| Abstract: |
| The Alber equation has been long used as a statistical model for ocean waves. Its bifurcation between a linearly stable and linearly unstable regime have long been linked with rogue waves. However, the nature of this bifurcation in the fully nonlinear problem (as opposed to the linearized one) has been much less understood. In this work, we use numerical simulation to explore that question. First of all, we focus on the lengthscales of the problem, as too short a computational domain can artificially suppress the instability. We show that, in the stable case, there is nonlinear Landau damping in the fully nonlinear problem. Moreover, we verify that inhomogeneities do start to grow exponentially in the unstable regime -- but, for intermediate intensities, their maxima plateau at values negligible with regard to the background. Thus, the presence of linear instability does not guarantee the appearance of localized maxima. A second bifurcation is observed, when the instability becomes large enough and localized extreme events are formed. Monte Carlo investigation shows that the qualitative behavior of the inhomogeneity depends only on the background spectrum, and not on the initial inhomogeneity. |
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