Parametric instability in the pure-quartic nonlinear Schrödinger equation

We study the nonlinear stage of modulation instability (MI) in the non-intergrable pure-quartic nonlinear Schrödinger equation where the fourth-order dispersion is modulated periodically. Using the three-mode truncation, we reveal the complex recurrence of parametric resonance (PR) breathers, where each recurrence is associated with two oscillation periods (PR period and internal oscillation period). The nonlinear stage of parametric instability admits the maximum energy exchange between the spectrum sidebands and central mode occurring outside the MI gain band.