  Solitons are observed to form in simulations of dark matter (DM) halos
consisting of bosonic fields. We use the extended Press-Schechter formalism to
compute the mass function of solitons, assuming various forms for the
relationship between halo mass and soliton mass. We further provide a new
calculation of the rate of soliton major mergers. Solitons composed of axion DM
are unstable above a critical mass, and decay to either relativistic axions or
photons, depending on the values of the coupling constants. We use the computed
soliton major merger rate to predict the enhanced DM decay rate due to soliton
instability. For certain values of currently allowed axion parameters, the
energy injection into the intergalactic medium from soliton decays to photons
is comparable to or larger than the energy injection due to core collapse
supernovae at $z>10$. A companion paper explores the phenomenology of such an
energy injection.
