Advancements in Nonlinear Dynamics: Lie Symmetry Applications in the Jaulent-Miodek Equation

This research presents a detailed analysis of the nonlinear Jaulent-Miodek (J-M) equation through the lens of Lie symmetries. Our primary objective is to comprehensively identify the symmetry group and the optimal systems of Lie sub-algebras pertinent to the J-M equation. We delve into the Lie invariants associated with symmetry generators and demonstrate their contribution to forming similarity-reduced equations that encapsulate the essence of the original equation. Moreover, the study introduces a two-step methodology for establishing the conservation laws relevant to the J-M equation. The initial phase involves identifying suitable multipliers essential for calculating these laws. Subsequently, we utilise symbolic computation to derive these conservation laws formally. This in-depth exploration of the equation’s symmetries and conservation laws not only enhances our understanding of the J-M equation’s intrinsic properties but also aids in simplifying and solving the equation under various conditions.