Baseband Modulation Instability as the Origin of Rogue Waves

TL;DR

This paper demonstrates that rogue waves in various nonlinear wave models originate from baseband modulation instability, which requires the presence of zero-frequency perturbations, and confirms this connection through numerical simulations.

Contribution

It establishes a direct link between rogue wave existence and baseband modulation instability across multiple nonlinear wave models.

Findings

Rogue waves occur only when baseband instability is present.

Numerical simulations confirm rogue waves are excited from weakly perturbed continuous waves.

Modulation instability leads to nonlinear periodic oscillations in the models.

Abstract

We study the existence and properties of rogue wave solutions in different nonlinear wave evolution models that are commonly used in optics and hydrodynamics. In particular, we consider Fokas-Lenells equation, the defocusing vector nolinear Schr\"odinger equation, and the long-wave-short-wave resonance equation. We show that rogue wave solutions in all of these models exist in the subset of parameters where modulation instability is present, if and only if the unstable sideband spectrum also contains cw or zero-frequency perturbations as a limiting case (baseband instability). We numerically confirm that rogue waves may only be excited from a weakly perturbed cw whenever the baseband instability is present. Conversely, modulation instability leads to nonlinear periodic oscillations.