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.
Highlights
Turbulence is one of the most recognizable forms of nonlinear motion that has been, and continues to be, the subject of very active research in classical fluid dynamics [1]
The definitive feature of dispersive hydrodynamics is the presence of two distinct spatiotemporal scales: the long scale specified by initial conditions and the short scale by the internal coherence length
This has been done from the perspective of dispersive hydrodynamics, a semiclassical theory of nonlinear dispersive waves exhibiting two distinct spatiotemporal scales: the long scale specified by initial conditions and the short scale by the internal coherence length [31]
Summary
Turbulence is one of the most recognizable forms of nonlinear motion that has been, and continues to be, the subject of very active research in classical (viscous) fluid dynamics [1]. The semiclassical, dispersive hydrodynamic approach describes the propagation regimes of a completely opposite nature compared to the regimes considered in the framework of wave turbulence theory This approach can be applied to the 1D-NLSE propagation if the initial scale of the fluctuations of the power of the complex field |ψ|2 are much larger than the one corresponding to the balance between nonlinearity and dispersion. The paper is organized as follows: In Sec. II, using the semiclassical approximation, we identify the initial stage of the 1D-NLSE development of partially coherent waves with the nonlinearity dominated, dispersionless regime and derive the general expression for the short-time evolution of the fourth-order moment κ4 as a power-series expansion in time t. Eq (11) can be rewritten as ρ uρx d x
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