Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves
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Athanassoulis2017_Article_LocalizedInstabilitiesOfTheWig.pdf
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Author(s)
Athanassoulis, A. G.
Athanassoulis, G. A.
Sapsis, Themistoklis P.
Date Issued
August 2017
Journal
Journal of Ocean Engineering and Marine Energy
Publisher
Springer
Citation
Athanassoulis, A. G. et al. “Localized Instabilities of the Wigner Equation as a Model for the Emergence of Rogue Waves.” Journal of Ocean Engineering and Marine Energy 3, 4 (August 2017): 353–372 © 2017 The Author(s)
Version
Final published version
Abstract
In this paper, we model Rogue Waves as localized instabilities emerging from homogeneous and stationary background wavefields, under NLS dynamics. This is achieved in two steps: given any background Fourier spectrum P(k), we use the Wigner transform and Penrose’s method to recover spatially periodic unstable modes, which we call unstable Penrose modes. These can be seen as generalized Benjamin–Feir modes, and their parameters are obtained by resolving the Penrose condition, a system of nonlinear equations involving P(k). Moreover, we show how the superposition of unstable Penrose modes can result in the appearance of localized unstable modes. By interpreting the appearance of an unstable mode localized in an area not larger than a reference wavelength λ0 as the emergence of a Rogue Wave, a criterion for the emergence of Rogue Waves is formulated. Our methodology is applied to δ spectra, where the standard Benjamin–Feir instability is recovered, and to more general spectra. In that context, we present a scheme for the numerical resolution of the Penrose condition and estimate the sharpest possible localization of unstable modes. Keywords: Rogue Waves; Wigner equation; Nonlinear Schrodinger equation; Penrose modes; Penrose condition
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
Terms of Use
Creative Commons Attribution 4.0 International license
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DOI of Published Version
https://doi.org/10.1007/S40722-017-0095-5
