  We investigate the dynamics of large dust grains in massive lopsided
transition discs via 2D hydrodynamical simulations including both gas and dust.
Our simulations adopt a ring-like gas density profile that becomes unstable
against the Rossby-wave instability and forms a large crescent-shaped vortex.
When gas self-gravity is discarded, but the indirect force from the
displacement of the star by the vortex is included, we confirm that dust grains
with stopping times of order the orbital time, which should be typically a few
centimetres in size, are trapped ahead of the vortex in the azimuthal
direction, while the smallest and largest grains concentrate towards the vortex
centre. We obtain maximum shift angles of about 25 degrees. Gas self-gravity
accentuates the concentration differences between small and large grains. At
low to moderate disc masses, the larger the grains, the farther they are
trapped ahead of the vortex. Shift angles up to 90 degrees are reached for 10
cm-sized grains, and we show that such large offsets can produce a
double-peaked continuum emission observable at mm/cm wavelengths. This
behaviour comes about because the large grains undergo horseshoe U-turns
relative to the vortex due to the vortex's gravity. At large disc masses, since
the vortex's pattern frequency becomes increasingly slower than Keplerian,
small grains concentrate slightly beyond the vortex and large grains form
generally non-axisymmetric ring-like structures around the vortex's radial
location. Gas self-gravity therefore imparts distinct trapping locations for
small and large dust grains which may be probed by current and future
observations, and which suggest that the formation of planetesimals in vortices
might be more difficult than previously thought.
