Higher-order Breathers as Quasi-rogue Waves on a Periodic Background

TL;DR

This paper explores higher-order breathers in the nonlinear Schrödinger equation, revealing that most appear as quasi-rogue waves on periodic backgrounds, with fully periodic breathers being rare and requiring precise conditions.

Contribution

It demonstrates that higher-order breathers are typically quasi-rogue waves on periodic backgrounds and identifies conditions for truly periodic breathers, highlighting their rarity.

Findings

Most higher-order breathers are quasi-rogue waves on periodic backgrounds.

Fully periodic breathers are rare and require harmonic parameter tuning.

Quasi-periodic breathers exhibit distorted side-peaks.

Abstract

We investigate higher-order breathers of the cubic nonlinear Schr\"odinger equation on an elliptic background. We find that, beyond first-order, any arbitrarily constructed breather is a single-peaked solitary wave on a disordered background. These "quasi-rogue waves" are also common on periodic backgrounds. We assume the higher-order breather is constructed out of constituent first-order breathers with commensurate periods (i.e., higher-order harmonic waves). In that case, one obtains "quasi-periodic" breathers with distorted side-peaks. Fully periodic breathers are obtained when their wavenumbers are harmonic multiples of the background and each other. They are truly rare, requiring finely-tuned parameters. Thus, on a periodic background, we arrive at the paradoxical conclusion that the apparent higher-order rogue waves are rather common, while the truly periodic breathers are exceedingly rare.