| We prove the existence and linear stability of breather solutions in a one-dimensional Hamiltonian infinite lattice system via the KAM technique. The extension of KAM theory to infinite-dimensional systems was initiated in the mid 1980's by Fröhlich, Spencer and Wayne and by Vittot and Bellissard (VB), it then found a firm ground in the PDEs setting in the works of Craig, Wayne, Bourgain, Pöschel and Kuksin. The use of KAM theory to find breather solutions (quasi-periodic in time and exponentially localized in space) in lattice systems (e.g., VB), was suggested by S. Aubry in the mid 1990's and later carried out by X. Yuan (2002). For a system of identical, weakly-coupled anharmonic oscillators with general on-site potentials and under the effect of long-range interaction, we establish the existence of quasi-periodic motions which, in the uncoupled limit, correspond to any number of N excited lattice sites oscillating altogether quasi-periodically, while all other sites remain at rest. The frequencies of the excited sites in the perturbed case are only slightly deformed from those of the uncoupled case, while the amplitudes of oscillation of all other sites decrease exponentially with the lattice index. This result follows as a corollary of an abstract KAM type of theorem whose proof we will outline and whose importance resides in its applicability to more general 1d lattice systems than the one we will describe in the application. Our KAM theorem is more general than Yuan's analogous theorem and our proof is made simpler for systems of physical interest. |
