Exploring the solitary wave solutions of Einstein's vacuum field equation in the context of ambitious experiments and space missions

In the present work, Einstein's vacuum field equation is investigated analytically to explore the solitary wave solutions. This equation arises in mathematical physics, having meaningful applications in the general theory of relativity. This concept is crucial for numerous challenging experiments and space missions. The generalized exponential rational function and modified auxiliary equation approaches are used to obtain the exact solitary wave solution. Various types of solutions are extracted, including exponential functions, hyperbolic functions, trigonometric functions, and rational forms. Additionally, a stability analysis for the Einstein vacuum field equation is conducted. Appropriate parameters are chosen to draw 3-D and corresponding contour plots of some solutions, which clearly demonstrate the solitary wave behaviors. The obtained results support the idea that applying these approaches is the most effective strategy for resolving any nonlinear issues that may arise in science and technology.