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arXiv · 2611.08885

KAM theory for quasi-periodic equilibrium of generalized Frenkel-Kontorova model in $\mathbb{Z}^{d}$

Xiaolong He·Zuhong You
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Abstract

We extend the KAM theory for one dimensional Frenkel-Kontorova model to multidimensional case in quasi-periodic media. By using the quasi-Newton method, we reduce the linearized operator with quasi-periodic coefficient in the Nash-Moser iteration to the composition of two diagonal operators and a near-identity operator, such that the small divisor problem can be overcome by imposing the simultaneous Diophantine condition. In contrast, we also discuss the application of Craig-Wayne-Bourgain method to the quasi-periodic equilibrium problem.

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Xiaolong He, Zuhong You. 2026-11-06. KAM theory for quasi-periodic equilibrium of generalized Frenkel-Kontorova model in $\mathbb{Z}^{d}$. https://arxiv.org/abs/2611.08885

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