The modified Benjamin–Bona–Mahony equation via extended direct algebraic method along with physical quantities

The study of nonlinear wave phenomena arising in mathematical physics has attracted considerable attention due to their wide range of applications in applied sciences. In this work, the extended algebraic method is employed to derive new analytical traveling wave solutions of the Modified Benjamin–Bona–Mahony equation, which is an important nonlinear evolution equation describing wave propagation in dispersive media. The proposed approach is based on the construction of a nonlinear auxiliary equation containing arbitrary parameters, which plays a crucial role in generating diverse families of solutions. By implementing this method, thirty-seven new exact solutions are successfully obtained, including soliton-type solutions and periodic wave solutions. The resulting solutions are systematically classified into twelve distinct groups according to their structural characteristics, such as hyperbolic function solutions, trigonometric function solutions, and rational function solutions. Under specific parameter conditions, one of the derived solutions reduces to a previously reported result, thereby validating the correctness and consistency of the obtained solutions. Furthermore, the associated conservation laws corresponding to the obtained solutions are derived and analyzed. The results confirm that the extended algebraic method provides a powerful and efficient framework for solving nonlinear partial differential equations. The proposed methodology can also be extended to investigate a broader class of nonlinear evolution equations arising in mathematical physics and related applied fields.