Comparative Study of Different Discretizations of the $ϕ^4$ Model

Roy, Ishani; Dmitriev, Sergey V.; Kevrekidis, Panayotis G.; Saxena, Avadh

Comparative Study of Different Discretizations of the $ϕ^4$ Model

Authors

Roy, Ishani

Dmitriev, Sergey V.

Kevrekidis, Panayotis G.

Saxena, Avadh

Description

We examine various recently proposed discretizations of the well-known $ϕ^4$ 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 and attempt to extract qualitative trends and issue pertinent warnings about some of the surprising resulting properties.
21 pages, 16 figures. Submitted for publication