Quantification and prediction of extreme events in a one-dimensional nonlinear dispersive wave model

Abstract

The aim of this work is the quantification and prediction of rare events characterized by extreme intensity in nonlinear waves with broad spectra. We consider a one-dimensional nonlinear model with deep-water waves dispersion relation, the Majda-McLaughlin-Tabak (MMT) model, in a dynamical regime that is characterized by broadband spectrum and strong nonlinear energy transfers during the development of intermittent events with finite-lifetime. To understand the energy transfers that occur during the development of an extreme event we perform a spatially localized analysis of the energy distribution along different wavenumbers by means of the Gabor transform. A stochastic analysis of the Gabor coefficients reveals i) the low-dimensionality of the intermittent structures, ii) the interplay between non-Gaussian statistical properties and nonlinear energy transfers between modes, as well as iii) the critical scales (or critical Gabor coefficients) where a critical amount of energy can trigger the formation of an extreme event. We analyze the unstable character of these special localized modes directly through the system equation and show that these intermittent events are due to the interplay of the system nonlinearity, the wave dispersion, and the wave dissipation which mimics wave breaking. These localized instabilities are triggered by random localizations of energy in space, created by the dispersive propagation of low-amplitude waves with random phase. Based on these properties, we design low-dimensional functionals of these Gabor coefficients that allow for the prediction of the extreme event well before the nonlinear interactions begin to occur.

Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves

“…The parameter D > 0 models dissipation (Slunyaev et al 2015;Segur et al 2005), which in most cases is very small, but physically present. The right hand side term f models applied pressure (e.g., by the wind) (Bühler et al 2016;Maleewong et al 2005), wave breaking (Cousins and Sapsis 2014), and other processes.…”

“…2.2. The same ideas can be used to derive the exact, closed nonlinear Wigner equation corresponding to a wide range of nonlinear dispersive equations, including the MMT equation (Majda et al 1997;Cousins and Sapsis 2014), Ginzburg-Landau models (Aranson and Kramer 2002), Hartree equations (Athanassoulis et al 2011), and the Szegö equation (Pocovnicu 2011). In particular, the Whitham pseudodifferential operator (Whitham 1967) can be treated in the place of the Laplacian, so that fully realistic dispersion is used.…”

Freak Wave Formation from Phase Coherence

“…However, most waves will not contribute to develop the freak wave. According to [27,28,29,30,31], a freak wave can be generated locally from clustered waves. Given a time signal, we investigate parts of the signal that may generate a freak wave based on the amount of the local energy, so-called group events [26].…”

Rogue Wave Modes for the Coupled Nonlinear Schrödinger System with Three Components: A Computational Study

“…The unexpectedly large displacements of the sea surface pose threats to maritime activities and offshore structures. Intensive research efforts are conducted to understand the physics of rogue wave and to develop measures to predict or detect such waves [4,5]. Although rogue waves originate from the context of water waves [1][2][3][6][7][8][9], these studies have been extended to other physical contexts like optical fibers [10][11][12] and Bose-Einstein condensates [13].…”