Asymptotics from scaling for nonlinear wave equations

TL;DR

This paper introduces a scaling method that simplifies the analysis of nonlinear wave equations with small initial data by approximating their long-term behavior with a linear wave equation featuring a distributional source.

Contribution

The paper develops a novel scaling technique that transforms nonlinear wave equations into linear ones with distributional sources, enabling asymptotic analysis of solutions.

Findings

Exact solutions approximate late-time behavior

Method applies to timelike and null directions

Simplifies analysis of nonlinear wave equations

Abstract

We present a scaling technique which transforms the evolution problem for a nonlinear wave equation with small initial data to a linear wave equation with a distributional source. The exact solution of the latter uniformly approximates the late-time behavior of solutions of the nonlinear problem in timelike and null directions.