  The nonlinear dynamics of a warped accretion disc is investigated in the
important case of a thin Keplerian disc with negligible viscosity and
self-gravity. A one-dimensional evolutionary equation is formally derived that
describes the primary nonlinear and dispersive effects on propagating bending
waves other than parametric instabilities. It has the form of a derivative
nonlinear Schroedinger equation with coefficients that are obtained explicitly
for a particular model of a disc. The properties of this equation are analysed
in some detail and illustrative numerical solutions are presented. The
nonlinear and dispersive effects both depend on the compressibility of the gas
through its adiabatic index Gamma. In the physically realistic case Gamma<3,
nonlinearity does not lead to the steepening of bending waves but instead
enhances their linear dispersion. In the opposite case Gamma>3, nonlinearity
leads to wave steepening and solitary waves are supported. The effects of a
small effective viscosity, which may suppress parametric instabilities, are
also considered. This analysis may provide a useful point of comparison between
theory and numerical simulations of warped accretion discs.
