Modified Wave Operators for the Wave-Klein-Gordon System

TL;DR

This paper analyzes the long-term behavior of a coupled wave-Klein-Gordon system in 3D, establishing modified wave operators and revealing the role of resonant interactions in the system's asymptotics.

Contribution

It introduces a novel approach to derive a resonant system that accurately describes the asymptotic dynamics of the coupled system, with precise error bounds.

Findings

Derived a resonant system governing asymptotic behavior.

Proved existence of modified wave operators for small data.

Provided detailed description of the system's long-time dynamics.

Abstract

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 and smooth data with mild decay at infinity. The key novelty comes from a crucial observation that the asymptotic dynamics are dictated by the resonant interactions. As a consequence, our main results include the derivation of a resonant system with good error bounds, and a detailed description of the asymptotic dynamics of such quasilinear evolution system of hyperbolic and dispersive type.