Normal form for travelling kinks in discrete Klein–Gordon lattices

We study travelling kinks in the spatial discretizations of the nonlinear Klein–Gordon equation, which include the discrete ϕ4 lattice and the discrete sine-Gordon lattice. The differential advance-delay equation for travelling kinks is reduced to the normal form, a scalar fourth-order differential equation, near the quadruple zero eigenvalue. We show numerically the non-existence of monotonic kinks (heteroclinic orbits between adjacent equilibrium points) in the fourth-order equation. Making generic assumptions on the reduced fourth-order equation, we prove the persistence of bounded solutions (heteroclinic connections between periodic solutions near adjacent equilibrium points) in the full differential advance-delay equation with the technique of centre manifold reduction. Existence of multiple kinks in the discrete sine-Gordon equation is discussed in connection to recent numerical results of Aigner et al. [A.A. Aigner, A.R. Champneys, V.M. Rothos, A new barrier to the existence of moving kinks in Frenkel–Kontorova lattices, Physica D 186 (2003) 148–170] and results of our normal form analysis.