  We consider the interaction of two-component bright-bright solitons with a
narrow potential barrier (splitter) in the framework of a system of two
Gross-Pitaevskii (nonlinear-Schr\"{o}dinger) equations modeling a binary
Bose-Einstein condensate, with self-attraction in each component and
cross-attraction between them. The objective is to study splitting of composite
solitons, which may be used in the design of two-component
soliton-interferometer schemes. We produce approximate analytic results,
assuming a weak barrier and applying the perturbation theory in the limit in
which the system is integrable and the solitary waves may be considered as
exact solitons. We do this in the case of negligible interspecies interactions,
and also when the nonlinearities are strongly asymmetric, allowing one to
neglect the self-interaction in one of the species. Then, we use systematic
simulations to study the transmissions of both components in regions outside
these approximations and, in particular, to compare numerical results with
their analytical countertparts. We concluded that there is an appreciable
parameter range where one component is almost entirely transmitted through the
barrier, while the other one is reflected. Excitation of internal vibrations in
the passing and rebounding solitons is explored too, with a conclusion that it
is weak in the regime of high-quality splitting.
