Increasing Potentials in Classical Non-Abelian and Abelian Gauge Theories

An exact solution for an SU(2) Yang–Mills field coupled to a scalar field is given, which has potentials with linear, 1/r and 1/r2 parts. This may be of some interest since some phenomenological QCD studies assume a linear plus Coulomb potential. We also show that in the Nielsen–Olesen Abelian model there is an exact solution in the BPS limit, which has a 1/r electromagnetic field and a logarithmically rising scalar field. Both of these solutions must be cutoff from above to avoid infinite field energy.