Wave collapse in a nonlinear medium occurs when the self-focusing effect overcomes diffraction.For example, in the 2D Kerr case, a beam of sufficient power will collapse in a radially-symmetric, self-similar fashion, maintaining a Townes profile [1] as it focuses, regardless of its initial shape [2].More complex responses, such as nonlocality, are known to prevent catastrophic collapse [3,4].Here, we report the surprising result that for a critical input power, nonlocal collapse can lead to a transverse profile with a peaked origin and exponentially-decaying tails.Unlike previous peakon/cuspon solutions, this profile is stable but not steady-state.For longer propagation distances, the beam undergoes focus ing/defocusing cycles.These observations are confirmed experimentally by demonstrating peakon profiles and beam oscillations in self-focusing lead glass.