Solutions And Gradient Of The Conformal Ricci Bourguignon Soliton On Vaidya Spacetime

In this work, we derive the complete and explicit solution for the conformal Ricci-Bourguignon soliton on Vaidya spacetime. We provide the closed-form expression for the vector field and establish the necessary conditions for the existence of the scalar potential, for which we also derive an explicit form. Our solution to the underlying system of linear partial differential equations proves that such solitons exist if and only if the mass function vanishes, forcing the metric to reduce to flat Minkowski spacetime (Schwarzschild, $m=0$). Synthesizing prior works, we show that the established classification of the soliton as shrinking, steady, or expanding is justified by the principles of linear stability. These findings refine the set of possible solitons within the non-linear theory of geometric flows by proving they are only admissible in the non-radiating vacuum limit, thereby enhancing the reliability of such models.