Novel analytical framework for Jacobian equations: Applications to ZK–BBM dynamics

This study proposes a new analytical framework for the Jacobian elliptic equation and derives periodic exact solutions using computational methods. The proposed approach solves these equations by transforming them into simpler forms. Many periodic wave solutions are obtained with this technique. The method is then applied to the Zakharov–Kuznetsov–Benjamin–Bona–Mahoney (ZK-BBM) equation. Nonlinear transformations and auxiliary equations are used in this process. A series of new exact periodic solutions are derived from the ZK-BBM equation. These solutions have not been reported in earlier studies. Numerical simulations are carried out to show the behavior of multi-periodic waves in the ZK-BBM system. The results reveal how these wave patterns evolve over time. This work provides useful information on nonlinear wave mechanics. The findings may apply to fluid dynamics and other physical systems where wave phenomena occur. The analytical tools developed here can be used for other nonlinear evolution equations. These equations appear in many areas of physics and engineering. The study expands the range of known solutions for the ZK-BBM equation. It also demonstrates the effectiveness of the proposed method for solving nonlinear partial differential equations. The combination of analytical derivations and numerical simulations offers a complete picture of wave dynamics in this system.