  The development of integrated, waveguide-based atom optical devices requires
a thorough understanding of nonlinear matter-wave mixing processes in confined
geometries. This paper analyzes the stability of counterpropagating
two-component Bose-Einstein condensates in such a geometry. The steady state
field equations of this system are solved analytically, predicting a
multivalued relation between the input and output field intensities. The
spatio-temporal linear stability of these solutions is investigated
numerically, leading to the prediction of a self-oscillation threshold that can
be expressed in terms of a matter-wave analog of the Fresnel number in optics.
