Dark matter-wave gap solitons of Bose-Einstein condensates trapped in optical lattices with competing cubic-quintic nonlinearities

Solitons are nonlinear self-sustained wave excitations and probably among the most interesting and exciting emergent nonlinear phenomenon in the corresponding theoretical settings. Bright solitons with sharp peak and dark solitons with central notch have been well known and observed in various nonlinear systems. The interplay of periodic potentials, like photonic crystals and lattices in optics and optical lattices in ultracold atoms, with the dispersion has brought about gap solitons within the finite band gaps of the underlying linear Bloch-wave spectrum and, particularly, the bright gap solitons have been experimentally observed in these nonlinear periodic systems, while little is known about the underlying physics of dark gap solitons. Here, we theoretically and numerically investigate the existence, property and stability of one-dimensional matter-wave gap solitons and soliton clusters of Bose-Einstein condensates trapped in optical lattices with competing cubic-quintic nonlinearity, the higher-order of which is self-defocusing and the lower-order (cubic) one is chosen as self-defocusing or focusing nonlinearities. By means of the conventional linear-stability analysis and direct numerical calculations with initial perturbations, we identify the stability and instability areas of the corresponding dark gap solitons and clusters ones.