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).