Singularity/scattering alternative for a discrete family of static critical solutions with various numbers of unstable eigenmodes

Abstract

This paper reports on the study of some solutions for a system of coupled nonlinear wave equations of a hyperbolic type, effectively dependent on one spatial (radial) variable and one time variable. This system of equations, which is a coupled system of the Yang-Mills equations with a scalar field, i.e., dilaton, belongs to the so-called class of supercritical systems, for which there exist solutions that inevitably lead to the formation of singularities at a point or even a whole region during a finite period of time at smooth initial distributions with finite energy. This system of equations has regular stationary solutions with finite energy. All such stationary solutions are unstable and can be parametrized by the number N, which is the number of their unstable eigenmodes in linear approximation, and N = 1, 2, 3, …, ∞. The self-consistent problem of the decay of such stationary solutions with N = 1, 2, 3, 4 on the independent excitation of their unstable eigenmodes was solved numerically in nonlinear regime. The corresponding initial-boundary problem was investigated numerically by means of an adaptive computation scheme based on a conservative finite-differential scheme with energy conservation. It has been found that for each considered stationary solution only the perturbation of its basic unstable eigenmode leads to the formation of the singularity/scattering alternative, the governing parameter in the choice of the alternative being the sign of the major unstable eigenmode. It is shown that the independent perturbation of all higher unstable eigenmodes necessarily leads to the formation of a singularity. It is also found that the nonlinear waves formed with the decay of the basic N = 1 solution on the perturbation of its single unstable mode may expose some properties peculiar to solitons.