Asymptotic completeness for a scalar quasilinear wave equation satisfying the weak null condition

TL;DR

This paper establishes the first asymptotic completeness result for a scalar quasilinear wave equation satisfying the weak null condition, using geometric reduced systems to analyze scattering and solution behavior.

Contribution

It introduces a novel approach employing geometric reduced systems to prove asymptotic completeness for such wave equations.

Findings

Proves asymptotic completeness for the first time in this setting

Shows that asymptotic variables satisfy the reduced system with small errors

Constructs matching solutions and recovers scattering data

Abstract

In this paper, we prove the first asymptotic completeness result for a scalar quasilinear wave equation satisfying the weak null condition. The main tool we use in the study of this equation is the geometric reduced system introduced in arXiv:2002.05355. Starting from a global solution $u$ to the quasilinear wave equation, we rigorously show that well chosen asymptotic variables solve the same reduced system with small error terms. This allows us to recover the scattering data for our system, as well as to construct a matching exact solution to the reduced system.