  A hidden symmetry of the nonlinear wave equation is exploited to analyse the
propagation of paraxial and uniform atom-laser beams in time-independent,
quadratic and cylindrical potentials varying smoothly along the propagation
axis. The quality factor and the paraxial ABCD formalism are generalized to
account exactly for mean-field interaction effects in such beams. Using an
approach based on moments, these theoretical tools provide a very simple and
yet exact picture of the interacting beam profile evolution. Guided atom laser
experiments are discussed. This treatment addresses simultaneously optical and
atomic beams in a unified manner, exploiting the formal analogy between
nonlinear optics and nonlinear paraxial atom optics.
