KAM theory for quasi-periodic equilibria in 1D quasi-periodic media: II. Long-range interactions

We consider Frenkel-Kontorova models corresponding to one-dimensional quasi-crystal with non-nearest neighbor and many-body interactions. We formulate and prove a KAM type theorem which establishes the existence of quasi-periodic solutions. The interactions we consider do not need to be of finite range or involve finitely many particles, but have to decay sufficiently fast with respect to the distance of the position of the atoms. The KAM theorem we present has an a posteriori format. We do not need to assume that the system is close to integrable. We just assume that there is an approximate solution for the functional equation which satisfies some non-degeneracy conditions.