On convergence towards a self-similar solution for a nonlinear wave equation - a case study

TL;DR

This paper investigates the asymptotic stability of a self-similar solution in a nonlinear wave equation related to Yang-Mills theory, combining spectral analysis and numerical validation.

Contribution

It introduces a spectral method using continued fractions to analyze stability and confirms the approach with numerical simulations.

Findings

Spectral analysis identifies the eigenvalues governing stability.

Numerical results match the theoretical predictions.

The self-similar attractor is asymptotically stable.

Abstract

We consider the problem of asymptotic stability of a self-similar attractor for a simple semilinear radial wave equation which arises in the study of the Yang-Mills equations in 5+1 dimensions. Our analysis consists of two steps. In the first step we determine the spectrum of linearized perturbations about the attractor using a method of continued fractions. In the second step we demonstrate numerically that the resulting eigensystem provides an accurate description of the dynamics of convergence towards the attractor.