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 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.
Keywords
Cite
@article{arxiv.math/0510474,
title = {Normal form for travelling kinks in discrete Klein-Gordon lattices},
author = {Gerard Iooss and Dmitry Pelinovsky},
journal= {arXiv preprint arXiv:math/0510474},
year = {2009}
}