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

arXiv comment10 pages

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Cited by in corpus (8)

• Bound state soliton gas dynamics underlying the noise-induced modulational instability
• Rogue waves on the double-periodic background in the focusing nonlinear Schrodinger equation
• Statistics of extreme events in integrable turbulence
• Effect of local Peregrine soliton emergence on statistics of random waves in the 1-D focusing Nonlinear Schrödinger equation
• Integrable turbulence developing from strongly nonlinear partially coherent waves
• Asymmetric integrable turbulence and rogue wave statistics for the derivative nonlinear Schrödinger equation
• Instantons, fluctuations and singularities in the supercritical stochastic nonlinear Schroedinger equation
• One-dimensional wave kinetic theory