Composite gap solitons in Rashba spin-orbit-coupled Bose-Einstein condensates within bi-periodic potentials

We investigate the existence, stability, and dynamical properties of composite gap solitons in Rashba spin-orbit-coupled Bose-Einstein condensates trapped in a bi-periodic optical potential. The system is modeled by two-component Gross-Pitaevskii equations incorporating cubic nonlinearities and Rashba-type spin-orbit coupling. We specifically focus on composite gap solitons formed by a fundamental soliton in the spin-down component and a dipole soliton in the spin-up component. The relationship between the total atom number and the chemical potential follows the anti-Vakhitov-Kolokolov criterion, establishing a necessary condition for stability. Through linear stability analysis and direct numerical simulations, we identify the stability domains of these state configurations. Notably, the inter-component (cross) interactions strongly dictate soliton stability: the repulsive cross-interaction yields significantly larger stability regions than the attractive, which often leads to narrower regions and soliton instability. These results provide a deeper insight into the behavior of multi-component gap solitons in spin-orbit-coupled systems with complex lattice geometries.