Recurrent nonlinear modulational instability via down-conversion in quadratic media

We investigate the induced modulation instability in second-harmonic generation beyond the early stage of the linearized growth of the modulation. We find a regime of recurrence (quasi-periodic conversion and back-conversion between the pump and the modulation) which is genuine of the parametric conversion process in quadratic media. Such recurrence is mainly driven by a process of non-degenerate downconversion, showing no analogy to the cascading regime which mimics the cubic (Kerr) nonlinearities. We consider two different steady states, i.e., a pure second-harmonic and a mixed fundamental/second-harmonic state. Both exhibit this dynamics, which we show to be amenable to a description in terms of reduced frequency-truncated models. The comparison with full numerical simulations of the starting model prove the validity and robustness of the reduced models in characterizing in a simple and elegant way a wide range of modulationally-unstable steady states.