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

Roberti, Giacomo, El, Gennady, Randoux, Stéphane and Suret, Pierre (2019) Early stage of integrable turbulence in the one-dimensional nonlinear Schrödinger equation: a semiclassical approach to statistics. Physical Review E. ISSN 2470-0045

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Abstract

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.

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