On the variational structure of breather solutions I: Sine-Gordon equation

In this paper we describe stability properties of the Sine-Gordon breather solution. These properties are first described by suitable variational elliptic equations, which also implies that the stability problem reduces in some sense to (i) the study of the spectrum of explicit linear systems, and (ii) the understanding of how bad directions (if any) can be controlled using low regularity conservation laws. Then we discuss spectral properties of a fourth-order linear matrix system. Using numerical methods, we confirm that all spectral assumptions leading to the H2×H1 stability of SG breathers are numerically satisfied, even in the ultra-relativistic, singular regime.