Cnoidal, snoidal, dnoidal wave solutions of the first extended 3D Vakhnenko–Parkes equation together with its conservation laws and various life applications

This paper significantly provides various analytical examinations of the first extended Vakhnenko–Parkes equation, which has never been explored before. To this model, abundant structures of various exact periodic solutions to this equation through the use of the extended Jacobi elliptic function expansion technique is abundantly computed for the first time. This approach employs three different auxiliary nonlinear differential equations to generate the solutions. Consequently, various cnoidal, snoidal, dnoidal, and complex snoidal wave solutions of notes were obtained. We further explore the wave dynamics of these periodic solutions in three and two dimensions using computer software. Moreover, Lie group analysis is utilized to generate the symmetries of the equation. Thereafter, conserved vectors associated with the equation are constructed through the application of Ibragimov’s conserved theorem using the formal Lagrangian of the model. The results, whose significance is demonstrated in physical sciences and technology in this research, can be very useful for researchers in relevant fields for further analysis and applications. All these results are new and unique, and they complement the work previously done on the model, thus underscoring the originality of this research.