# Dynamics of Domain Wall Strings

The following movies correspond to some of the simulations performed for the paper:

- “
**Dynamics of Domain Wall Strings**“, Jose J. Blanco-Pillado, Daniel Jiménez-Aguilar, José M. Queiruga and Jon Urrestilla, e-Print:

**Radiation from Standing Wave Domain wall strings**

We show in this movie the field theory simulation of a standing wave for a domain wall string. The frequency of the oscillation is below the threshold of the mass of the scalar field radiation in the bulk. This suppresses heavily the radiation from the oscillating string which follows the Nambu-Goto prediction very accurately. We show for comparison the Nambu-Goto prediction (in white) on top of the field theory description.

We can extract the position of the domain wall string directly from the field theory simulation and compare it to the Nambu-Goto prediction. We show in the following movie how the two approaches give exactly the same dynamics.

Going to higher frequencies for the standing waves, we see a slight departure between field theory and Nambu-Goto.

**Propagating wiggles on the domain wall string**

The transverse excitations of the domain wall string propagate at the speed of light on a straigh segment. One can easily show this to be the case in the Nambu-Goto action. Moreover, there is a full non-linear solution of field theory that describes this excitations. Here we show in this movie how one can construct these solutions and how they can preserve their form without any radiation present. The agreement between the field theory motion and the NG prediction is clear from this simulation.

**Collision of wiggles on the domain wall string**

The collision between wiggles on the domain wall string can have different behaviour depending whether or not the collision induces the presence of large curvature regions. Let us show some of these differences in our numerical experiments.

If the initial wiggles have a sufficiently small amplitude the collision does not produce any radiation and their shape emerges from the collision without any change. However, there is a small delay on their travelling velocity compared to the pure wave equation. This is the sole effect of the non-linear interaction of the effective NG theory on the worldsheet. We show an example of such simulation where we present together the field theory simulation, the NG description (white) and the pure linear theory (green).

For wiggles with a larger amplitude, their overlap leads to the formation of a higher curvature region that produces non-perturbative radiation. This produces a departure between the position inferred from field theory and the one predicted in NG. This process keeps happening until the wiggles have decreased their amplitude enough by sheding part of their energy in each collision. After a few such collisions the wiggles are again mild enough and their collision does not lead to modifications of the NG dynamics. We demonstrate this by reconstructing the NG string after the first 6 collisions and show that the subsequent evolution in field theory is identical to the NG.

**Parametric resonance excitation of a standing wave**

In the following video we show the lattice field theory simulation of a domain wall string initially excited with a homogeneous bound state of amplitude A(t=0)= . We see how the subsequent evolution of this excited state triggers the resonant aplification of a standing wave of the position of the domain wall string.

If our starting point is an excited domain wall string with a bound state with some finite wavelength along the string one can show that the subsequent evolution leads to the parametric amplification of 2 different modes of the position of the string. (See the paper for an analytic understanding of this process).