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

We study both analytically and numerically a coupled system of spherically symmetric SU(2) Yang-Mills-dilaton equations 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 a 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 (YM) function. According to the performed linear perturbation analysis, the lowest solution with $N=0\mathrm{}$ 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 $N=0\mathrm{}$ stable self-similar profile, which can pretend to be a global stable attractor in the Cauchy problem. The solutions reside 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.