Caustic free completion of pressureless perfect fluid and k-essence
  Both k-essence and the pressureless perfect fluid develop caustic
singularities at finite time. We further explore the connection between the two
and show that they belong to the same class of models, which admits the caustic
free completion by means of the canonical complex scalar field. Specifically,
the free massive/self-interacting complex scalar reproduces dynamics of
pressureless perfect fluid/shift-symmetric k-essence under certain initial
conditions in the limit of large mass/sharp self-interacting potential. We
elucidate a mechanism of resolving caustic singularities in the complete
picture. The collapse time is promoted to complex number. Hence, the
singularity is not developed in real time. The same conclusion holds for a
collection of collisionless particles modelled by means of the Schroedinger
equation, or ultra-light axions (generically, coherent oscillations of bosons
in the Bose--Einstein condensate state).
