Nonlinear gap modes and compactons in a lattice model for spin-orbit coupled exciton-polaritons in zigzag chains

We consider a system of generalized coupled Discrete Nonlinear Schr\"{o}dinger (DNLS) equations, derived as a tight-binding model from the Gross-Pitaevskii-type equations describing a zigzag chain of weakly coupled condensates of exciton-polaritons with spin-orbit (TE-TM) coupling. We focus on the simplest case when the angles for the links in the zigzag chain are $\pm \pi/4$ with respect to the chain axis, and the basis (Wannier) functions are cylindrically symmetric (zero orbital angular momenta). We analyze the properties of the fundamental nonlinear localized solutions, with particular interest in the discrete gap solitons appearing due to the simultaneous presence of spin-orbit coupling and zigzag geometry, opening a gap in the linear dispersion relation. In particular, their linear stability is analyzed. We also find that the linear dispersion relation becomes exactly flat at particular parameter values, and obtain corresponding compact solutions localized on two neighboring sites, with spin-up and spin-down parts $\pi/2$ out of phase at each site. The continuation of these compact modes into exponentially decaying gap modes for generic parameter values is studied numerically, and regions of stability are found to exist in the lower or upper half of the gap, depending on the type of gap modes.