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

TL;DR

This paper analyzes the existence and stability of multi-breather solutions in the discrete Klein-Gordon equation using a spatial dynamics approach, Fourier mode projection, and numerical validation.

Contribution

It introduces an analytical method combining Fourier mode projection and Lin's method to construct and analyze multi-breathers in the discrete Klein-Gordon equation.

Findings

Eigenmodes are accurately predicted by the analytical matrix equation.

Spectral stability correlates well with numerical Floquet spectrum results.

Numerical experiments support the analytical spectral predictions.

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. To obtain analytical results, we project the system onto a finite-dimensional Hilbert space consisting of the first $M$ Fourier modes, for arbitrary $M$. On this approximate system, we then take a spatial dynamics approach and use Lin's method to construct multi-breathers from a sequence of well-separated copies of the 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. Expressions for these eigenmodes for the approximate, finite-dimensional 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.