Characteristic approach to the soliton resolution

TL;DR

This paper uses a characteristic initial boundary value problem approach to analyze the soliton resolution in a toy model of the 4d equivariant Yang-Mills equation, highlighting the role of outgoing null foliations and identifying attractors.

Contribution

It introduces a characteristic approach with outgoing null foliations to study soliton resolution, demonstrating the attractors and convergence rates in a toy Yang-Mills model.

Findings

Static half-kink and superposed configurations are generic attractors.

Numerical evidence of basins of attraction and a codimension-one boundary.

Detailed analysis of convergence rates to attractors.

Abstract

As a toy model for understanding the soliton resolution phenomenon we consider a characteristic initial boundary value problem for the 4$d$ equivariant Yang-Mills equation outside a ball. Our main objective is to illustrate the advantages of employing outgoing null (or asymptotically null) foliations in analyzing the relaxation processes due to the dispersal of energy by radiation. In particular, within this approach it is evident that the endstate of evolution must be non-radiative (meaning vanishing flux of energy at future null infinity). In our toy model such non-radiative configurations are given by a static solution (called the half-kink) plus an alternating chain of $N$ decoupled kinks and antikinks. We show numerically that the configurations $N=0$ (static half-kink) and $N=1$ (superposition of the static half-kink and the antikink which recedes to infinity) appear as generic attractors and we determine a codimension-one borderline between their basins of attraction. The rates of convergence to these attractors are analyzed in detail.