  We consider an ultracold bosonic binary mixture confined in a one-dimensional
double-well trap. The two bosonic components are assumed to be two hyperfine
internal states of the same atom. We suppose that these two components are
spin-orbit coupled between each other. We employ the two-mode approximation
starting from two coupled Gross-Pitaevskii equations and derive a system of
ordinary differential equations governing the temporal evolution of the
inter-well population imbalance of each component and that between the two
bosonic species. We study the Josephson oscillations of these spin-orbit
coupled Bose-Einstein condensates by analyzing the interplay between the
interatomic interactions and the spin-orbit coupling and the self-trapped
dynamics of the inter-species imbalance. We show that the dynamics of this
latter variable is crucially determined by the relationship between the
spin-orbit coupling, the tunneling energy, and the interactions.
