Normal form for travelling kinks in discrete Klein-Gordon lattices

Details Normal form for travelling kinks in discrete Klein-Gordon lattices Normal form for travelling kinks in discrete Klein-Gordon lattices

2005-10-21

arxiv.org/abs/math/0510474v1

Mathematics

Dynamical Systems

Scientific paper

10.1016/j.physd.2006.03.012

We study travelling kinks in the spatial discretizations of the nonlinear Klein--Gordon equation, which include the discrete $\phi^4$ 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 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 advanced-delay equation with the technique of center manifold reduction. Existence and persistence of multiple kinks in the discrete sine--Gordon equation are discussed in connection to recent numerical results of \cite{ACR03} and results of our normal form analysis.

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