Competition between attractive and repulsive interactions in two-component Bose-Einstein condensates trapped in an optical lattice

We consider effects of inter-species attraction on two-component gap solitons (GSs) in the binary BEC with intra-species repulsion, trapped in the one-dimensional optical lattice (OL). Systematic simulations of the coupled Gross-Pitaevskii equations (GPEs) corroborate an assumption that, because the effective mass of GSs is negative, the inter-species attraction may \emph{split} the two-component soliton. Two critical values, $\kappa_{1}$ and $\kappa_{2}$, of the OL strength ($\kappa$) are identified. Two-species GSs with fully overlapping wave functions are stable in strong lattices ($\kappa >\kappa_{1}$). In an intermediate region, $\kappa_{1}>\kappa >\kappa_{2}$, the soliton splits into a double-humped state with separated components. Finally, in weak lattices ($\kappa <\kappa_{2}$%), the splitting generates a pair of freely moving single-species GSs. We present and explain the dependence of $\kappa_{1}$ and $\kappa_{2}$ on thenumber of atoms (total norm), and on the relative strength of the competing inter-species attraction and intra-species repulsion. The splitting of asymmetric solitons, with unequal norms of the two species, is briefly considered too. It is found and explained that the splitting threshold grows with the increase of the asymmetry.