  X-waves are an example of a localized wave packet solution of the homogeneous
wave equation, and can potentially arise in any area of physics relating to
wave phenomena, such as acoustics, electromagnetism, or quantum mechanics. They
have been predicted in condensed matter systems such as atomic Bose-Einstein
condensates in optical lattices, and were recently observed in
exciton-polariton condensates. Here we show that polariton X-waves result from
an interference between two separating wave packets that arise from the
combination of a locally hyperbolic dispersion relation and nonlinear
interactions. We show that similar X-wave structures could also be observed in
expanding spin-orbit coupled Bose-Einstein condensates.
