Stability analysis of plane wave solutions of the generalized Ablowitz-Ladik system

We report in this paper a detailed analysis of modulational instability of a plane wave solution of discrete dissipative structures. A generalized Ablowitz-Ladik (AL) equation with a cubic and quintic nonlinearity is then considered, and analysed via the full linear stability analysis of the nonlinear plane wave solution. We derive analytical expressions for the domain of existence as well as the gain of modulational instability of moving plane waves. In particular, we find that discreteness drastically modifies the stability condition as well as the quintic nonlinearity. The analytical and numerical stability analyses are fully supported by an extensive series of numerical simulations of the full model. The generation of pulse trains with high repetition rate predicted by analytical study is then exhibited.