Interaction-Phase Dynamics and Spectral Organization in Damped Higher-Order Nonlinear Schrödinger Models

Abstract

We investigate the dynamical mechanisms underlying the contrasting nonlinear Floquet spectral evolutions observed in viscously damped and in nonlinear mean-flow damped higher-order nonlinear Schrödinger models. A reduced five-mode carrier–sideband truncation is derived and reformulated in amplitude–phase variables to isolate the principal interaction phases associated with the dominant four-wave interaction products. Within this framework, viscous damping acts modewise and does not contribute explicitly to the leading interaction-phase equations, whereas nonlinear mean-flow damping acts through the coupled mean-flow interactions and, in the carrier–sideband regime, generates terms of the form $-{\kappa }_j\sin \left({\psi }_j\right)$. The reduced system is used as a mechanism-identification model rather than as a quantitatively accurate approximation of the full PDE. To interpret the resulting interaction-phase evolution, we examine recurrent nonlinear Schrödinger (NLS) benchmark solutions whose modulation dynamics and Floquet spectrum are independently characterized. These benchmarks show that substantial interaction-phase evolution may coexist with recurrent modulation and invariant Floquet-band structure. Full-PDE diagnostics are then used to compare the phase dynamics within the spectral regimes previously identified for the two damped systems. Under nonlinear mean-flow damping, the interaction phases develop a persistent large-scale evolution, with the larger phase changes remaining tied to recurrent focusing events while the Floquet-band configuration persists. Under viscous damping, the large-scale phase trends change repeatedly without a systematic association with recurrent focusing, while the Floquet spectrum undergoes repeated critical-point crossings and band reconfiguration. A Fourier-energy assessment further shows that this contrast cannot be attributed simply to broader excitation of Fourier modes in the viscous system. Instead, the results are consistent with a structural distinction in how the two dissipative mechanisms act on the dominant carrier–sideband interactions.