Characteristic triangular spectra of extreme localised structures: Insight from optics into rogue wave early warning

The field of optical rogue wave physics began in 2007 and has since become a major international research effort involving many international groups [1]. A central challenge in ongoing studies in the field is to determine whether the emergence of extreme localized states is associated with particular early-time field properties that can be used as an early-warning indicator. In an optical context such indicators are important for identifying regimes of noise-driven dynamics in supercontinuum generation whilst, in the oceanic context, there are important consequences for developing wave forecasting and early-warning methods. Some progress has already been made in this regard, and when considering modulation instability seeded by weak perturbations, the presence of particular modulation frequencies in the input field has been shown to be strongly correlated with an increased likelihood of rogue wave emergence [2]. In this paper, we discuss a complementary indicator of rogue wave presence and show that information on the presence of rogue wave structures can in fact also be contained in the shape of the spectrum of the propagating field. In particular, we show that large classes of rational localized soliton solutions to the nonlinear Schrodinger equation (NLSE) are associated with a characteristic triangular spectrum when measured (as is commonly the case) on a logarithmic scale. The presence of such log-triangular structure has already been associated with Akhmediev breather localization in the early stage of supercontinuum generation [3], and the results here generalize this to a wider class of solution.