Extensions of the Alber equation and statistical modulation instability analysis for crossing seas

About the project

It is well known that nonlinear dispersive equations exhibit modulation instability, a phenomenon linked to extreme events such as rogue waves. Modulation instability was first discovered as the linear instability of plane-wave solutions to nonlinear equations, but realistic problems demand a more flexible framework. In that context, the Alber equation was introduced as a stochastic moment system, providing a framework for the statistical analysis for the stability of a homogeneous wave-field. Alber equations have several novel features, and their rigorous study is only now reaching maturity, with several questions still open. Moreover, the classical Alber equation only covers quasi-unidirectional problems, excluding in particular crossing seas (wave systems arriving to the same area from different directions), which have recently been identified as crucial settings for the appearance of rogue waves. In this project, an extended two-dimensional Alber equation, including crossing seas, will be derived and studied.