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

Agissilaos Athanassoulis (Lead / Corresponding author), Gerassimos Athanassoulis, Themistoklis Sapsis

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6 Citations (Scopus)
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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 Ppkq, 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 Ppkq. 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
Original languageEnglish
Pages (from-to)353-372
Number of pages20
JournalJournal of Ocean Engineering and Marine Energy
Volume3
Issue number4
Early online date10 Aug 2017
DOIs
Publication statusPublished - Nov 2017

Keywords

  • Rogue Waves
  • Wigner equation
  • Nonlinear Schrodinger Equation
  • Penrose modes
  • Penrose condition

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