Instabilities and self-socillations in atomic four-wave mixing
  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. 
J. Heurich, H. Pu, M. G. Moore and P. Meystre