High Order Asymptotic Expansions of a Good-Bad-Ugly Wave Equation

TL;DR

This paper develops a heuristic method to derive high-order asymptotic expansions for a non-linear wave system modeling Einstein equations near null infinity, revealing logarithmic terms and introducing stratified null forms.

Contribution

It introduces a novel heuristic approach for high-order asymptotic expansions in a non-linear wave system mimicking Einstein equations, including polyhomogeneous solutions and stratified null forms.

Findings

Logarithmic terms appear at every order in the bad field

The model admits polyhomogeneous asymptotic solutions

Stratified null forms generalize standard null forms without changing solution types

Abstract

A heuristic method to find asymptotic solutions to a system of non-linear wave equations near null infinity is proposed. The non-linearities in this model, dubbed good-bad-ugly, are known to mimic the ones present in the Einstein field equations (EFE) and we expect to be able to exploit this method to derive an asymptotic expansion for the metric in General Relativity (GR) close to null infinity that goes beyond first order as performed by Lindblad and Rodnianski for the leading asymptotics. For the good-bad-ugly model, we derive formal expansions in which terms proportional to the logarithm of the radial coordinate appear at every order in the bad field, from the second order onward in the ugly field but never in the good field. The model is generalized to wave operators built from an asymptotically flat metric and it is shown that it admits polyhomogeneous asymptotic solutions. Finally we define stratified null forms, a generalization of standard null forms, which capture the behavior of different types of field, and demonstrate that the addition of such terms to the original system bears no qualitative influence on the type of asymptotic solutions found.