The expansion of a dense plasma through a more rarefied ionized medium has been studied by means of two-dimensional particle-in-cell simulations. The initial conditions involve a density jump by a factor of 100, located in the middle of an otherwise equally dense electron–proton plasma with uniform proton and electron temperatures of 10 eV and 1 keV, respectively. Simulations show the creation of a purely electrostatic collisionless shock together with an ion-acoustic soliton tied to its downstream region. The shock front is seen to evolve in filamentary structures consistently with the onset of the ion–ion instability. Meanwhile, an un-magnetized drift instability is triggered in the core part of the dense plasma. Such results explain recent experimental laser–plasma experiments, carried out in similar conditions, and are of intrinsic relevance to non-relativistic shock scenarios in the solar and astrophysical systems.