Abstract

It has long been known that plane wave solutions of the cubic NLS are linearly unstable. This fact is widely known as modulation instability (MI), and sometimes referred to as Benjamin-Feir instability in the context of water waves. While instances of MI have been recreated in wave tanks, the relevance of plane wave solutions in realistic ocean waves is seen as debatable. In this talk we will present recent results about the Alber equation, which predict a stable and unstable regime for sea states of known power spectra, with the classical MI as the limit in the case of asymptotically narrow spectra. Crucially, in contrast to plane wave solutions, realistic sea states may be stable or unstable, i.e. a bifurcation exists with no counterpart in the classical theory of MI. In the unstable case, there are clear analogies with the MI, hence this regime can be called generalised MI. The stable regime, where the vast majority of realistic spectra belong, is mathematically extremely similar to Landau damping. Finally, a new numerical investigation of the bifurcation from Landau damping to generalised MI in JONSWAP sea states is presented, inspired by the analysis. In particular, the (in)stability of inhomogeneities is for the first time tracked directly.  Key findings include the fact that nonlinear evolution is surprisingly analogous to that of the classical MI, and allows to make more precise the relationship between generalised MI and rogue waves. Includes work joint with G. Athanassoulis (NTUA), T. Sapsis (MIT), M. Ptashnyk (Heriot-Watt) and O. Gramstad (DNV).