  We study the dynamics of domain wall solitons in $(2+1)d$ field theories.
These objects are extended along one of the spatial directions, so they also
behave as strings; hence the name of domain wall strings. We show analytically
and numerically that the amount of radiation from the propagation of wiggles on
these objects is negligible except for regions of high curvature. Therefore, at
low curvatures, the domain wall strings behave exactly as the Nambu-Goto action
predicts. We show this explicitly with the use of several different numerical
experiments of the evolution of these objects in a lattice. We then explore
their dynamics in the presence of internal mode excitations. We do this again
by performing field theory simulations and identify an effective action that
captures the relevant interactions between the different degrees of freedom
living on the string. We uncover a new parametric resonance instability that
transfers energy from the internal mode to the position of the domain wall. We
show that this instability accelerates the radiation of the internal mode
energy. We also explore the possibility of exciting the internal mode of the
soliton with the collision of wiggles on the domain wall. Our numerical
experiments indicate that this does not happen unless the wiggles have already
a wavelength of the order of the string thickness. Finally, we comment on the
possible relevance of our findings to cosmological networks of defects. We
argue that our results cast some doubts on the significance of the internal
modes in cosmological applications beyond a brief transient period right after
their formation. This, however, should be further investigated using
cosmological simulations of our model.
