Solitary Waves in Quadratically Nonlinear Resonators

We identify two-dimensional stable and unstable bright solitary waves or localized structures in a planar resonator with a quadratically nonlinear medium driven by a field at the fundamental frequency only. These waves are extremely localized while the nonlocal interaction between the fundamental and second harmonics prevents a collapse. To a certain extent they can be regarded as residuals of coexisting hexagon patterns.