We study particle dynamics in self-gravitating gaseous discs with a simple cooling law prescription via two-dimensional simulations in the shearing sheet approximation. It is well known that structures arising in the gaseous component of the disc due to a gravitational instability can have a significant effect on the evolution of dust particles. Previous results have shown that spiral density waves can be highly efficient at collecting dust particles, creating significant local over-densities of particles. The degree of such concentrations has been shown to be dependent on two parameters: the size of the dust particles and the rate of gas cooling. We expand on these findings, including the self-gravity of dust particles, to see how these particle over-densities evolve. We use the PENCIL CODE to solve the local shearing sheet equations for gas on a fixed grid together with the equations of motion for solids coupled to the gas through an aerodynamic drag force. We find that the enhancements in the surface density of particles in spiral density wave crests can reach levels high enough to allow the solid component of the disc to collapse under its own self-gravity. This produces many gravitationally bound collections of particles within the spiral structure. The total mass contained in bound structures appears nearly independent of the cooling time, suggesting that the formation of planetesimals through dust particle trapping by self-gravitating density waves may be possible at a larger range of radii within a disc than previously thought. So, density waves due to gravitational instabilities in the early stages of star formation may provide excellent sites for the rapid formation of many large, planetesimal-sized objects.
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