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Normal form for travelling kinks in discrete Klein-Gordon lattices

Gerard Iooss·Dmitry Pelinovsky
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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 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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Gerard Iooss, Dmitry Pelinovsky. 2005-10-21. Normal form for travelling kinks in discrete Klein-Gordon lattices. https://doi.org/10.1016/j.physd.2006.03.012

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