  We examine integrable turbulence (IT) in the framework of the defocusing
cubic one-dimensional nonlinear Schr\"{o}dinger equation. This is done
theoretically and experimentally, by realizing an optical fiber experiment in
which the defocusing Kerr nonlinearity strongly dominates linear dispersive
effects. Using a dispersive-hydrodynamic approach, we show that the development
of IT can be divided into two distinct stages, the initial, pre-breaking stage
being described by a system of interacting random Riemann waves. We explain the
low-tailed statistics of the wave intensity in IT and show that the Riemann
invariants of the asymptotic nonlinear geometric optics system represent the
observable quantities that provide new insight into statistical features of the
initial stage of the IT development by exhibiting stationary probability
density functions.
