Blowup/scattering alternative for a discrete family of static critical solutions with various number of unstable eigenmodes

TL;DR

This study investigates the nonlinear evolution of static solutions in the SU(2) Yang-Mills-dilaton system, revealing their role as thresholds between blowup and scattering, with higher unstable modes leading to blowup and the lowest mode exhibiting soliton-like decay.

Contribution

It provides the first detailed numerical analysis of the decay of a discrete family of static solutions with various unstable eigenmodes in the YMd system, highlighting their threshold nature.

Findings

All static solutions act as local thresholds between blowup and scattering.

Only the main unstable eigenmodes influence the blowup/scattering outcome.

Higher unstable modes always lead to finite-time blowup.

Abstract

Decay of regular static spherically symmetric solutions in the SU(2) Yang-Mills-dilaton (YMd) system of equations under the independent excitation of their unstable eigenmodes has been studied self-consistently in the nonlinear regime. The considered regular YMd solutions form a discrete family and can be parametrised by the number $N=1,2,3,4...$ of their unstable eigenmodes in linear approximation. We have obtained strong numerical evidences in favour of the following statements: i) all static YMd solutions are distinct local threshold configurations, separating blowup and scattering solutions; ii) the main unstable eigenmodes are only those responsible for the blowup/scattering alternative; iii) excitation of higher unstable eigenmodes always leads to finite-time blowup; iv) the decay of the lowest N=1 static YMd solution via excitation of its unique unstable mode is an exceptional case because the resulting waves propagate as a whole without energy dispersion revealing features peculiar to solitons. Applications of the obtained results to Type-I gravitational collapse of massless fields are briefly discussed.