  Nonlinear waves in defocusing media are investigated in the framework of the
hydrodynamic description of light as a photon fluid. The observations are
interpreted in terms of an emergent curved spacetime generated by the waves
themselves, which fully determines their dynamics. The spacetime geometry
emerges naturally as a result of the nonlinear interaction between the waves
and the self-induced background flow. In particular, as observed in real
fluids, different points of the wave profile propagate at different velocities
leading to the self-steepening of the wave front and to the formation of a
shock. This phenomenon can be associated to a curvature singularity of the
emergent metric. Our analysis offers an alternative insight into the problem of
shock formation and provides a demonstration of an analogue gravity model that
goes beyond the kinematic level.
