The random and deterministic dynamics of ‘rogue waves’ in unidirectional, deep-water wave trains

Abstract

‘Rogue,’ ‘freak’ or ‘giant’ sea waves have been reported in a wide variety of contexts and studied from many points of view in recent years. In some sense, one might agree that these waves may be ‘generic’ and the availability of an analytical theory to explain their behavior could be quite useful. Here I provide new perspective on the nonlinear wave dynamics for this problem in the particular case of unidirectional wave propagation. My results are based upon the well-known nonlinear Schroedinger equation that governs the space/time dynamics of narrow banded, deep-water wave trains to leading order in cubic nonlinearity. I use the nonlinear Fourier formulation of the inverse scattering transform (IST) to provide new insight into the behavior of ‘rogue’ waves which are here assumed to arise from the Benjamin–Feir instability. I give a physical explanation of these phenomena in terms of their deterministic and random dynamics and discuss a number of examples of simple ‘rogue wave’ behavior.