Nonlinear stage of Benjamin-Feir instability in forced/damped deep water waves

摘要

We study a three-wave truncation of a recently proposed damped/forcedhigh-order nonlinear Schr\"odinger equation for deep-water gravity waves underthe effect of wind and viscosity. The evolution of the norm (wave-action) andspectral mean of the full model are well captured by the reduced dynamics.Three regimes are found for the wind-viscosity balance: we classify themaccording to the attractor in the phase-plane of the truncated system and tothe shift of the spectral mean. A downshift can coexist with both net forcingand damping, i.e., attraction to period-1 or period-2 solutions. Upshift isassociated with stronger winds, i.e., to a net forcing where the attractor isalways a period-1 solution. The applicability of our classification toexperiments in long wave-tanks is verified.