We study modulational instability of two-component Bose-Einstein condensates in an optical lattice, which is modelled as a coupled discrete nonlinear Schrödinger equation. The excitation spectrum and the modulational instability condition of the total system are presented analytically. In the long-wavelength limit, our results agree with the homogeneous two-component Bose-Einstein condensates case. The discreteness effects result in the appearance of the modulational instability for the condensates in miscible region. The numerical calculations confirm our analytical results and show that the interspecies coupling can transfer the instability from one component to another.

Based on a two-mode boson model, we study nonclassical properties of the atom-molecule Bose-Einstein condensate. The effects of nonlinear collisions on the dynamics of the molecular formation is studied both in classical and quantum treatments. We find that the conversion from atoms to molecules can be suppressed strongly due to nonlinearity induced localization of the atomic population. In addition, we study statistical properties of the atom-molecule condensed system by calculating the intensity correlation functions numerically. We find that the effect of nonlinearity leads to the appearance of superchaotic molecular pulses, while maintaining the atomic field sub-Poissonian. The joint quantum statistical properties of the atoms and the molecules always show antibunching.

Physical properties of the ideal Bose gas with non-integer dimensions between D = 2 and D = 3 are theoretically investigated. Calculation shows that there exists a hierarchy of condensation transitions with changing fractional dimensionality. The results obtained on the specific heat and the condensed density in non-integer dimensions are similar to those for liquid helium-4 in porous media. This results suggest that the geometrical factor may be important for the physical properties of liquid helium-4 in porous media.

The superfluid fraction of a charged boson fluid at non-integer dimensions between 2 and 3 is studied. Interaction between particles is assumed to be Coulombic, and Bogoliubov approximation is applied for a weak coupling regime. The theoretical results are compared with the superfluid densities of superconducting films, showing an universal splitting behavior. The qualitative similarity between the charged boson fluid and the superconducting films gives a strong clue for the origin of the universality observed in the superconducting films.