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. 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.

arXiv comment36 pages, 18 figures

References in corpus (16)

• Discrete Breathers
• Discrete breathers in a forced-damped array of coupled pendula: Modeling, Computation and Experiment
• Dynamics of Lattice Kinks
• When Does Linear Stability Not Exclude Nonlinear Instability ?
• Demonstration of the stability or instability of multibreathers at low coupling
• On the stability of multibreathers in Klein-Gordon chains
• Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations
• Normal form for travelling kinks in discrete Klein-Gordon lattices
• Energy criterion for the spectral stability of discrete breathers
• Asymptotic stability of breathers in some Hamiltonian networks of weakly coupled oscillators
• Multibreather and vortex breather stability in Klein--Gordon lattices: Equivalence between two different approaches
• Existence and spectral stability of multi-pulses in discrete Hamiltonian lattice systems
• Multi-peaked localized states of DNLS in one and two dimensions
• Stationary multi-kinks in the discrete sine-Gordon equation
• Vortex Pairs in the Discrete Nonlinear Schrödinger Equation
• Some breathers and multi-breathers for FPU-type chains