On the Long-Time Behavior of a Wave-Klein-Gordon Coupled System.

We consider a coupled Wave-Klein-Gordon system in 3D, which is a simplified model for the global nonlinear stability of the Minkowski space-time for self-gravitating massive fields. In this paper we study the large-time asymptotic behavior of solutions to such systems, and prove modified wave operators for small data. The key novelty comes from a crucial observation that the modified asymptotic dynamics are dictated by the resonant interactions of the equations. As a consequence, our main results include the derivation of a resonant system with good error bounds, and a detailed description of the global behavior of solutions to the Wave-Klein-Gordon system.