Several traveling wave solutions of the modified Benjamin-Bona-Mahony equation using the Kumar-Malik method

Abstract This research investigates the modified Benjamin-Bona-Mahony (mBBM) equation, a crucial model within nonlinear wave dynamics, which effectively characterizes long-wave propagation in dispersive media. By applying the Kumar-Malik method, the study obtains novel exact solutions to the mBBM equation, represented through diverse mathematical forms, including Jacobi elliptic, hyperbolic, trigonometric, and exponential functions. The flexibility of this approach facilitates the construction of various traveling wave solutions, including periodic, singular periodic, bright, dark, kink, anti-kink, and singular waveforms. The graphical visualization of these solutions in multiple dimensions elucidates their propagation behavior and stability, thereby reinforcing the reliability of the proposed methodology. This investigation enhances the knowledge of mathematical techniques for solving nonlinear differential equations and demonstrates their applicability to other nonlinear wave models across various scientific fields. Additionally, the findings not only provide deeper insights into the dynamics of the mBBM equation but also offer new opportunities for studying nonlinear phenomena in diverse physical systems, such as hydromagnetic waves in cold plasma, coastal engineering, nonlinear optics, fluid dynamics, plasma physics, and optical illusions. This work highlights the Kumar-Malik method as a powerful analytical tool, significantly contributing to exploring and comprehending complex wave phenomena within mathematical physics and the applied sciences.