  Ringed structures have been observed in a variety of protoplanetary discs.
Among the processes that might be able to generate such features, the Secular
Gravitational Instability (SGI) is a possible candidate. It has also been
proposed that the SGI might lead to the formation of planetesimals during the
non-linear phase of the instability. In this context, we employ two-fluid
hydrodynamical simulations with self-gravity to study the non-axisymmetric,
non-linear evolution of ringed perturbations that grow under the action of the
SGI. We find that the non-linear evolution outcome of the SGI depends mainly on
the initial linear growth rate. For SGI growth rates smaller than typically
$\sigma \gtrsim 10^{-4}-10^{-5}\Omega$, dissipation resulting from dust
feedback introduces a $m=1$ spiral wave in the gas, even for Toomre gas
stability parameters $Q_g>2$ for which non-axisymmetric instabilities appear in
a purely gaseous disc. This one-armed spiral subsequently traps dust particles
until a dust-to-gas ratio $\epsilon \sim 1$ is achieved. For higher linear
growth rates, the dust ring is found to undergo gravitational collapse until
the bump in the surface density profile becomes strong enough to trigger the
formation of dusty vortices through the Rossby Wave Instability (RWI).
Enhancements in dust density resulting from this process are found to scale
with the linear growth rate, and can be such that the dust density is higher
than the Roche density, leading to the formation of bound clumps. Fragmentation
of axisymmetric rings produced by the SGI might therefore appear as a possible
process for the formation of planetesimals.
