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

Research output: Contribution to journalArticlepeer-review


External departments

  • Université de Lille


Original languageEnglish
Article number032212
Number of pages10
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Issue number3
Publication statusPublished - 19 Sep 2019
Publication type

Research output: Contribution to journalArticlepeer-review


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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