This is a supplementary material for
arxiv:1303.3044,
These results were shown at the workshop
Quantised Flux in Tightly Knotted and Linked Systems,
held in Cambridge, UK (Dec. 37 2012).
See the corresponding talk of
M. S. Volkov.
See the published version in Phys. Rev. Lett.
Introduction
Description of the simulations:
Movies show the relativistic evolution (wave equation) in 3+1 with a semiimplicit
time discretization using $\beta$Newmark method. The spatial discretization is
done with a FiniteElement decomposition over spherical domain,
equipped with tetrahedral mesh obtain by DelaunayVoronoi tetrahedrizations.
The initial configuration by minimization of the axially symmetric configuration.
At the boundary, the fields are fixed to their initial (vacuum) value, that is
$\partial_t\phi_a=0$. Thus the integration domain is a `confining box'.
This means that the vorton cannot escape the box.
Stable regimes is run much longer than unstable ones, just to make sure they are not just longerlived.
The displayed quantities are: The isosurfaces are surfaces of constant density of the second scalar field $\phi_2^2$. Blue to Red colors shows small to big densities. The arrows represent the associated Noether current (supercurrent) $~\mathrm{Im}(\phi_2^*\partial_k\phi_2)$. Some simulations just show one (outter) isosurface of $\phi_2^2$ (in blue).
Stable regimes is run much longer than unstable ones, just to make sure they are not just longerlived.
The displayed quantities are: The isosurfaces are surfaces of constant density of the second scalar field $\phi_2^2$. Blue to Red colors shows small to big densities. The arrows represent the associated Noether current (supercurrent) $~\mathrm{Im}(\phi_2^*\partial_k\phi_2)$. Some simulations just show one (outter) isosurface of $\phi_2^2$ (in blue).
1. Unstable vortons :
This shows the time evolution of thin unstable vortons.
Note that the evolution reflects the fact that simulation is done in a closed box.
Indeed, after destruction due to the pinching instability, the three lumps
should escape. But since they live in a closed box, they can bounce and recombine to form
a bigger lump.
This shows the time evolution of a thin unstable vorton. It corresponds
to the first line displayed in Fig. 3 of tha paper. This is a
$m=6$, $Q=24000$ vorton.
This shows the time evolution of a thin unstable vorton. It corresponds
to the second line displayed in Fig. 3 of tha paper. This is a
$m=3$, $Q=12000$ vorton.
2. Stable vorton :
This shows the (relativistic) dynamical evolution of a thick, stable vorton.
This shows the time evolution of a thick stable vorton. It correspond
to the third line displayed in Fig. 3 of tha paper. This is a
$m=1$, $Q=6000$ vorton.
References and useful readings

J. Garaud, E. Radu and M. S. Volkov,
Stable cosmic vortons
Phys. Rev. Lett. 111, 171602, (2013). Link , arXiv 
E. Radu and M. S. Volkov,
Existence of stationary, nonradiating ring solitons in field theory: knots and vortons.
Phys. Rept. 468, 101151, (2008). Link , arXiv 
R. A. Battye and P. M. Sutcliffe,
Vorton construction and dynamics.
Nucl. Phys. B 814, 180194, (2009). Link , arXiv