  We study the emergence of dissipative localized states in phase mismatched
singly resonant optical parametric oscillators.
  These states arise in two different bistable configurations due to the
locking of fronts waves connecting the two coexisting states. In one of these
configurations the bistabiity is mediated by the coexistence of two uniform
states. Here the localized states are organized in a collapsed snaking
bifurcation structure. Moreover, these states undergo oscillatory instabilities
which lead to a breathing behavior. When the the bistability is related with
the coexistence of an uniform state and a spatially periodic pattern, localized
states are organized in a bifurcation structure similar to the standard
homoclinic snaking. Performing an exhaustive bifurcation analysis, we
characterize in detail the previous structures, their linear stability and the
modification of their dynamics as a function of the control parameters of the
system.
