Asymptotics and Scattering for Wave-Klein-Gordon Systems

TL;DR

This paper analyzes the long-term behavior and scattering properties of coupled wave-Klein-Gordon systems, revealing detailed asymptotics and inverse scattering results relevant to Einstein-Klein-Gordon models.

Contribution

It provides a detailed asymptotic analysis and inverse scattering results for coupled wave-Klein-Gordon systems, extending understanding of their long-term dynamics.

Findings

Asymptotics of Klein-Gordon fields involve modified phases and homogeneous functions.

Wave equation asymptotics include radiation fields and interior solutions.

Existence of solutions matching prescribed asymptotic behaviors at infinity.

Abstract

We study the coupled wave-Klein-Gordon systems, introduced by LeFloch-Ma and then Ionescu-Pausader, to model the nonlinear effects from the Einstein-Klein-Gordon equation in harmonic coordinates. We first go over a slightly simplified version of global existence based on LeFloch-Ma, and then derive the asymptotic behavior of the system. The asymptotics of the Klein-Gordon field consist of a modified phase times a homogeneous function, and the asymptotics of the wave equation consist of a radiation field in the wave zone and an interior homogeneous solution coupled to the Klein-Gordon asymptotics. We then consider the inverse problem, the scattering from infinity. We show that given the type of asymptotic behavior at infinity, there exist solutions of the system that present the exact same behavior.