准周期介质中的共振平衡位形：KAM理论
Resonant equilibrium configurations in quasi-periodic media: KAM theory
摘要：
当解的频率与基底的频率共振时，我们发展了一个关于拟周期Frenkel-Kontorova模型中拟周期解平衡方程的后验KAM理论.我们发展的KAM理论在方法和结论上都与更常用的哈密顿系统的KAM理论或准周期介质中的KAM理论有很大的不同，后者的解的频率是关于介质频率的丢番图。主要的困难是，我们不能使用变换（如在哈密顿的情况下），也没有沃德身份（如在丢番图的频率与那些媒体的情况下）。我们使用的技术是添加一个额外的方程，以确保平衡方程的线性化。为了解这个额外的方程，需要一个额外的反项。我们将这种现象与KAM理论中的其他现象进行了比较。这种技术似乎可以用于其他几个问题。作为结论，我们得到了当位势空间中的余维流形上的位势时，前一篇文章\cite{SuZL15}中的扰动展开式收敛.证明的方法也导致有效的（低存储要求和低运算计数）算法来计算拟周期解。
英文摘要：
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 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 \cite{SuZL15} 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.
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