Revisiting Multi-breathers in the discrete Klein-Gordon equation: A Spatial Dynamics Approach

Abstract

We consider the existence and spectral stability of multi-breather structures in the discrete Klein-Gordon equation, both for soft and hard symmetric potentials. By taking a spatial dynamics approach and an appropriate approximation of the system on a finite-dimensional Hilbert space, we use Lin's method to construct multi-breathers from a sequence of well-separated copies of a primary, single-site breather. We then locate the eigenmodes in the Floquet spectrum associated with the interaction between the individual breathers of such multi-breather states by reducing the spectral problem to a matrix equation. Analytical expressions for these eigenmodes for the approximate system are obtained in terms of the primary breather and its kernel eigenfunctions, and these are found to be in very good agreement with the numerical Floquet spectrum results. This is supplemented with results from numerical timestepping experiments, which are interpreted using the spectral computations.