  Stationary waves in the condensate of electron-hole pairs in the $n-p$
bilayer system are studied. The system demonstrates the transition from a
uniform (superfluid) to a nonuniform (supersolid) state. The precursor of this
transition is the appearance of the roton-type minimum in the collective mode
spectrum. Stationary waves occur in the flow of the condensate past an
obstacle. It is shown that the roton-type minimum manifests itself in a rather
complicated stationary wave pattern with several families of crests which cross
one another. It is found that the stationary wave pattern is essentially
modified under variation in the density of the condensate and under variation
in the flow velocity. It is shown that the pattern is formed in the main part
by shortwave modes in the case of a point obstacle. The contribution of
longwave modes is clearly visible in the case of a weak extended obstacle,
where the stationary wave pattern resembles the ship wave pattern.

Mesoscale and Nanoscale Physics