Twenty years of progresses in oceanic rogue waves: the role played by weakly nonlinear models

Here we discuss some of the progresses that have been made in the last 20 years in the field of oceanic rogue waves, focusing on the role played by leading order equations such as the nonlinear Schrödinger and the Korteweg-De Vries equations. For such equations, it is possible, as shown in Onorato et al. (Origin of heavy tail statistics in equations of the nonlinear Schrödinger type: an exact result, 2016. arXiv:1601.04317), to derive a very simple relation in which the variation of the third (for the KdV) and fourth (for the NLS) moment of the probability density function of the wave field can be related to the variation of the spectral bandwidth. These relations give some new perspectives on the formation of rogue waves in a random sea state.