We examine various recently proposed translationally invariant discretizations of the well-known ${\varphi }^4$ field theory. We compare and contrast the properties of their fundamental solutions including the nature of their kink-type solitary waves and the spectral properties of the linearization around such waves. We study these features as a function of the lattice spacing $h$, as one deviates from the continuum limit of $h\to 0$. We then proceed to a more “stringent” comparison of the models, by discussing the scattering properties of a kink-antikink pair for the different discretizations. These collisions are well known to possess properties that quite sensitively depend on the initial speed even at the continuum limit. We examine how typical model behaviors are modified in the presence (and as a function) of discreteness. One of the surprising trends that we observe is the increasing elasticity of kink collisions with deviation from the continuum limit. Another general feature is that the most inelastic kink collisions are observed in the classical discrete ${\varphi }^4$ model, while they are more elastic in the four studied translationally invariant models.

• Received 18 August 2006

DOI:https://doi.org/10.1103/PhysRevE.76.026601