  Symbiotic solitons (SS) and quantum droplets (QD) are self-trapped localized
modes emerging in binary Bose-gas mixtures with intra-component repulsion and
inter-component attraction. We have shown that two-dimensional SS can be
stabilized against collapse or decay by means of a quasi-one-dimensional
optical lattice (OL). Mobility of SSs along the free direction of the potential
allows us to explore interactions and collisions of SSs moving in the same
channel and neighboring channels of the quasi-1D OL. For the case of equal atom
numbers in both components of the binary Bose-Einstein condensate (BEC) we have
developed a variational approach that showed the stability of SS. For parameter
settings when the SS stays on the verge of collapse instability, we take into
account the Lee-Huang-Yang quantum fluctuations (QF) term in the coupled
Gross-Pitaevskii equations. Repulsive QF prevents the mean-field collapse and
gives rise to the formation of 2D QDs with peculiar properties such as
incompressibility and surface tension, which are inherent to liquids. The
proposed model of binary BEC loaded in a quasi-1D OL allows us to demonstrate
the manifestations of the incompressibility and surface tension of 2D QDs. The
formation of QD in imbalanced binary BEC with different numbers of atoms of the
components in the presence of a quasi-1D OL has been investigated. The possible
application of the proposed model to studies of anisotropic superfluidity is
discussed.
