Modulational instability of Bose–Einstein condensates with inter-spin–orbit coupling in deep optical lattice

We present a comprehensive study of modulational instability (MI) in a binary Bose–Einstein condensate with spin–orbit coupling, confined to a deep optical lattice. The system is modeled by a set of discrete Gross–Pitaevskii equations. Using linear stability analysis, we derive the explicit MI conditions for the system, elucidating the critical and distinct roles played by spin–orbit coupling, inter-species nonlinearity, and intra-species nonlinearity. Our analysis, conducted for both unstaggered and staggered fundamental modes, reveals markedly different instability landscapes for these two configurations. The analytical predictions are confirmed by extensive numerical simulations of the full nonlinear dynamics, which vividly illustrate the spatiotemporal evolution of wave amplitudes, phase coherence, and energy localization during the instability process. The numerical results, obtained via a fourth-order Runge–Kutta method, show excellent agreement with the linear stability theory and provide a complete picture of the MI-induced pattern formation.