Breather continuation from infinity in nonlinear oscillator chains

TL;DR

This paper proves the existence of large-amplitude, localized, time-periodic breathers in a discrete Klein-Gordon chain with compact support on the potential derivative, using a novel method that captures solutions beyond classical approaches.

Contribution

It introduces a new analytical method for finding breather solutions in nonlinear oscillator chains, extending beyond the classical anti-continuum limit.

Findings

Breathers exist at small coupling with nonresonance conditions.

Amplitude and period of breathers tend to infinity as coupling approaches zero.

The method captures solution branches not accessible by traditional techniques.

Abstract

Existence of large-amplitude time-periodic breathers localized near a single site is proved for the discrete Klein--Gordon equation, in the case when the derivative of the on-site potential has a compact support. Breathers are obtained at small coupling between oscillators and under nonresonance conditions. Our method is different from the classical anti-continuum limit developed by MacKay and Aubry, and yields in general branches of breather solutions that cannot be captured with this approach. When the coupling constant goes to zero, the amplitude and period of oscillations at the excited site go to infinity. Our method is based on near-identity transformations, analysis of singular limits in nonlinear oscillator equations, and fixed-point arguments.