  The $\varphi^4$-theory is ubiquitous as a low-energy effective description of
processes in all fields of physics ranging from cosmology and particle physics
to biophysics and condensed matter theory. The topological defects, or kinks,
in this theory describe stable, particle-like excitations. In practice, these
excitations will necessarily encounter impurities or imperfections in the
background potential as they propagate. Here, we describe the interaction
between kinks and various types of realistic impurity models. We find that
realistic impurities behave qualitatively like the well-studied, idealized
delta function impurities, but that significant quantitative differences appear
in both the characteristics of localized impurity modes, and in the collision
dynamics. We also identify a particular regime of kink-impurity interactions,
in which kinks loose all of their kinetic energy upon colliding with an
impurity.
