2005/10/21 by Gerard Iooss, Dmitry Pelinovsky · 1 citation
Mathematics · #math.DS #math.CA #msc:37K60
paper · pdf · doi:10.1016/j.physd.2006.03.012
arxiv created 2005/10/21 · arxiv updated 2009/12/01
We study travelling kinks in the spatial discretizations of the nonlinear Klein--Gordon equation, 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 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 \citeACR03 and results of our normal form analysis.