Origin of heavy tail statistics in equations of the Nonlinear Schr\"odinger type: an exact result

TL;DR

This paper derives an exact identity linking kurtosis evolution to spectral width change in nonlinear dispersive systems, providing insights into rogue wave formation in incoherent media.

Contribution

It presents an exact relation between kurtosis and spectral width evolution in nonlinear Schrödinger-type equations, elucidating the origin of heavy tails and rogue waves.

Findings

Derived an exact identity for kurtosis and spectral width dynamics.

Confirmed the theoretical results with numerical simulations.

Provided new understanding of rogue wave formation mechanisms.

Abstract

We study the formation of extreme events in incoherent systems described by envelope equations, such as the Nonliner Schr\"odinger equation. We derive an identity that relates the evolution of the kurtosis (a measure of the relevance of the tails in a probability density function) of the wave amplitude to the rate of change of the width of the Fourier spectrum of the wave field. The result is exact for all dispersive systems characterized by a nonlinear term of the form of the one contained in the Nonlinear Schr\"odinger equation. Numerical simulations are also performed to confirm our findings. Our work sheds some light on the origin of rogue waves in incoherent dispersive nonlinear media ruled by local cubic nonlinearity.