  By direct numerical simulation of the time-dependent Gross-Pitaevskii
equation using the split-step Fourier spectral method we study different
aspects of the localization of a cigar-shaped interacting binary
(two-component) Bose-Einstein condensate (BEC) in a one-dimensional
bi-chromatic quasi-periodic optical-lattice potential, as used in a recent
experiment on the localization of a BEC [Roati et al., Nature 453, 895 (2008)].
We consider two types of localized states: (i) when both localized components
have a maximum of density at the origin x=0, and (ii) when the first component
has a maximum of density and the second a minimum of density at x=0. In the
non-interacting case the density profiles are symmetric around x=0. We
numerically study the breakdown of this symmetry due to inter-species and
intra-species interaction acting on the two components. Where possible, we have
compared the numerical results with a time-dependent variational analysis. We
also demonstrate the stability of the localized symmetry-broken BEC states
under small perturbation.
