Resonant radiation emitted by solitary waves via cascading in quadratic media

We present a systematic investigation of the resonant radiation emitted by localized soliton-like wave-packets supported by second-harmonic generation in the cascading regime. We emphasize a general mechanism which allows for the resonant radiation to grow without the need for higher-order dispersion, primarily driven by the second-harmonic component, while radiation is also shed around the fundamental-frequency component through parametric down-conversion processes. The ubiquity of such a mechanism is revealed with reference to different localized waves such as bright solitons (both fundamental and second-order), Akhmediev breathers, and dark solitons. A simple phase matching condition is put forward to account for the frequencies radiated around such solitons, which agrees well with numerical simulations performed against changes of material parameters (say, phase mismatch, dispersion ratio). The results provide explicit understanding of the mechanism of soliton radiation in quadratic nonlinear media.