Heteroclinic-structure transition of the pure quartic modulation instability

TL;DR

This paper investigates the modulation instability in pure-quartic systems, revealing heteroclinic-structure transitions at critical frequencies that significantly alter the dynamics of nonlinear waves, including FPU recurrences and Akhmediev breathers.

Contribution

It uncovers the heteroclinic-structure transitions in pure-quartic systems and their impact on nonlinear wave behaviors, providing new insights into the dynamics of modulation instability.

Findings

Identification of two critical frequencies {} c1 and {} c2 for heteroclinic-structure transitions.

Discovery of various FPU recurrences and Akhmediev breathers in the system.

Demonstration of pattern switching in wave behaviors across critical frequencies.

Abstract

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.