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Stabilizing the Benjamin-Feir instability. (English) Zbl 1120.76022

The authors show that for waves with narrow bandwidth and moderate amplitude, dissipation stabilizes the instability. Using appropriate definitions of stability, they show that the dissipation is crucial in a modulational instability like that of Benjamin and Feir. More precisely, they prove that in the presence of dissipation, growth of perturbations might be bounded by dissipation before nonlinearity comes into play. The authors demonstrate that the so-called Stokes solution is a nonlinearly stable solution of a damped version of nonlinear Schrödinger equation with periodic boundary conditions. Furthermore, they compare predictions of this model with experiments and find good agreement.

MSC:

76E17 Interfacial stability and instability in hydrodynamic stability
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76E30 Nonlinear effects in hydrodynamic stability
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