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

TL;DR

The study advances the understanding of rogue-wave dynamics in higher-order NLS systems by leveraging Floquet spectral analysis to distinguish soliton-like rogue waves (SRWs) from diffuse events across three models: HONLS, V-HONLS, and NLD-HONLS. By classifying SRWs via spectral band contraction |γ| < 0.025 together with a strength threshold S ≥ 2.2 and tracking phase coherence through PVD, the authors reveal a strong dependence on the dissipation mechanism: nonlinear mean-flow damping (NLD-HONLS) promotes highly coherent, long-lived SRWs and a close link between SRWs and permanent downshift, while viscous damping (V-HONLS) yields a more disordered background with fewer SRWs and a slower, decoupled downshift; the conservative HONLS exhibits intermittent SRWs within a fluctuating multi-mode background. Across two steepness classes of initial data, the Floquet framework uncovers a spectrum-based pathway to pre–wave-breaking coherence in NLD-HONLS, contrasting with the diffusion-dominated dynamics in HONLS and V-HONLS. These results establish spectral diagnostics as powerful tools for predicting the onset and character of extreme events in near-integrable wave systems and elucidate how damping type shapes the interplay between rogue waves and frequency downshifting.

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). 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.

Figures (23)

• Figure 1: $|u(x,t)|$ (solid line) versus $\left|\beta u {\cal H}\left(|u|^2_x\right)\right|$ (dashed red line) at $t = t^*$. The nonlinear damping term is steepness dependent and is effective only near the crest of the envelope.
• Figure 2: HONLS evolution with SPB initial data (\ref{['SPB_ic']}): (a) $|u(x,t)|$ for $0\leq t\leq 200$. (b) the strength $S(t)$ with the dashed red line indicating the threshold for a rogue wave; (c) the phase variance PVD$(t)$ with red dots marking rogue wave events.
• Figure 3: HONLS evolution with SPB initial data (\ref{['SPB_ic']}): Floquet spectra at (a) $t = 0$, (b ) $t = 0.5$ (where the characteristic asymmetric spectrum is clearly visible), (c) $t = 1.5$ (one-mode soliton-like state), and (d) $t = 3.5$ (two-mode SRW event).
• Figure 4: HONLS evolution with SPB initial data (\ref{['SPB_ic']}): Floquet spectra at (a) $t= 3.9$, (b) $t=4.0$, (c)$t = 4.1$, (d) = $t = 5.5$, (e) $t = 32.8$, (f) $t=39.4$, (g) $t = 39.5$, (h) $t = 39.6$, and (i) the band lengths $|\gamma_1(t)|$ and $|\gamma_2(t)|$ with the horizontal line indicating the soliton-like threshold and red dots marking rogue wave events.
• Figure 5: HONLS evolution ($\Gamma = 0$) and V-HONLS evolutions for varying damping parameter $\Gamma$: Timelines using SPB initial data showing (a) SRW and generic rogue wave events (Magenta dots: 2-mode SRWs, blue diamonds: 1-mode SRWs, and stars: generic rogue waves), and (b) critical point transitions: blue dots for real, red '$\bigtimes$' for complex; (c) average PVD for SPB (light blue) and Stokes (green) initial data.

Theorems & Definitions (1)

• Definition 3.1