Discrete solitons dynamics in $\cal{PT}$-symmetric oligomers with complex-valued couplings

We consider an array of double oligomers in an optical waveguide device. A mathematical model for the system is the coupled discrete nonlinear Schr\"odinger (NLS) equations, where the gain-and-loss parameter contributes to the complex-valued linear coupling. The array caters to an optical simulation of the parity-time ($\cal{PT}$)-symmetry property between the coupled arms. The system admits fundamental bright discrete soliton solutions. We investigate their existence and spectral stability using perturbation theory analysis. These analytical findings are verified further numerically using the Newton-Raphson method and a standard eigenvalue-problem solver. Our study focuses on two natural discrete modes of the solitons: single- and double-excited-sites, also known as onsite and intersite modes, respectively. Each of these modes acquires three distinct configurations between the dimer arms, i.e., symmetric, asymmetric, and antisymmetric. Although both intersite and onsite discrete solitons are generally unstable, the latter can be stable, depending on the combined values of the propagation constant, horizontal linear coupling coefficient, and gain-loss parameter.