This study explores the synchronization dynamics of a one-dimension edge-emitting laser array monolithically-integrated with an external cold cavity comprehensively, aiming to achieve an in-phase mode optical field. By employing the optical feedback rate equation, the impact of mutual feedback coefficient and coupling coefficient on the synchronization process are investigated thoroughly. The proposed external cold cavity, designed according to the Talbot effect, could significantly diminish the reflectivity of front facet through separated electrode structure, therefore facilitating the phase-locking process. Consequently, the study uncovers the effective regime for establishing in-phase mode operation. Additionally, the numerical analysis also reveals the vivid synchronization dynamic from chaotic state to partial locking, then fully-phase-locked, and ultimately periodic oscillation. Based on the simulation results, our works could offer valuable insights for steering the on-chip optical field and developing novel laser arrays with high beam quality.