  The equilibrium and stability properties of a coupled two-component BEC is
studied using a variational method and the one-dimensional model of Williams
and collaborators. The variational parameters are the population fraction,
translation and scaling transformation of the condensate densities, assumed to
have a Gaussian shape. We study the equilibrium and stability properties as a
function of the strength of the laser field and the traps displacement. We find
many branches of equilibrium configurations, with a host of critical points. In
all cases, the signature of the onset of criticality is the collapse of a
normal mode which is a linear combination of the out of phase translation and
an in phase breathing oscillation of the condensate densities. Our calculations
also indicate that we have symmetry breaking effects when the traps are not
displaced
