Rogue waves, dissipation, and downshifting

We investigate the effects of dissipation on the development of rogue waves and downshifting by adding nonlinear and linear damping terms to the one-dimensional Dysthe equation. Significantly, rogue waves do not develop after the downshifting becomes permanent. Thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves. Using the inverse spectral theory of the NLS equation, simulations of the damped Dysthe equation for sea states characterized by JONSWAP spectrum consistently show that rogue wave events are well-predicted by proximity to homoclinic data, as measured by the spectral splitting distance δδ. The cut off distance δcutoffδcutoff decreases as the strength of the damping increases, indicating that for stronger damping the JONSWAP initial data must be closer to homoclinic data for rogue waves to occur.

► Rogue waves and downshifting for 1D damped Dysthe equation. ► Rogue waves are found to not develop after the downshifting becomes permanent. ► Permanent downshifting indicates sufficient damping to prevent further rogue waves. ► Spectral diagnostics predict well JONSWAP rogue waves in the presence of damping.