We present optical fiber experiments in which we study the nonlinear propagation of partially coherent waves. We focus our attention on systems of random waves that are well described by the integrable one dimensional nonlinear Schrödinger equation. We investigate both experimentally and numerically the statistical evolution of power fluctuations of random waves in the normal and anomalous dispersion regime of the optical fiber.

Using fast detection techniques, our experiments show that nonlinear propagation strongly influences the statistics of the partially coherent waves by producing deviations from the normal distribution. In particular, in anomalous dispersion regime, the probability of emergence of extreme events is significantly enhanced whereas it is strongly reduced in the normal dispersion regime.

Moreover, these experimental results are reproduced in a quantitive way by our numerical simulations of the one dimensional nonlinear Schrödinger equation. Futhermore, we observe the emergence of coherent structures embedded in the random fluctuations which are similar to solitons on finite background in the case of the anomalous regime of the optical fiber.