  (abridged) Vortices are believed to play a role in the formation of km-sized
planetesimals. However, vortex dynamics is commonly studied in
non-self-gravitating discs. The main goal here is to examine the effects of
disc self-gravity on vortex dynamics. For this purpose, we employ the 2D
self-gravitating shearing sheet approximation. A simple cooling law with a
constant cooling time is adopted, such that the disc settles down into a
quasi-steady gravitoturbulent state. In this state, vortices appear as
transient structures undergoing recurring phases of formation, growth to sizes
comparable to a local Jeans scale and eventual shearing and destruction due to
the combined effects of self-gravity and background Keplerian shear. Each phase
typically lasts about 2 orbital periods or less. As a result, in
self-gravitating discs the overall dynamical picture of vortex evolution is
irregular consisting of many transient vortices at different evolutionary
stages and, therefore, with various sizes up to the local Jeans scale. Vortices
generate density waves during evolution, which turn into shocks. Therefore, the
dynamics of density waves and vortices are coupled implying that, in general,
one should consider both vortex and spiral density wave modes in order to get a
proper understanding of self-gravitating disc dynamics. Our results suggest
that given such an irregular and rapidly varying character of vortex evolution
in self-gravitating discs, it may be difficult for such vortices to effectively
trap dust particles. Further study of the behaviour of dust particles embedded
in a self-gravitating gaseous disc is, however, required to strengthen this
conclusion.
