Quasibreathers in the MMT model

Abstract

We report numerical detection of new type of localized structures in the frame of Majda-McLaughlin-Tabak (MMT ) model adjusted for description of essentially nonlinear gravity waves on the surface of ideal deep water. These structures -quasibreathers, or oscillating quasisolitons -can be treated as groups of freak waves closely resembling experimentally observed "Three Sisters" wave packs on the ocean surface. The MMT model has quasisolitonic solutions. Unlike NLSE solitons, MMT quasisolitons are permanently backward radiating energy, but nevertheless do exist during thousands of carrier wave periods. Quasisolitons of small amplitude are regular and stable, but large-amplitude ones demonstrate oscillations of amplitude and spectral shape. This effect can be explained by periodic formation of weak collapses, carrying out negligibly small amount of energy. We call oscillating quasisolitons "quasibreathers".

Quantification and prediction of extreme events in a one-dimensional nonlinear dispersive wave model

“…This fact holds as long as the exponent of L in (2) is positive, meaning β < 1 − α, or simply β < 1/2 using the standard value of α = 1/2. Pushkarev and Zakharov [19] use β = −3 to study extreme waves, which is small enough to ensure that this relationship between localization and energy criticality still holds. In fact, with β = −3 we have r crit (L) = L 7/2 r crit (1), so this relationship (r crit decreasing as L decreases) would presumably be even stronger than the β = 0 case we consider.…”

Emergence of coherent wave groups in deep-water random sea

“…We also note that, in order to shed light on the structures embedded in a random sea, it is necessary to follow the evolution of wave groups over large spatial and temporal scales jointly. As also other authors remark [27], this represents a challenging task for laboratory and field investigations, whereby less information is available in regard to this particular aspect of extreme-wave dynamics.…”

Mathematical Modeling of Rogue Waves, a Review of Conventional and Emerging Mathematical Methods and Solutions

“…This behavior of the solutions may be particularly compatible with the simulation of rogue wave events occurring in regions with strong counter-wind currents, such as in the Aghulas-current [62] or in the regions of the Irish sea [2], which are heavily populated by rogue events, on the passage of the warm waters of the Gulf stream when encountering the frequent low-pressure systems over the Irish sea with counter-wave winds. The quasibreathers or quasisolitons [58], have the root function similar to the Dysthe-type solutions given in (23). Zakharov and Pushkarev [58] approach the solutions in the form:…”

“…The energy of the rogue wave is transferred to the surroundings and experiences a complete zero-point state, merging completely in the background [21]. The MMT model shows also the formation of quasisolitons which appear in triple-wave packets, as modelled by Zakharov and Pushkarev [58] and differ from regular solitons in that they radiate the energy backwards towards the preceding amplitudes. This behavior of the solutions may be particularly compatible with the simulation of rogue wave events occurring in regions with strong counter-wind currents, such as in the Aghulas-current [62] or in the regions of the Irish sea [2], which are heavily populated by rogue events, on the passage of the warm waters of the Gulf stream when encountering the frequent low-pressure systems over the Irish sea with counter-wave winds.…”

“…The MMT equation is a one-dimensional nonlinear dispersion equation which was originally proposed by Majda, McLaughlin and Tabak [57]. The MMT equation gives soliton-like solutions which have been analyzed in detail by Zhakarov [58][59][60] and gives four-wave resonant interaction between waves, which, when coupled with large scale forces and small-scale damping, yields a family of solutions which exhibit direct and inverse cascades [21]. The MMT equation is given by:…”