Hilbert transform view of water-wave theory

Abstract

A general rethinking of the mathematical foundations of water surface waves from the perspective of the Hilbert transform uncovers shortcomings of the standard multiple-scale approach as well as elucidates the interplay of non-local and dispersive effects. Application of the Hilbert transforms to planar and cylindrical settings allows us to deduce new weakly nonlinear models, including an alternative to Zakharov's equation and an envelope equation for cylindrical waves on deep water, as well as to highlight the crucial differences between these geometries.