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Quantification and prediction of extreme events in a one-dimensional nonlinear dispersive wave model
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 non- linear 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 non- linear 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 statis- tical
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.Comment: 21 pages, 14 figure
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