The stability of the electrostatic, fast, solitary structures that are observed frequently on auroral field lines is analyzed as a three-dimensional dynamic system and investigated numerically with the help of test-particle and particle-in-cell simulations. The recently observed large-amplitude, localized potential structures with a flat-top shape and separated electric field spikes parallel to the external magnetic field are shown to be supported by a particular self-consistent charge distribution function that differs qualitatively from the distribution that supports the previously investigated Gaussian potentials. The evolution of the solitary structures is related to changes in the trajectories of the supporting, trapped electrons. It is shown that ignoring the interaction with the ions, the planar, isolated spikes propagating into a plasma with realistic, open boundary conditions, are stable in three dimensions. The addition of a perturbation to the large-amplitude primary structure due to a smaller, faster secondary propagating spike results in destruction of the perturbing spike by modifying electron orbits of its trapped population, while the main spike gains an average momentum without any noticeable deformation. The analysis indicates robustness of the solitary spikes on the auroral field lines over significantly long times and their ability to propagate over long distances.