  A quasi-one-dimensional Bose-Einstein condensate loaded into a quasi-periodic
potential created by two sub-lattices of comparable amplitudes and
incommensurate periods is considered. Although the conventional tight-binding
approximation is not applicable in this setting, the description can still be
reduced to a discrete model that accounts for the modes below the mobility
edge. In the respective discrete lattice, where no linear hopping exists,
solutions and their dynamics are governed solely by nonlinear interactions.
Families of nonlinear modes, including those with no linear limit, are
described with a special focus on dimers, which correspond to breather
solutions of the Gross-Pitaevskii equation with a quasi-periodic potential. The
breathers are found to be stable for negative scattering lengths. Localization
and stable propagation of breathers are also observed for positive scattering
lengths at relatively weak and moderate nonlinearities.
