D.E. Pelinovsky, T. Penati, and S. Paleari

Existence and stability of Klein–Gordon breathers in the small-amplitude limit

Mathematics of Wave Phenomena, Editors: W. Dörfler, M. Hochbruck, D. Hundertmark, W. Reichel, A. Rieder, R. Schnaubelt, and B. Schörkhuber, Trends in Mathematics (Birkhäuser Basel) (2020), 251-278

Abstract:
We consider a discrete Klein–Gordon (dKG) equation on Z^d in the limit of the discrete nonlinear Schrödinger (dNLS) equation, for which small-amplitude breathers have precise scaling with respect to the small coupling strength. 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, is obtained via the normal form technique, together with higher order approximations of the KG breather through perturbations of the corresponding dNLS soliton.

Keywords:
discrete Klein-Gordon equation, discrete nonlinear Schrodinger equation, breathers, existence and stability, normal form theorem.