  We consider the tunneling of localized excitations (many boson bound states)
in the presence of a bosonic bath. We show both analytically and numerically
that the bath influence results in a dramatical enhancement of the amplitude of
the excitation tunneling. The order of the bosonic flow in the course of the
tunneling process is obtained. On the background of the giant tunneling
enhancement we observe and describe additional resonant enhancement and {\sl
suppression} of tunneling due to avoided level crossings.
