Equilibrium Quasi-periodic Configurations with Resonant Frequencies in Quasi-periodic Media Ii: Kam Theory

We develop an a-posteriori KAM theory for the equilibrium equations for quasi-periodic solutions in a quasi-periodic Frenkel-Kontorova model when the frequency of the solutions resonates with the frequencies of the substratum. The KAM theory we develop is very different both in the methods and in the conclusions from the more customary KAM theory for Hamiltonian systems or from the KAM theory in quasi-periodic media for solutions with frequencies Diophantine with respect to the frequencies of the media. The main difficulty is that we cannot use transformations (as in the Hamiltonian case) nor Ward identities (as in the case of frequencies Diophantine with those of the media). The technique we use is to add an extra equation to make the linearization of the equilibrium equation factorize. This requires an extra counterterm. We compare this phenomenon with other phenomena in KAM theory. It seems that this technique could be used in several other problems. As a conclusion, we obtain that the perturbation expansions developed in the previous paper [SZdlL14] converge when the potentials are in a codimension one manifold in a space of potentials. The method of proof also leads to efficient (low storage requirements and low operation count) algorithms to compute the quasi-periodic solutions. Quasi-periodic Frenkel-Kontorova models, resonant frequencies, equilibria, quasicrystals, Lindstedt series, counterterms, KAM theory [2000] 70K43, 37J40, 52C23