Normal form for travelling kinks in discrete Klein-Gordon lattices, Phys

We study travelling kinks in the spatial discretizations of the nonlinear Klein–Gordon equa-tion, 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 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 per-sistence of bounded solutions (heteroclinic connections between periodic solutions near adjacent equilibrium points) in the full differential advanced-delay equation with the technique of center manifold reduction. Existence and persistence of multiple kinks in the discrete sine–Gordon equa-tion are discussed in connection to recent numerical results of [ACR03] and results of our normal form analysis.