Characteristic approach to the soliton resolution

Abstract

As a toy model for understanding the soliton resolution phenomenon we consider a characteristic initial boundary value problem for the 4d 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.