# Floquet Analysis of Kuznetsov--Ma breathers: A Path Towards Spectral Stability of Rogue Waves

## TL;DR

This paper investigates the spectral stability of Peregrine solitons by analyzing the Floquet multipliers of their periodic generalizations, the Kuznetsov--Ma breathers, to infer stability properties of rogue waves.

## Contribution

It introduces a Floquet analysis approach to study the spectral stability of Peregrine solitons via their periodic counterparts, the Kuznetsov--Ma breathers.

## Key findings

- Multiple unstable modes are enhanced near the Peregrine limit.
- No new unstable eigenmodes appear as the limit is approached.
- Numerical simulations support the spectral stability analysis.

## Abstract

In the present work, we aim at taking a step towards the spectral stability analysis of Peregrine solitons, i.e., wave structures that are used to emulate extreme wave events. Given the space-time localized nature of Peregrine solitons, this is a priori a non-trivial task. Our main tool in this effort will be the study of the spectral stability of the periodic generalization of the Peregrine soliton in the evolution variable, namely the Kuznetsov--Ma breather. Given the periodic structure of the latter, we compute the corresponding Floquet multipliers, and examine them in the limit where the period of the orbit tends to infinity. This way, we extrapolate towards the stability of the limiting structure, namely the Peregrine soliton. We find that multiple unstable modes of the background are enhanced, yet no additional unstable eigenmodes arise as the Peregrine limit is approached. We explore the instability evolution also in direct numerical simulations.