Parametric localized patterns and breathers in dispersive quadratic cavities

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.