  In this work we experimentally demonstrate for the first time spontaneous
generation of two-dimensional exciton-polariton X-waves. X-waves belong to the
family of localized packets, which are capable of sustaining their shape with
no spreading even in the linear regime. This allows to keep the packet shape
and size for very low densities and very long times compared, for instance, to
soliton waves, which always necessitate a nonlinearity to compensate the
diffusion. Here we exploit the polariton nonlinearity and unique structured
dispersion, comprising both positive- and negative-mass curvatures, to trigger
an asymmetric four wave mixing in the momentum space. This ultimately enables
self-formation of a spatial X-wave front. By means of ultrafast imaging
experiments we observe the early reshaping of the initial Gaussian packet into
the X-pulse and its propagation even for vanishing small densities. This allows
us to outline the crucial effects and parameters driving the phenomena and to
tune the degree of peak superluminal propagation, which we found to be in a
good agreement with numerical simulations.
