We examine the general question of statistical changes experienced byensembles of nonlinear random waves propagating in systems ruled by integrableequations. In our study that enters within the framework of integrableturbulence, we specifically focus on optical fiber systems accurately describedby the integrable one-dimensional nonlinear Schr\"odinger equation. We considerrandom complex fields having a gaussian statistics and an infinite extension atinitial stage. We use numerical simulations with periodic boundary conditionsand optical fiber experiments to investigate spectral and statistical changesexperienced by nonlinear waves in focusing and in defocusing propagationregimes. As a result of nonlinear propagation, the power spectrum of the randomwave broadens and takes exponential wings both in focusing and in defocusingregimes. Heavy-tailed deviations from gaussian statistics are observed infocusing regime while low-tailed deviations from gaussian statistics areobserved in defocusing regime. After some transient evolution, the wave systemis found to exhibit a statistically stationary state in which neither theprobability density function of the wave field nor the spectrum change with theevolution variable. Separating fluctuations of small scale from fluctuations oflarge scale both in focusing and defocusing regime, we reveal the phenomenon ofintermittency; i.e., small scales are characterized by large heavy-taileddeviations from Gaussian statistics, while the large ones are almost Gaussian.
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