On the ground of the Landau criterion we study the behavior of critical velocities in a superfluid two-component Bose gas. It is found that under motion of the components with different velocities the velocity of each component should not be lower than a minimum phase velocity of elementary excitations (s − ). The Landau criterion yields a relation between the critical velocities of the components (v c1 , v c2 ). The velocity of one or even both components may exceed s − . The maximum value of the critical velocity of a given component can be reached when the other component does not move. The approach is generalized for a two-component condensate confined in a cylindrical harmonic potential.

A stationary wave pattern occurring in a flow of a two-component Bose-Einstein condensate past an obstacle is studied. We consider the general case of unequal velocities of two superfluid components. The Landau criterium applied to the two-component system determines a certain region in the velocity space in which superfluidity may take place. Stationary waves arise out of this region, but under the additional condition that the relative velocity of the components does not exceed some critical value. Under increase of the relative velocity the spectrum of the excitations becomes complex valued and the stationary wave pattern is broken. In case of equal velocities two sets of stationary waves that correspond to the lower and the upper Bogolyubov mode can arise. If one component flows and the other is at rest only one set of waves may emerge. Two or even three interfere sets of waves may arise if the velocities approximately of equal value and the angle between the velocities is close to π/2. In two latter cases the stationary waves correspond to the lower mode and the densities of the components oscillate out-of-phase. The ratio of amplitudes of the components in the stationary waves is computed. This quantity depends on the relative velocity, is different for different sets of waves, and varies along the crests of the waves. For the cases where two or three waves interfere the density images are obtained.

The state with a spontaneous interlayer phase coherence in a graphene based bilayer quantum Hall system is studied. This state can be considered as a gas of superfluid electron-hole pairs with the components of the pair belonging to different layers. Superfluid flux of such pairs is equivalent to two electrical supercurrents in the layers. It is shown that the state with the interlayer phase coherence emerges in the graphene system if a certain imbalance of the Landau level filling factors of the layers is created. We obtain the temperature of transition into the superfluid state, the maximum interlayer distance at which the phase coherence is possible, and the critical values of the supercurrent. The advantages of use of graphene systems instead of GaAs heterostructures for the realization of the bilayer electron-hole superconductivity is discussed.