What makes the Peregrine soliton so special as a prototype of freak waves?

Abstract

The formation of breathers as prototypes of freak waves is studied within the framework of the classic 'focussing' nonlinear Schrödinger (NLS) equation. The analysis is confined to evolution of localised initial perturbations upon an otherwise uniform wave train. For a breather to emerge out of an initial hump, a certain integral over the hump, which we refer to as the "area", should exceed a certain critical value. It is shown that the breathers produced by the critical and slightly supercritical initial perturbations are described by the Peregrine soliton which represents a spatially localised breather with only one oscillation in time and thus captures the main feature of freak waves: a propensity to appear out of nowhere and disappear without trace. The maximal amplitude of the Peregrine soliton equals exactly three times the amplitude of the unperturbed uniform wave train. It is found that, independently of the proximity to criticality, all small-amplitude supercritical humps generate the Peregrine solitons to leading order. Since the criticality condition requires the spatial scale of the initially small perturbation to be very large (inversely proportional to the square root of the smallness of the hump magnitude), this allows one to predict a priori whether a freak wave could develop judging just by the presence/absence of the corresponding scales in the initial conditions. If a freak wave does develop, it will be most likely the Peregrine soliton with the peak amplitude close to three times the background level. Hence, within the framework of the one-dimensional NLS equation the Peregrine soliton describes the most likely freak-wave patterns. The prospects of applying the findings to real-world freak waves are also discussed.

Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation

“…[27], [5], [10] and references therein). Some of them relate the rogue wave appearance to the development of modulational instability of the plane wave due to small perturbations (see, e.g., [30], [1], [17]) or large-scale initial modulations [34], [18]. Other proposed mechanisms involve interactions of individual solitons [14], [32] or interaction of solitons with the plane wave [41].…”

The Hydrodynamic Nonlinear Schrödinger Equation: Space and Time

“…The validity of the NLS has been experimentally confirmed even in the modelling of extreme localisations, beyond its well-known asymptotic limitations [4][5][6][7][8], and due to its interdisciplinary character analogies being able to be built into other nonlinear dispersive media, such as in optics [9], a research field in which several NLS applications have found strong interest [9][10][11][12][13][14]. Furthermore, the NLS admits basic models for the description of oceanic extreme events known as breathers [15][16][17]. Indeed, the family of Akhmediev breathers (ABs) [18] and Peregrine breathers [18,19] are strongly connected to the modulation instability (MI), also known as Benjamin-Feir instability [20], of Stokes waves [21].…”

Rogue waves for a system of coupled derivative nonlinear Schrödinger equations

“…Wave profiles and dynamics are discussed in Section IV, where special attention is paid to the possibility of reaching higher than normal RW amplification ratio. A gauge transformation connecting the present system 6 and other models studied in the literature is presented in Section V, together with a discussion on applications to optics and plasma physics. Conclusions are drawn in Section VI.…”