Early stage of integrable turbulence in the one-dimensional nonlinear Schrödinger equation: a semiclassical approach to statistics

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@article{ba4c5a81ff1f4f778d9cad068e6dddc4,
title = "Early stage of integrable turbulence in the one-dimensional nonlinear Schr{\"o}dinger equation: a semiclassical approach to statistics",
abstract = "We examine statistical properties of integrable turbulence in the defocusing and focusing regimes of one-dimensional small-dispersion nonlinear Schr{\"o}dinger 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 semiclassical 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 (respectively, low) tails of the statistical distribution occurring in the focusing (respectively, defocusing) regime of 1D-NLSE.",
author = "Giacomo Roberti and Gennady El and St{\'e}phane Randoux and Pierre Suret",
year = "2019",
month = sep,
day = "19",
doi = "10.1103/PhysRevE.100.032212",
language = "English",
volume = "100",
journal = "Physical review. E",
issn = "2470-0045",
publisher = "American Physical Society",
number = "3",

}

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TY - JOUR

T1 - Early stage of integrable turbulence in the one-dimensional nonlinear Schrödinger equation

T2 - a semiclassical approach to statistics

AU - Roberti, Giacomo

AU - El, Gennady

AU - Randoux, Stéphane

AU - Suret, Pierre

PY - 2019/9/19

Y1 - 2019/9/19

N2 - We examine statistical properties of integrable turbulence in the defocusing and focusing regimes of one-dimensional small-dispersion nonlinear Schrödinger 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 semiclassical 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 (respectively, low) tails of the statistical distribution occurring in the focusing (respectively, defocusing) regime of 1D-NLSE.

AB - We examine statistical properties of integrable turbulence in the defocusing and focusing regimes of one-dimensional small-dispersion nonlinear Schrödinger 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 semiclassical 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 (respectively, low) tails of the statistical distribution occurring in the focusing (respectively, defocusing) regime of 1D-NLSE.

UR - http://www.scopus.com/inward/record.url?scp=85072985098&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.100.032212

DO - 10.1103/PhysRevE.100.032212

M3 - Article

VL - 100

JO - Physical review. E

JF - Physical review. E

SN - 2470-0045

IS - 3

M1 - 032212

ER -