Excitation of Peregrine-Type Waveforms from Vanishing Initial Conditions in the Presence of Periodic Forcing

Abstract

We show by direct numerical simulations that spatiotemporally localized wave forms, strongly reminiscent of the Peregrine rogue wave, can be excited by vanishing initial conditions for the periodically driven nonlinear Schrödinger equation. The emergence of the Peregrine-type waveforms can be potentially justified, in terms of the existence and modulational instability of spatially homogeneous solutions of the model, and the continuous dependence of the localized initial data for small time intervals. We also comment on the persistence of the above dynamics, under the presence of small damping effects, and justify, that this behavior should be considered as far from approximations of the corresponding integrable limit.

Extreme wave events for a nonlinear Schrödinger equation with linear damping and Gaussian driving

“…This is so for certain spatiotemporal scales where the spatial width of the driver dominates over its temporal width -even far from the integrable limit. This phenomenology differs nontrivially from the one observed in [19] in the presence of a time-periodic forcing. Therein, PRW-type waveforms emerge on the top of a finite background as explained in terms of the modulation instability of continuous wave (cw) solutions of the model.…”

“…Panel (c) of Fig. 2 offers another view of the extreme event occurring at t = 1.74, highlighting a novel feature which totally differs from what was observed before: the extreme event occurs on the top of an emergent decaying support of the solution, as opposed to the uniform background of the exact PRW waveforms, or even of the PRWwaveforms observed in the presence of gain/loss, time-periodic driver, or higher-order effects [18,19,50,51]. This support appears due to the existence of the specific type of driving -see discussion below.…”

“…In the present study on Eq. (2), we continue our explorations [18,19] on the robustness of extreme wave events and rogue waves [20][21][22][23], in the presence of non-integrable perturbations of the integrable NLS (1). We recall that in this limit, the manifestation of such events is described by the class of rational solutions such as the Peregrine rogue wave (PRW) and the Akhmediev and Kuznetsov-Ma (KMb) breathers [24][25][26][27][28].…”

The linearly damped nonlinear Schrödinger equation with localized driving: spatiotemporal decay estimates and the emergence of extreme wave events

“…In our recent works [1,2], we demonstrated, mainly by direct numerical simulations, the emergence of spatiotemporally localized wave forms reminiscent of the Peregrine rogue wave (PRW) [3], for the linearly damped and driven Schrödinger (NLS) equation…”

“…Recalling some of the numerical results of [1] for the Gaussian driver, we proceed to the study of the dynamics in the presence of the weakly localized forcing. In the present study, we consider both cases, of algebraically and exponentially decaying initial conditions (2).…”