Understanding dark matter interactions is pivotal in astrophysics, as it constitutes most of the universe's mass. The objective of this research is to explore dark matter interactions within galaxy clusters using gravitational lensing as a probe. We analyze data from the Hubble Space Telescope and the Atacama Large Millimeter Array, focusing on gravitational lensing patterns in galaxy cluster Abell 1689. We apply a combination of weak and strong lensing techniques to extract detailed mass distribution maps. Our findings reveal anomalies in lensing profiles that suggest possible dark matter self-interactions beyond the standard collisionless cold dark matter model. The results provide stronger constraints on interaction cross-sections, offering a novel insight into the dark matter particle nature. Conclusively, this study presents evidence that dark matter exhibits complex behaviors that may require new physics to fully comprehend. Further research is encouraged to refine these observations and explore their implications in the broader cosmological context.