In this study, we obtain solitary wave solutions and other exact wave solutions for Davey–Stewartson equation (DSE), which explains how waves move through water with a finite depth while being affected by gravity and surface tension. The study is conducted with the aid of the modified extended mapping method (MEMM). A variety of distinct traveling wave solutions are furnished. The obtained solutions comprise dark, bright, and singular solitary wave solutions. Additionally, Jacobi elliptic function solutions, exponential wave solutions, singular periodic wave solutions, rational wave solutions, and periodic wave solutions are also offered. To help readers physically grasp the acquired solutions, graphical representations of some of the extracted solutions are provided.
