Wave propagation to the doubly dispersive equation and the improved Boussinesq equation

In this paper, we examine the optical solitons for the nonlinear doubly dispersive equation and the modified Boussinesq equation, which explain the flow of shallow water in a small-amplitude surface system. We realize a variety of solitons using the Sardar sub-equation approach, including bright solitons, dark solitons, singular solitons, mixed bright-singular solitons, periodic, exponential, and rational solutions. The generated optical solutions can be used to simulate water waves and the free movement of a fluid surface, both of which are important in computing models of nonlinear partial differential equations in science, engineering, and mathematical physics. For the physical interpretation of the data, the well-known symbolic software Mathematica 12 was employed.