Beyond the surface: mathematical insights into water waves and quantum fields

This paper examines the complex characteristics of the modified Benjamin–Bona–Mahony equation (\(\text {m}{\mathcal {BBM}}\)) and the Klein–Gordon (\({\mathcal{K}\mathcal{G}}\)) equation in the field of mathematical physics. The \(\text {m}{\mathcal {BBM}}\) equation is a basic model used to describe surface water waves, especially in shallow water situations. It provides valuable information on wave propagation, stability, and the formation of solitons. The applications of this instrument are wide-ranging, including fields such as oceanography, where it plays a crucial role in comprehending wave behavior. On the other hand, the \({\mathcal{K}\mathcal{G}}\) equation is of utmost importance in quantum field theory since it sheds light on the dynamics and interactions of scalar fields such as mesons. Within the field of particle physics, it offers substantial insights into basic concepts, acting as a fundamental basis for comprehending particle behavior. The primary goal of our work is to develop strong analytical techniques for solving these problems. In order to tackle these issues, we use two novel methodologies: the extended simple equation technique and the generalized Kudryashov method. Furthermore, we validate our results by using the extended cubic–B–spline approach for numerical computations. The work effectively solves these intricate equations, resulting in encouraging results. The presented approaches demonstrate their effectiveness, providing significant advances to mathematical physics. This work has inherent worth by presenting innovative analytical methods and perspectives, particularly designed to solve complex nonlinear equations such as the \(\text {m}{\mathcal {BBM}}\) and \({\mathcal{K}\mathcal{G}}\) equations. The finding has significant ramifications that extend across several scientific fields, offering novel approaches to tackle complex issues in mathematical physics. To summarize, our paper introduces innovative analytical techniques designed to solve nonlinear equations in the field of mathematical physics, with a particular emphasis on the \(\text {m}{\mathcal {BBM}}\) and \({\mathcal{K}\mathcal{G}}\) equations.