Akhmediev breathers and Peregrine solitary waves in a quadratic medium

I investigate the formation of optical localized nonlinear structures, evolving upon a non-zero background plane wave in a dispersive quadratic medium. I show the existence of quadratic Akhmediev breathers and Peregrine solitary waves in the regime of cascading second-harmonic generation. This finding opens a novel path for the excitation of extreme rogue waves and for the description of modulation instability in quadratic nonlinear optics.