KAM Theory for Quasi-periodic Equilibria in One-Dimensional Quasi-periodic Media

Abstract

We consider Frenkel--Kontorova models corresponding to one-dimensional quasi crystals. We present a KAM theory for quasi-periodic equilibria. The theorem presented has an a posteriori format. We show that, given an approximate solution of the equilibrium equation, which satisfies some appropriate nondegeneracy conditions, there is a true solution nearby. This solution is locally unique. Such a posteriori theorems can be used to validate numerical computations and also lead immediately to several consequences: (a) existence to all orders of perturbative expansion and their convergence, (b) bootstrap for regularity, (c) an efficient method to compute the breakdown of analyticity. Since the system does not admit an easy dynamical formulation, the method of proof is based on developing several identities. These identities also lead to very efficient algorithms. We note that the quasi-periodic solutions considered here correspond to the correctors considered in homogenization theory. In contrast with the one frequency case, the variational construction of solutions may fail to exist.