beam of cool electrons into the hot plasma. This
double layer reduces the electron temperature gradient. The presence of the
low-pressure plasma modifies the proton dynamics compared to the plasma
expansion into a vacuum. The jump in the thermal pressure develops into a
primary shock. The fast protons, which move from the hot into the cold plasma
in form of a beam, give rise to the formation of phase space holes in the
electron and proton distributions. The proton phase space holes develop into a
secondary shock that thermalizes the beam.
