A Class of Solitary and Periodic Wave Structures Related to a Dual‐Mode Benjamin‐Bona‐Mahony Equation in Fluid Flow

This study presents the Benjamin‐Bona‐Mahony equation, a new mathematical model for nonlinear wave propagation in medium with just spatial dispersion. The suggested model only considers spatial derivatives, hence representing pure spatial dispersion, in contrast to the traditional formulation that incorporates mixed space‐time derivatives in its dispersion. In this study, the dual‐mode Benjamin‐Bona‐Mahony equation is proposed. This model describes the propagation of two‐way waves in two and three dimensions. In two‐ and three‐dimensional nonlinear dispersive systems, bidirectional wave dynamics are particularly well suited for this kind of approach. The newly developed model incorporates three new physical factors: dispersion factor, phase velocity, and nonlinearity. Using the modified F expansion methodology and the Sardar sub‐equation method, several results are achieved for solitary wave solutions of many types, including singular, bright, dark, Kink soliton, and periodic wave solutions. These novel findings for solitary and periodic waves have significant applications in applied sciences, fluid mechanics, engineering, and fiber optics. Additionally, the dark, bright, periodic‐singular, periodic solutions, and Kink‐shaped solutions are shown using 2D and 3D graphs created using Mathematica. Understanding the wave behavior inside the dual‐mode Benjamin‐Bona‐Mahony equation is enhanced by the graphical depiction, which makes it evident how the phase velocity component impacts the solutions. The resulting soliton solutions are exceptional, one‐of‐a‐kind, and very effective; they have not been used in any prior research.