  We study the formation of localized patterns arising in doubly resonant
dispersive optical parametric oscillators. They form through the locking of
fronts connecting a continuous-wave and a Turing pattern state. This type of
localized pattern can be seen as a slug of the pattern embedded in a
homogeneous surrounding. They are organized in terms of a homoclinic snaking
bifurcation structure, which is preserved under the modification of the control
parameter of the system. We show that, in the presence of phase mismatch,
localized patterns can undergo oscillatory instabilities which make them
breathe in a complex manner.
