Normal form for travelling kinks in discrete Klein–Gordon lattices

Abstract

We study travelling kinks in the spatial discretizations of the nonlinear Klein–Gordon equation, which include the discrete ϕ 4 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.