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The data recorded in optical fiber [1] and in hydrodynamic [2] experimentsreported the pioneering observation of nonlinear waves with spatiotemporallocalization similar to the Peregrine soliton are examined by using nonlinearspectral analysis. Our approach is based on the integrable nature of theone-dimensional focusing nonlinear Schrodinger equation (1D-NLSE) that governsat leading order the propagation of the optical and hydrodynamic waves in thetwo experiments. Nonlinear spectral analysis provides certain spectralportraits of the analyzed structures that are composed of bands lying in thecomplex plane. The spectral portraits can be interpreted within the frameworkof the so-called finite gap theory (or periodic inverse scattering transform).In particular, the number N of bands composing the nonlinear spectrumdetermines the genus g = N - 1 of the solution that can be viewed as a measureof complexity of the space-time evolution of the considered solution. Withinthis 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 wellapproximated by the Peregrine solitons, nonlinear spectral analysis of thebreathers observed both in the optical fiber and in the water tank experimentsreveals that they exhibit spectral portraits associated with more general,genus 4 finite-gap NLSE solutions. Moreover, the nonlinear spectral analysisshows that the nonlinear spectrum of the breathers observed in the experimentsslowly changes with the propagation distance, thus confirming the influence ofunavoidable perturbative higher order effects or dissipation in theexperiments.

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