Adiabatic Invariance of Oscillons/I-balls
  Real scalar fields are known to fragment into spatially localized and
long-lived solitons called oscillons or $I$-balls. We prove the adiabatic
invariance of the oscillons/$I$-balls for a potential that allows periodic
motion even in the presence of non-negligible spatial gradient energy. We show
that such potential is uniquely determined to be the quadratic one with a
logarithmic correction, for which the oscillons/$I$-balls are absolutely
stable. For slightly different forms of the scalar potential dominated by the
quadratic one, the oscillons/$I$-balls are only quasi-stable, because the
adiabatic charge is only approximately conserved. We check the conservation of
the adiabatic charge of the $I$-balls in numerical simulation by slowly varying
the coefficient of logarithmic corrections. This unambiguously shows that the
longevity of oscillons/$I$-balls is due to the adiabatic invariance.
