Existence, linear stability and long-time nonlinear stability of Klein-Gordon breathers in the small-amplitude limit

In this paper we consider a discrete Klein-Gordon (dKG) equation on $\ZZ^d$ in the limit of the discrete nonlinear Schrodinger (dNLS) equation, for which small-amplitude breathers have precise scaling with respect to the small coupling strength $\eps$. By using the classical Lyapunov-Schmidt method, we show existence and linear stability of the KG breather from existence and linear stability of the corresponding dNLS soliton. Nonlinear stability, for an exponentially long time scale of the order $\mathcal{O}(\exp(\eps^{-1}))$, is also obtained via the normal form technique, together with higher order approximations of the KG breather through perturbations of the corresponding dNLS soliton.