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 lambda _0 as the emergence of a Rogue Wave, a criterion for the emergence of Rogue Waves is formulated. Our methodology is applied to delta 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.
Highlights
Rogue Waves in the ocean are often defined as waves larger than twice the significant wave height, 2Hs, loosely speaking waves “much larger than the waves around them”
Many authors point out that Benjamin–Feir instabilities of focusing nonlinear waves must play a key role in the formation of Rogue Waves (Chabchoub et al 2015; Kibler et al 2015; Onorato et al 2013b)
We study the Wigner transform of the nonlinear Schrödinger equation, and carry out a linear stability analysis of the resulting exact, nonlinear Wigner equation in phase space
Summary
Rogue Waves in the ocean are often defined as waves larger than twice the significant wave height, 2Hs, loosely speaking waves “much larger than the waves around them”. A key novelty of our analysis is that it can be applied to any background Fourier spectrum, not just plane waves. Another key point is that we quantify how spatially-periodic Benjamin– Feir-type modes can combine to yield persistently localized. It must be mentioned that our approach can be applied to a large family of nonlinear pseudodifferential equations (including in particular realistic Whitham operators). This is due to the pseudodifferential calculus available for the Wigner transform (Athanassoulis et al 2011; Gérard et al 1997)
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