Dynamics of closed-form invariant solutions and formal Lagrangian approach to a nonlinear model generated by the Jaulent–Miodek hierarchy

Abstract This study focuses on a (3 + 1)-dimensional nonlinear evolution model derived from the Jaulent–Miodek hierarchy. Our research employed analytical tools to scrutinize the invariance characteristics of this model. However, our primary emphasis is on utilizing the potent Lie group method, which effectively reveals the inherent symmetries within the model and explores solutions that remain invariant under group transformations using symmetry algebra. Furthermore, we investigated the application of Ibragimov’s approach to examine the conservation laws relevant to the model under consideration. This theorem is employed to identify and analyze the conservation laws associated with the (3 + 1)-dimensional nonlinear evolution model, which is essential for understanding the behavior of the system. Our research is significant as it contributes to exploring this particular model and addresses a specific gap in the group theoretic approach within this context.