Heteroclinic-structure transition of the pure quartic modulation instability

We show that, in the pure-quartic systems, modulation instability (MI) undergoes heteroclinic-structure transitions (HSTs) at two critical frequencies of {\omega} c1 and {\omega} c2 ( {\omega} c2 > {\omega} c1 ), which indicates that there are significantchanges of the spatiotemporal behavior in the system. The complicated heteroclinic structure of instability obtained by the mode truncation method reveals all possible dynamic trajectories of nonlinear waves, which allows us to discover the various types of Fermi-Pasta-Ulam (FPU) recurrences and Akhmediev breathers (ABs). When the modulational frequency satisfies {\omega} < {\omega} c2 , the heteroclinic structure encompasses two separatrixes corresponding to the ABs and the nonlinear wave with a modulated final state, which individually separate FPU recurrences into three different regions. Remarkably, crossing critical frequency {\omega} c1 , both the staggered FPU recurrences and ABs essentially switch their patterns. These HST behaviors will give vitality to the study of MI.