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We examine various recently proposed discretizations of the well-known$\phi^4$ field theory. We compare and contrast the properties of theirfundamental solutions including the nature of their kink-type solitary wavesand the spectral properties of the linearization around such waves. We studythese features as a function of the lattice spacing $h$, as one deviates fromthe continuum limit of $h \to 0$. We then proceed to a more ``stringent''comparison of the models, by discussing the scattering properties of akink-antikink pair for the different discretizations. These collisions arewell-known to possess properties that quite sensitively depend on the initialspeed even at the continuum limit. We examine how typical model behaviors aremodified in the presence (and as a function) of discreteness and attempt toextract qualitative trends and issue pertinent warnings about some of thesurprising resulting properties.Comment: 21 pages, 16 figures. Submitted for publicatio

physical review. e, statistical, nonlinear and soft matter physics

Comparative study of different discretizations of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>ϕ</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:math>model

Comparative study of different discretizations of theϕ4model