  In our previous work, we found new types of the cosmic string solutions in
the Abelian-Higgs model with an enhanced $U(1)$ global symmetry. We dubbed
those solutions as the compensated/uncompensated strings. The compensated
string is similar to the conventional cosmic string in the
Abrikosov-Nielsen-Olesen (ANO) string, around which only the would-be
Nambu-Goldstone (NG) boson winds. Around the uncompensated string, on the other
hand, the physical NG boson also winds, where the physical NG boson is
associated with the spontaneous breaking of the enhanced symmetry. Our previous
simulation in the 2+1 dimensional spacetime confirmed that both the
compensated/uncompensated strings are formed at the phase transition of the
symmetry breaking. Non-trivial winding of the physical NG boson around the
strings potentially causes the so-called axion domain-wall problem when the
model is applied to the axion model. In this paper, we perform simulation in
the 3+1 dimensional spacetime to discuss the fate of the uncompensated strings.
We observe that the evolution of the string-network is highly complicated in
the 3+1 dimensional simulation compared with that seen in the previous
simulation. Despite such complications, we find that the number of the
uncompensated strings which could cause can be highly suppressed at late times.
Our observation suggests that the present setup can be applied to the axion
model without suffering from the axion domain-wall problem.
