Wave number downshift in modulated wavetrain through a nonlinear damping effect

The time evolution of a nonlinear modulated wavetrain is investigated using the Dysthe equation with an additional nonlocal nonlinear damping term. It is found that the additional damping term has no significant effect on qualitative features of the modulational instability of the Stokes wave. Numerical results show that for appropriate values of the coefficient of the damping term, it is practically effective only near the crest of envelope, and causes the wave number downshift. These results suggested that two factors are essential to cause the downshift. The first is the enough nonlinearity to produce the asymmetry in spectral distribution. The second is the nonlinear dissipation that affects especially the higher components only when the wavetrain is strongly modulated.