Nonlinear spiral density waves - An inviscid theory

It is pointed out that the theory of spiral density waves, invented to explain the spiral structure of disk galaxies, has also been found useful for the study of planetary rings. The linear theory is by now well developed, while the nonlinear theory is less complete. Analytical calculations which include self-gravitation have, so far, obtained results only in the slightly nonlinear regime, or have concentrated on partial effects which are not of primary importance to the physical problem at hand. In the present paper, it is attempted to remedy these shortcomings. The simplest asymptotic ordering which can still yield useful results is adopted. Attention is given to the reduction to a nonlinear integral equation in a single variable, the use of the Wentzel-Kramers-Brillouin-Jeffreys theory, and the replacement of an equation by another which is easier to handle numerically.