Soliton-like Rogue Wave Dynamics in Dissipative Higher-Order NLS Models: A Floquet Spectral Perspective

TL;DR

This paper uses Floquet spectral analysis to study rogue wave formation in dissipative higher-order nonlinear Schrödinger models, revealing how different damping mechanisms influence rogue wave structures and spectral dynamics.

Contribution

It introduces a spectral perspective to distinguish rogue wave types and shows how nonlinear and viscous damping differently affect rogue wave formation and spectral organization.

Findings

Nonlinear damping supports stable, soliton-like rogue waves with coherent spectra.

Viscous damping leads to disordered spectra and less localized rogue waves.

Rogue wave events are linked to spectral downshifting in the nonlinear damping model.

Abstract

We investigate rogue wave formation and spectral downshifting in the higher-order nonlinear Schr\"odinger (HONLS) equation and its dissipative extensions: the nonlinear mean-flow damping model (NLD-HONLS) and the viscous damping model (V-HONLS). By applying Floquet spectral analysis, we characterize i) the structural organization of the dynamical background and ii) the nature of the rogue waves that appear, distinguishing sharply localized, soliton-like structures from more diffuse, spatially extended waveforms with mixed mode characteristics. In the conservative HONLS, soliton-like rogue waves (SRWs) arise only for steep initial data, with the dynamics intermittently switching between periods of SRW formation and periods dominated by a disordered multi-mode background. For moderately steep initial data, only broader, less coherent rogue waves form. Nonlinear damping in the NLD-HONLS model suppresses disorder and supports a stable, well-organized Floquet spectra that reflects a sustained soliton-like state from which SRWs emerge, along with strong phase coherence. In contrast, viscous damping in the V-HONLS model leads to a disordered Floquet spectral evolution with broader, less localized rogue waves and increased phase variability. Furthermore, the NLD-HONLS model shows a close link between rogue wave events and the time of permanent downshift, whereas these phenomena appear decoupled in the V-HONLS model. These results clarify how dissipation type and wave steepness interact to shape extreme events in near-integrable wave systems and highlight the value of spectral diagnostics for studying nonlinear wave dynamics.