Dynamical structures in Einstein’s vacuum field equation: Symmetry-derived conservation laws, stability analysis, and nonlinear wave solutions

This paper investigates the solutions to the variable-coefficient Einstein’s field equation (EFE) in vacuum to elucidate wave propagation phenomena in diverse physical systems. As a foundational framework in theoretical physics, this model holds profound implications across multiple disciplines, including astrophysics, quantum mechanics, observational astronomy, and cosmological studies. Employing the Lie group method, we analyze the variable-coefficient EFE, determine its symmetry properties, obtain soliton and periodic wave solutions, and derive a distinct form of exact solution. Graphical representations of the solutions are presented to illustrate the equation’s dynamical characteristics. Finally, conservation laws are constructed, and a linear stability analysis is performed to verify the robustness of the vacuum field background.