Asymptotic analysis of small energy breathers for the nonlinear Klein-Gordon equation
Michał Kowalczyk, Yvan Martel
TL;DR
This work analyzes small-energy breathers for a one-dimensional nonlinear Klein-Gordon equation with a localized potential. By rescaling and performing a nonlinear profile decomposition, the authors prove that any sequence of breathers with vanishing energy decomposes into a finite sum of decoupled, periodically modulated sine-Gordon breathers, with common amplitude and periods that converge to those of the sine-Gordon family; the centers of these breathers separate to infinity as energy vanishes. A compactness framework is used to pass to a limit where the dominant Fourier mode satisfies a stationary nonlinear system whose solutions are given by scaled Q-profiles, yielding explicit emergence of sine-Gordon-like components. A Fermi golden rule argument shows that under a natural nonresonance condition on the potential (namely, $\\widehat{U}(\\sqrt{3})\\neq 0$), none of the breathers remain bounded in space, forcing all centers to escape to infinity. Overall, the paper establishes a precise, quantitative rigidity for small breathers and clarifies how near-Sine-Gordon dynamics organize into a structured multi-breather configuration.
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.