What Becomes of Vortices in Theories with Flat Directions
  In many theories with flat directions of scalar potential, static vortex solutions do not exist for a generic choice of vacuum. In two Euclidean dimensions, we find their substitutes --- constrained instantons consisting of compact core formed by Abrikosov--Nielsen--Olesen vortex and long-ranged cloud of modulus field. In (3+1) dimensions, an initial compact configuration of string topology evolves in such a way that at every point in space the modulus relaxes, in a universal manner, to one and the same value characteristic to the theory. 
A.A. Penin, V.A. Rubakov, P.G. Tinyakov and S.V. Troitsky