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

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.