  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.
