Statistics of extreme events in integrable turbulence

TL;DR

This paper develops a spectral kinetic theory approach to analyze the likelihood of extreme rogue waves in integrable turbulence governed by the focusing nonlinear Schrödinger equation, linking soliton gas properties to wave statistics.

Contribution

It introduces a stochastic inverse scattering transform framework to analytically compute kurtosis in soliton gases, explaining rogue wave statistics in integrable turbulence.

Findings

Analytical kurtosis matches numerical simulations for bound state soliton gases.

The approach explains rogue wave statistics in modulational instability scenarios.

Insights into virial theorem validity from non-bound state soliton gases.

Abstract

We use the spectral kinetic theory of soliton gas to investigate the likelihood of extreme events in integrable turbulence described by the one-dimensional focusing nonlinear Schr\"odinger equation (fNLSE). This is done by invoking a stochastic interpretation of the inverse scattering transform for fNLSE and analytically evaluating the kurtosis of the emerging random nonlinear wave field in terms of the spectral density of states of the corresponding soliton gas. We then apply the general result to two fundamental scenarios of the generation of integrable turbulence: (i) the asymptotic development of the spontaneous (noise induced) modulational instability of a plane wave, and (ii) the long-time evolution of strongly nonlinear, partially coherent waves. In both cases, involving the bound state soliton gas dynamics, the analytically obtained values of the kurtosis are in perfect agreement with those inferred from direct numerical simulations of the the fNLSE, providing the long-awaited theoretical explanation of the respective rogue wave statistics. Additionally, the evolution of a particular non-bound state gas is considered providing important insights related to the validity of the so-called virial theorem.