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G. Iooss and D.E. Pelinovsky

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Normal form for travelling breathers in discrete Klein-Gordon lattices

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Physica D 216, 327-345 (2006)

**Abstract:**

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 [ACR03] and results
of our normal form analysis.

**Keywords:**

DISCRETE KLEIN-GORDON EQUATIONS, TRAVELLING KINKS, PHI-4 MODEL,
SINE-GORDON MODEL, NORMAL FORMS, CENTER MANIFOLD REDUCTIONS,
PERSISTENCE ANALYSIS, NUMERICAL ANALYSIS