Nonlinear spectral analysis of Peregrine solitons observed in optics and in hydrodynamic experiments

Abstract

The data recorded in optical fiber and in hydrodynamic experiments reported the pioneering observation of nonlinear waves with spatiotemporal localization similar to the Peregrine soliton are examined by using nonlinear spectral analysis. Our approach is based on the integrable nature of the one-dimensional focusing nonlinear Schrödinger equation (1D-NLSE) that governs at leading order the propagation of the optical and hydrodynamic waves in the two experiments. Nonlinear spectral analysis provides certain spectral portraits of the analyzed structures that are composed of bands lying in the complex plane. The spectral portraits can be interpreted within the framework of the so-called finite gap theory (or periodic inverse scattering transform). In particular, the number N of bands composing the nonlinear spectrum determines the genus g=N-1 of the solution that can be viewed as a measure of complexity of the space-time evolution of the considered solution. Within this setting the ideal, rational Peregrine soliton represents a special, degenerate genus 2 solution. While the fitting procedures previously employed show that the experimentally observed structures are quite well approximated by the Peregrine solitons, nonlinear spectral analysis of the breathers observed both in the optical fiber and in the water tank experiments reveals that they exhibit spectral portraits associated with more general, genus 4 finite-gap NLSE solutions. Moreover, the nonlinear spectral analysis shows that the nonlinear spectrum of the breathers observed in the experiments slowly changes with the propagation distance, thus confirming the influence of unavoidable perturbative higher-order effects or dissipation in the experiments.

Rogue waves on the double-periodic background in the focusing nonlinear Schrödinger equation

“…Formation of particular rogue waves such as Akhmediev breathers, Peregrine solution, and Kuznetsov-Ma breathers have been modeled from different initial data such as local condensates [22], multi-soliton gases [23,34], and periodic perturbations [2,3]. Experimental observations of rogue waves have been confirmed both in hydrodynamical and optical laboratories [28,5]. Statistical analysis of rogue waves has recently been developed [29] (see also [2,23]).…”

From modulational instability to focusing dam breaks in water waves

“…Our experimental results are shown to be in good quantitative agreement with predictions of the 1D-NLSE semi-classical theory [54,55], confirming the robustness of the observed dynamical scenario with respect to perturbative higher-order nonlinear effects inevitably present in a water wave experiment. We have also shown that nonlinear spectral analysis [85] can be used to advantage to determine the soliton content of the generated wavepackets while also providing useful information about the local wave dynamics in terms of the number of fundamental nonlinear wave modes (the genus) comprising the observed structure at a given space-time point.…”

“…For instance, it has been shown in ref. [85] that dissipative effects occurring in a water tank produce some slow modulation of the spectral (IST) portrait of the Peregrine soliton recorded in water wave experiments reported in ref. [91].…”

“…[76] and already applied for the analysis of experimental signals in ref. [85]. In the approach used for local IST analysis, the analyzed coherent structure is isolated by truncating the wave profile over some given time interval.…”

Branched flow and caustics in nonlinear waves

“…This suggests that the structures present in the branched flow are some kind of realization of these solutions. Different studies from optics have compared peaks appearing due to randomly perturbed phases of the initial wave and found that the shapes match known solutions quite well [12,13], though their complexity is higher [34].…”