  In string theory, the simultaneous existence of many Axion-Like Particles
(ALPs) are suggested over a vast mass range, and a variety of potentials have
been developed in the context of inflation. In such potentials shallower than
quadratic, the prominent instability can produce localized dense objects,
oscillons. Because of the approximate conservation of their adiabatic
invariant, oscillons generally survive quite long, maybe up to the current age
of the universe in the case of ultra-light ALPs with $m \sim 10^{-22}\ {\rm
eV}$. Such oscillons can have significant effects on the evolution of the
recent universe. In this paper, we investigate the oscillons of the
pure-natural type potential by classical lattice simulation to explore the key
quantities necessary for phenomenological application: the number density of
oscillons, the oscillon mass distribution, the energy ratio of oscillons to the
ALP field, and the power spectrum. Then, we evolve these values in
consideration of the analytic decay rate.
