  Single-armed, stationary density waves can propagate in very weakly
self-gravitating gas disks dominated by a central mass. Examples include
circumstellar disks of protostars and molecular disks in galactic nuclei. We
explore the linear and nonlinear dynamics of such waves. Variational methods
yield nonlinear 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. We also find a
series of nonlinear logarithmic spirals which is exact in the limit of small
disk mass and which extends to large pitch angle.
