Resonant Equilibrium Configurations in Quasi-Periodic Media: KAM Theory

Abstract

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 main motivation for the study of these solutions is that they are present when there are constant external forces while the nonresonant solutions disappear. 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 which are 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 that ensures the linearization of the equilibrium equation factorizes. To solve the extra equation 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 [R. de la Llave, X. Su, and L. Zhang, J. Stat. Phys., 162 (2016), pp. 1522--1538] 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.