# Early stage of integrable turbulence in 1D NLS equation: the semi-classical approach to statistics

## TL;DR

This paper investigates the early statistical evolution of integrable turbulence in the 1D nonlinear Schrödinger equation using a semi-classical approach, deriving explicit formulas and validating them with numerical simulations.

## Contribution

It introduces a semi-classical method to analytically describe early-stage statistical properties of 1D-NLSE turbulence, focusing on the evolution of the fourth moment.

## Key findings

- Excellent agreement between analytical formulas and numerical simulations.
- Identification of heavy tails in focusing regime and low tails in defocusing regime.
- Explicit formula for early evolution of wave amplitude statistics.

## Abstract

We examine statistical properties of integrable turbulence in the defocusing and focusing regimes of one-dimensional small-dispersion nonlinear Schrodinger equation (1D-NLSE). Specifically, we study the 1D-NLSE evolution of partially coherent waves having Gaussian statistics at time t=0. Using short time asymptotic expansions and taking advantage of the scale separation in the semi-classical regime we obtain a simple explicit formula describing an early stage of the evolution of the fourth moment of the random wave field amplitude, a quantitative measure of the "tailedness" of the probability density function. Our results show excellent agreement with numerical simulations of the full 1D-NLSE random field dynamics and provide insight into the emergence of the well-known phenomenon of heavy (resp. low) tails of the statistical distribution emerging in the focusing (resp. defocusing) regime of 1D-NLSE.