15-23 Lei Zhang, Xifeng Su, Rafael de la Llave
Equilibrium quasi-periodic configurations with resonant frequencies in quasi-periodic media II: KAM theor configurations with resonant frequencies in quasi-periodic media II: KAM theory} (276K, pdf) Mar 11, 15
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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 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 th e media. The main difficulty is that we cannot use transformations (as in the Ha miltonian 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 th e 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 [SuZL14] converge when the potentials are in a codimension one manifo ld 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.

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