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. Comment: 17 pages, 9 Figures, accepted for publication in plasma physics and controlled fusion