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

Abstract

We investigate the dynamical mechanisms underlying the contrasting Floquet spectral evolutions observed in viscously damped and in nonlinear mean-flow damped higher-order nonlinear Schrödinger models. Motivated by the persistent organized Floquet-band structure under nonlinear mean-flow damping and the repeated Floquet band reconnection observed under viscous damping, we derive a reduced five-mode carrier-sideband truncation and reformulate the dynamics in amplitude-phase variables to isolate the dominant interaction phases associated with the principal four-wave interaction products. Within this framework, viscous damping acts primarily modewise and does not directly modify the leading interaction-phase dynamics. By contrast, nonlinear mean-flow damping contributes directly to the interaction-phase evolution through interaction-dependent dissipative corrections. In the carrier-sideband regime, these corrections generate terms of the form $-κ_j \sin(ψ_j)$, introducing dissipative feedback into the dominant carrier-sideband interaction dynamics. To interpret the resulting interaction-phase evolution, we examine recurrent finite-gap NLS benchmark solutions whose modulation dynamics are independently understood. These benchmarks show that substantial interaction-phase evolution and localized restructuring may occur even within organized quasiperiodic dynamics possessing invariant Floquet spectral structure. Numerical diagnostics show that the nonlinear mean-flow damped system exhibits persistent recurrent carrier-sideband focusing dynamics together with organized Floquet evolution despite substantial interaction-phase restructuring, whereas the viscous system exhibits progressively diffuse modulation dynamics together with repeated Floquet reconfiguration and weakening persistence of the recurrent carrier-sideband interaction structure.