Dark gap solitons in composite incommensurate linear-nonlinear lattices

We investigate dark gap solitons in one-dimensional composite incommensurate linear-nonlinear lattices, which serve as a simplified analogue of moiré-like configurations. The linear potential is single-periodic and gives rise to a Bloch band structure, while the cubic defocusing nonlinearity introduces an additional detuned spatial periodicity that enables the formation of two-component oscillatory structures. Families of fundamental (single) dark solitons and soliton clusters are thus generated, and their existence is characterized through propagation constant-power defect relationships. Linear stability analysis reveals that most states remain stable throughout the first bandgap, with instabilities emerging only near the band edges—a result further confirmed by direct numerical simulations. Both the propagation constant and the detuned period of the nonlinear lattice strongly influence the background structure and the soliton properties. Our findings demonstrate that composite incommensurate linear-nonlinear lattices offer a versatile platform for the analysis and stabilization of dark gap solitons.