Non-linear single-armed spiral density waves in nearly Keplerian discs

Single-armed, stationary density waves can exist even in discs with only weak self-gravity, provided that the rotation curve is dominated by a central mass. Such waves could play a significant role in the transport of angular momentum. By variational methods, we derive non-linear versions of the dispersion relation, angular momentum flux and propagation velocity in the tight-winding limit. The pitch angle increases with amplitude until the tight-winding approximation breaks down. By other methods, we find a series of non-linear logarithmic spirals which is exact in the limit of small disc mass and which extends to large pitch angle. These waves may be supported by low-mass protoplanetary discs, and perhaps by compact molecular discs in galactic nuclei.