Translationally invariant discrete kinks from one-dimensional maps
  For most discretisations of the $\phi^4$ theory, the stationary kink can only be centered either on a lattice site or midway between two adjacent sites. We search for exceptional discretisations which allow stationary kinks to be centered anywhere between the sites. We show that this translational invariance of the kink implies the existence of an underlying one-dimensional map $\phi_{n+1}=F(\phi_n)$. A simple algorithm based on this observation generates three different families of exceptional discretisations. 
I.V. Barashenkov, O.F. Oxtoby, Dmitry E. Pelinovsky