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

Source abstract

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 . 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.