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 c2 ( > ), which indicates that there are significantchanges spatiotemporal behavior system. The complicated heteroclinic structure obtained by mode truncation method reveals all possible dynamic trajectories nonlinear waves, allows us to discover various types Fermi-Pasta-Ulam (FPU) recurrences Akhmediev breathers (ABs). When modulational frequency satisfies < , encompasses separatrixes corresponding ABs wave with a modulated final state, individually separate FPU into three different regions. Remarkably, crossing both staggered essentially switch their patterns. These HST behaviors will give vitality study MI.