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

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 λ 0 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.B A. G. Athanassoulis

Strong solutions for the Alber equation and stability of unidirectional wave spectra

“…In fact, the lack of a dramatic bifurcation was seen as a challenge for the validity Alber equation in the aformentioned works. However our proof here (and the heuristic results of [4] for the unstable case) show that indeed the Alber equation only predicts a gradual transition.…”

“…There is broad agreement with [13,29], but we find somewhat fewer unstable sea states. Modulationally unstable sea states are the prime suspects for rogue waves [8,4,10,11,14,25], and we find that such sea states are very unlikely but nevertheless they do exist, with an estimated total likelihood of 2 ¤ 10 ¡3 . This is broadly consistent with the record of observations of rogue waves.…”

“…On the other hand, in the unstable case a very slow rate of growth would make the instability irrelevant; moreover, a very small bandwidth of unstable wavenumbers X would make the resulting extreme events supported over unrealistically large regions (e.g. thousands of wavelengths) [4]; but there are no energy transport mechanisms to support such events. In other words, to really observe the modulation instability a fast enough rate of growth and a large enough bandwidth of unstable wavenumbers are required.…”

“…This would be the "Landau damping" / stable regime. However, in those exceptional cases where the inhomogeneity wpx, k, tq is allowed to feed on the (infinite) energy of the power spectrum and grow significantly, then localized extreme events such as Rogue Waves become possible [4,8,10,11,13,14,23,25,29]. This would be the "modulation instability" (MI) / unstable regime, and it can be thought of as a generalization of the standard MI of the NLS [7,37,38] to continuous spectra.…”

A Unifying Framework for Describing Rogue Waves

“…Because the instanton calculus proposed in this paper uses as limiting parameter the maximal wave amplitude itself, without condition on model parameters or regimes in the NLSE, it allows us to assess the validity of the quasideterministic and semiclassical theories by comparing them to the results of our approach in appropriate regimes. Our approach could also be useful in the context of other nonlinear theories for rogue waves based on NLSE, like statistical approaches based on the Alber and the Wigner equations [21][22][23][24][25][26]. We also stress that the method proposed here can be generalized to the full two-dimensional setting, as well as other relevant physical systems where an understanding of extreme events is important [27,28] but made challenging by the complexity of the models involved combined with the stochasticity of their evolution and the uncertainty of their parameters [27,[29][30][31][32].…”