The misalignment mechanism for axion-like particles (ALPs) is a leading explanation for dark matter. In this work we investigate ALPs with non-periodic potentials, which allow for large misalignment of the field from the minimum and make it possible for ALPs to match the relic density of dark matter in a large part of the parameter space. Such potentials give rise to self-interactions which can trigger an exponential growth of fluctuations in the ALP field via parametric resonance, leading to the fragmentation of the field. The fluctuations later collapse to halos that can be dense enough to produce observable gravitational effects. These effects would provide a probe of dark matter even if it does not couple to the Standard Model (or too feebly). We determine the relevant regions of parameter space in the (ALP mass, decay constant)-plane and compare predictions in different axion fragmentation models. These proceedings are a short version of [1].