Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations

TL;DR

This paper establishes a comprehensive spectral stability theorem for multi-site breathers in Klein-Gordon lattices, analyzing how phase, distance, and amplitude influence stability, and uncovers bifurcations in soft nonlinear potentials.

Contribution

It provides the most general stability theorem for multi-site breathers, extending previous work to include non-adjacent sites, large amplitudes, and bifurcation phenomena.

Findings

Stability depends on phase difference and site separation.

Large-amplitude breathers in soft potentials can become unstable.

Identifies symmetry-breaking bifurcations near 1:3 resonance.

Abstract

We prove the most general theorem about spectral stability of multi-site breathers in the discrete Klein-Gordon equation with a small coupling constant. In the anti-continuum limit, multi-site breathers represent excited oscillations at different sites of the lattice separated by a number of "holes" (sites at rest). The theorem describes how the stability or instability of a multi-site breather depends on the phase difference and distance between the excited oscillators. Previously, only multi-site breathers with adjacent excited sites were considered within the first-order perturbation theory. We show that the stability of multi-site breathers with one-site holes change for large-amplitude oscillations in soft nonlinear potentials. We also discover and study a symmetry-breaking (pitchfork) bifurcation of one-site and multi-site breathers in soft quartic potentials near the points of 1:3 resonance.