  In the present work we discuss non-linear physics problems such as
Nielsen-Olesen strings, superconducting bosonic straight strings and static
vortex rings. We start with a toy model. We search for antiperiodic solitons of
the Goldstone model on a circle. Such models provide the basis as well as
useful hints for further research on three-dimensional more realistic problems.
We proceed with a full research on a $U(1)$ model which admits stable straight
string solutions in a small, numerically determined area. That model has a
Ginzburg-Landau potential with a cubic term added to it and can be found in
condensed matter problems as well. The next part of our research, has to do
with a $U(1) \times U(1)$ model which is the main subject of our interest.
There, we search for stable axially symmetric solutions which are solitons,
which can represent particles, the mass of which is of the order of TeV. The
confirmation or rejection of the existence of those defects is of great
interest if we consider that LHC will work in the same energy range. In our
study, we find out that due to current quenching, these vortex rings seem to be
unstable. We also extend the model of vortex rings by adding higher derivative
terms which are favorable for stability. After the extensive analysis we
performed, we conclude that these objects donât seem to be stable. The reasons
which brought us to this conclusion are explained.
