Soliton‐Like Rogue Wave Dynamics in Dissipative Higher‐Order Nonlinear Schrödinger Models: A Floquet Spectral Perspective

ABSTRACT We investigate rogue wave formation and spectral downshifting in the higher‐order nonlinear Schrödinger (HONLS) equation and its dissipative extensions: the nonlinear mean‐flow damping model (NLD‐HONLS) and the viscous damping model (V‐HONLS). Using Floquet spectral analysis, we characterize (i) the organization of the underlying dynamical background and (ii) the structure of the emergent rogue waves, distinguishing soliton‐like rogue waves (SRWs)—which are sharply localized and spectrally coherent—from more diffuse rogue‐wave events that lack coherent localization. In the conservative HONLS, SRWs arise only for sufficiently steep initial data, with the evolution intermittently switching between SRW formation and disordered multi‐mode dynamics, while moderately steep initial data produce only broader, less coherent rogue waves. Nonlinear mean‐flow damping in the NLD‐HONLS model suppresses disorder and promotes a sustained, well‐organized Floquet spectrum that supports persistent soliton‐like states from which SRWs emerge. In contrast, viscous damping in the V‐HONLS model generates a disordered Floquet spectral evolution, broader rogue wave events, and enhanced phase variability. Furthermore, the NLD‐HONLS exhibits a close link between rogue wave events and the time of permanent spectral downshift, whereas these phenomena appear decoupled in the V‐HONLS model. These results clarify how dissipation type and wave steepness interact to regulate coherence, rogue wave formation, and spectral downshifting in near‐integrable wave systems. Moreover, the persistent coherent states identified in the NLD‐HONLS align with the focused, highly organized wave groups observed immediately prior to wave breaking in laboratory experiments, providing a Floquet spectral framework that is conceptually relevant to pre‐breaking wave dynamics.