An Investigation of the Influence of Time Evolution on the Solution Structure Using Hyperbolic Trigonometric Function Methods

In attempting to mathematically create traveling wave solutions to the (2+1)-dimensional integro-differential Jaulent–Miodek equation, the extended rational \(\sinh \) $sinh$ - \(\cosh \) $cosh$ technique and the modified extended \(\tanh \) $tanh$ function approach have been effectively employed in this study. To achieve this, the Jaulent–Miodek model is integrally transformed into a fourth-order nonlinear partial differential equation. Subsequently, the fourth-order nonlinear ordinary differential equation of the integro-differential Jaulent–Miodek equation was obtained using an appropriate traveling wave transformation. For the under-examination model, several novel exact solitary, singular, and periodic wave solutions are deduced in the interpretations of a mixture of complicated hyperbolic, periodic, trigonometric, and rational functions. To illustrate the physical significance of some of the previously unidentified solutions, two- and three-dimensional figures have been generated with the required arbitrary parameters.