  We study superfluid (SF) states of strongly interacting Bose-Bose mixtures
with equal mass and intra-component interaction in optical lattices both in the
presence and absence of a barrier potential (BP). We show that the SF order
parameters obey the two-component nonlinear Schroedinger equation (NLSE) with
not only cubic but also quintic nonlinearity in the vicinity of the first-order
transitions to the Mott insulators with even fillings. In the case of no BP, we
analyze solitary-wave (SW) solutions of the cubic-quintic NLSE. When the SF
state changes from a ground state to a metastable one, a standard dark SW turns
into a bubble-like dark SW, which has a non-vanishing density dip and no pi
phase kink even in the case of a standing SW. It is shown that the former and
latter SW are dynamically unstable against an out-of-phase fluctuation and an
in-phase fluctuation, respectively, and the dynamical instabilities are
weakened when one approaches the transition point. We find that the size and
the inertial mass of the SW diverge at the first-order transition point. We
suggest that the divergence of the inertial mass may be detected through
measurement of the relation between the velocity and the phase jump of the SW.
In the presence of BP, we reveal that when the barrier strength exceeds a
certain critical value, the SF state that was metastable without the barrier is
destabilized towards complete disjunction of the SF. The presence of the
critical BP strength indicates that the strong BP qualitatively changes the
criticality near the metastability limit of the SF state. We derive critical
behaviors of the density, the compressibility, and the critical current near
the metastability limit induced by the BP. It is also found that the relation
between the supercurrent and the phase jump across the BP exhibits a peculiar
behavior, owing to the non-topological nature of the bubble-like SW.
