Self-similarity and singularity formation in a coupled system of Yang-Mills-dilaton evolution equations

TL;DR

This paper investigates the self-similar solutions and singularity formation in a coupled SU(2) Yang-Mills-dilaton system, revealing a stable self-similar attractor and conditions leading to finite-time singularities.

Contribution

It introduces a hidden scale invariance in the system, finds a family of self-similar solutions, and demonstrates their stability and role in singularity formation through analytical and numerical methods.

Findings

Existence of a countable family of self-similar solutions.

The N=0 solution is stable and acts as an attractor.

Finite-time singularity formation observed in supercritical regimes.

Abstract

We study both analytically and numerically a coupled system of spherically symmetric SU(2) Yang-Mills-dilaton equation in 3+1 Minkowski space-time. It has been found that the system admits a hidden scale invariance which becomes transparent if a special ansatz for the dilaton field is used. This choice corresponds to transition to a frame rotated in the $\ln r-t$ plane at a definite angle. We find an infinite countable family of self-similar solutions which can be parametrized by the $N$ - the number of zeros of the relevant Yang-Mills function. According to the performed linear perturbation analysis, the lowest solution with N=0 only occurred to be stable. The Cauchy problem has been solved numerically for a wide range of smooth finite energy initial data. It has been found that if the initial data exceed some threshold, the resulting solutions in a compact region shrinking to the origin, attain the lowest N=0 stable self-similar profile, which can pretend to be a global stable attractor in the Cauchy problem. The solutions live a finite time in a self-similar regime and then the unbounded growth of the second derivative of the YM function at the origin indicates a singularity formation, which is in agreement with the general expectations for the supercritical systems.