postscript
1 Introduction...
Abstract...
3 Vortices with Non-symmetric ...
We consider a model with a 
symmetry explicitly broken to a 
. This
breaking can be realized by the Lagrangian density (Vilenkin & Shellard (1985), Axenides & Perivolaropoulos (1997))
where 
is a complex scalar field. After a rescaling
The corresponding equation of motion for the field 
is
The potential takes the form
For 
it has the shape of a ``saddle hat'' potential i.e. at 
there is a local minimum in the 
direction but a local maximum in
the 
(Fig.1). For this range of values of 
the
equation of motion admits the well known static kink solution
It corresponds to a symmetric domain wall since in the core of the
soliton the full symmetry of the Lagrangian is manifest (
) and the
topological charge arises as a consequence of the behaviour of the field at
infinity (
).
For 
the local minimum in the 
direction becomes a local
maximum but the vacuum manifold remains disconnected, and the 
symmetry
remains. This type of potential may be called a ``Napoleon hat'' potential in
analogy to the Mexican hat potential that is obtained in the limit 
and corresponds to the restoration of the 
vacuum
manifold.
Figure 1: (a) The domain wall potential has a local maximum
at 
in the 
direction. (b) For 
(
)
this point is a local maximum (minimum) in the 
direction.
The form of the potential however implies that the symmetric wall solution may
not be stable for 
since in that case the potential energy favours a
solution with 
. However, the answer is not obvious because for
, 
would save the wall some potential energy but
would cost additional gradient energy as 
varies from a constant value
at 
to 0 at infinity. Indeed a stability analysis was performed by
introducing a small perturbation about the kink solution reveals the presence
of negative modes for 
For the
range of values 
the potential takes the shape of a ``High
Napoleon hat''. We study the full non-linear static field equations obtained
from (6) for a typical value of 
with boundary conditions
Figure 2: Field configuration for a symmetric wall with
.
Figure 3: Field configuration for a non-symmetric wall with
.
Using a relaxation method based on collocation at Gaussian points
(Press et al. (1993)) to solve the system (6) of second order non-linear equations we
find that for 
the solution relaxes to the expected form
of (7) for 
while 
(Fig.2). For 
we find 
and 
(Fig.3) obeying
the boundary conditions (13), (14) and giving the explicit solution for the
non-symmetric domain wall. In both cases we also plot the analytic solution (7)
stable only for 
for comparison (bold dashed line). As
expected the numerical and analytic solutions are identical for 
(Fig.2).
We now proceed to present results of our study on the evolution of bubbles of a
domain wall. We constructed a two dimensional simulation of the field evolution
of domain wall bubbles with both symmetric and non-symmetric core. In
particular we solved the non-static field equation (6) using a leapfrog
algorithm (Press et al. (1993)) with reflective boundary conditions. We used an 
lattice and in all runs we retained 
thus satisfying the Cauchy stability criterion for the timestep
and the lattice spacing 
. The initial conditions were those
corresponding to a spherically symmetric bubble with initial field ansatz
where 
and 
is the initial radius of the bubble.
Energy was conserved to within 2% in all runs. For 
in the region of
symmetric core stability the imaginary initial fluctuation of the field 
decreased and the bubble collapsed due to tension in a spherically
symmetric way as expected.
Figure 4: Initial field configuration for a non-symmetric
spherical bubble wall with 
.
Figure 5: Evolved field configuration (
, 90
timesteps) for a non-symmetric initially spherical bubble wall with 
.
For 
in the region of values corresponding to having a non-symmetric
stable core the evolution of the bubble was quite different. The initial
imaginary perturbation increased but even though dynamics favoured the increase
of the perturbation, topology forced the 
to stay at zero along a
line on the bubble: the intersections of the bubble wall with the 
-axis
(Figs 4 and 5). Thus in the region of these points, surface
energy (tension) of the bubble wall remained larger than the energy on other
points of the bubble. The result was a non-spherical collapse with the
-direction of the bubble collapsing first (Fig.5).