Title: Floquet Analysis of Kuznetsov--Ma breathers: A Path Towards Spectral Stability of Rogue Waves,
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.