Asymptotic analysis of small energy breathers for the nonlinear Klein-Gordon equation

TL;DR

This paper analyzes the behavior of small energy breathers in nonlinear Klein-Gordon equations, showing they decompose into decoupled solitons resembling sine-Gordon breathers, with implications for their spatial localization.

Contribution

It provides an asymptotic decomposition of small energy breathers into explicit sine-Gordon-like solitons and establishes conditions for their spatial distribution.

Findings

Breathers decompose into decoupled solitons as energy approaches zero

Distance between solitons increases with decreasing energy

Breathers are not localized in bounded regions under certain conditions

Abstract

For a class of nonlinear Klein-Gordon equations, we prove that in the small energy limit, any sequence of breathers decomposes into a finite sum of decoupled, periodically modulated canonical solitons. Each of these solitons is asymptotically equal to an explicit sine-Gordon breather and the distance between them grows to infinity as the energy decreases to 0. Finally we prove that none of these breathers is centered in a bounded set provided that a certain non resonance condition holds.