Optical soliton solutions of the coupled equation in a stratified deep sea environment with engineering application

• Simulates deep-sea wave interactions employing the coupled NLS equation. • Improves the examination of wave interactions in marine and optical systems. • Employs symbolic computation to enhance exactness, efficiency, and flexibility in wave analysis. In maritime transport, the motion of fluids, and transmission of energy, comprehending the movement of deep-sea waves presents an important difficulty in ocean science. The mathematical description of solitary wave movements in a 2-layer stratified deep-sea atmosphere results in the NLS, while the relationship of two such waves is described by the coupled NLS equation. This study utilizes the modified subequation method (MSEM) and new kudryashov method (NKM) to derive accurate optical solutions based on the Manakov model of the coupled NLS equation. The suggested approaches not only enables the examination of singular wave solutions but also produces generic optical solutions. These methods facilitates the derivation of solitary wave solutions, including many forms such as dark soliton , twofold bright soliton, bell soliton, each trigonometric waves possessing numerous potential uses in physical sciences and engineering. Furthermore, previously established solutions can be obtained for particular cases by carefully picking parameters acquired from the resolution of mathematical systems. Symbolical computation improves the effectiveness and versatility of the method, rendering it an effective strategy for examining wave connections in oceanic and optical systems.