Analysis of the integro-differential Jaulent–Miodek evolution equation by nonlinear self-adjointness and Lie theory

The current exploration is related to the extraction of the exact explicit solutions of the Integrodifferential Jaulent–Miodek evolution (IDJME) equation. In general, the Jaulent–Miodek equation has many applications in many divisions of physics, for example optics and fluid dynamics. The symmetries of this (2 + 1)-dimensional (IDJME) equation are derived, and it admits the 8th Lie algebra. The similarity transformation method is considered to convert the NLPDE to the nonlinear ODE by using the translational symmetries. Then, we calculated the traveling-wave solutions by an analytical method, namely the extended direct algebraic method. Some obtained solutions are represented by giving suitable parameter values to understand their physical interpretation. The results obtained from this method involve various types of functions, for example, exponential, logarithmic, hyperbolic, and trigonometric. The self-adjointness theory is employed to classify and help us to compute the conserved quantities of the assumed model. A similar work related to this does not exist in the literature.