Characterization of the emergence of rogue waves from given spectra through a Wigner equation approach

The Wigner transform can be used to derive equations directly for the evolution of the autocorrelation of the sea elevation. This has been known in the literature as the derivation of the Alber equation, and applies to envelope equations. Wigner-Alber equations have been used to characterise spectra as either stable or unstable, and to predict Fermi-Pasta-Ulam recurrent dynamics for the unstable ones. Here we show that a systematic study of Wigner equations can improve this analysis in several respects, including: (i) the incorporation of accurate dispersion and (simple) wave breaking effects; and (ii) the characterization of the space and time scales over which localized extreme events emerge. More broadly this approach can be seen as a full modulation instability analysis for any measured spectrum. This work builds upon recent joint work with G. Athanassoulis and T. Sapsis.