We consider a system of two Gross-Pitaevskii (GP) equations, in the presenceof an optical-lattice (OL) potential, coupled by both nonlinear and linearterms. This system describes a Bose-Einstein condensate (BEC) composed of twodifferent spin states of the same atomic species, which interact linearlythrough a resonant electromagnetic field. In the absence of the OL, we findplane-wave solutions and examine their stability. In the presence of the OL, wederive a system of amplitude equations for spatially modulated states which arecoupled to the periodic potential through the lowest-order subharmonicresonance. We determine this averaged system's equilibria, which representspatially periodic solutions, and subsequently examine the stability of thecorresponding solutions with direct simulations of the coupled GP equations. Wefind that symmetric (equal-amplitude) and asymmetric (unequal-amplitude)dual-mode resonant states are, respectively, stable and unstable. The unstablestates generate periodic oscillations between the two condensate components,which is possible only because of the linear coupling between them. We alsofind four-mode states, but they are always unstable. Finally, we brieflyconsider ternary (three-component) condensates.
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