Discrete kinks in Klein-Gordon equations typically have two equilibrium configurations, an unstable one with maximum potential energy and a stable one with minimal energy. The difference between the kink energies in these two configurations gives the height of the Peierls-Nabarro potential. The maximal gradient of this potential gives the minimum force needed to set the kink in motion. It has been shown that some exceptional, non-integrable discretizations of the Klein-Gordon equation have zero static Peierls-Nabarro potential. An arbitrarily small external force in such models results in kink acceleration. Here several methods that give discrete Klein-Gordon models with zero static Peierls-Nabarro potential will be reviewed. Conservation laws which are satisfied for these discrete equations will be mentioned.
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