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

TL;DR

This paper uses nonlinear spectral analysis to examine Peregrine solitons observed in optics and hydrodynamics, revealing their spectral complexity and effects of perturbations on their evolution.

Contribution

It introduces a spectral analysis approach based on the integrable NLSE framework to characterize Peregrine solitons and related structures in experimental data.

Findings

Peregrine solitons correspond to genus 2 solutions in ideal cases.

Observed breathers exhibit spectral signatures of genus 4 solutions.

Spectral spectra evolve slowly, indicating perturbative effects.

Abstract

The data recorded in optical fiber [1] and in hydrodynamic [2] 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 Schrodinger 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 employed in [1] and [2] 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.