An infinite-dimensional Kolmogorov theorem and the construction of almost periodic breathers
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911554721546240 |
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| author | Tong, Zhicheng Li, Yong |
| author_facet | Tong, Zhicheng Li, Yong |
| contents | In this paper, we present two infinite-dimensional Kolmogorov theorems based on non-resonant frequencies of Bourgain's Diophantine type or even weaker conditions. To be more precise, under a Legendre-type nondegeneracy condition for an infinite-dimensional Hamiltonian system, we prove the persistence of a full-dimensional KAM torus with a universally prescribed frequency independent of any spectral asymptotics. As an application, we prove that for a class of perturbed networks with weakly coupled oscillators described by \[\frac{{{{\rm d}^2}{x_n}}}{{{\rm d}{t^2}}} + V'\left( {x_n} \right) = \varepsilon_n {W'\left( {x_{n + 1} - {x_n}} \right) - \varepsilon_{n-1}W'\left( {{x_n} - {x_{n - 1}}} \right)} ,\quad n \in \mathbb{Z},\] or even for more general perturbed networks, frequency-preserving almost periodic breathers do persist, provided that the local potential $ V $ and the coupling potential $ W $ satisfy certain assumptions. In particular, this yields the first frequency-preserving result for the Aubry--MacKay conjecture [MA94,Aub95]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13501 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An infinite-dimensional Kolmogorov theorem and the construction of almost periodic breathers Tong, Zhicheng Li, Yong Dynamical Systems Mathematical Physics 37K60, 82C20, 37L60, 37K55, 37C15 In this paper, we present two infinite-dimensional Kolmogorov theorems based on non-resonant frequencies of Bourgain's Diophantine type or even weaker conditions. To be more precise, under a Legendre-type nondegeneracy condition for an infinite-dimensional Hamiltonian system, we prove the persistence of a full-dimensional KAM torus with a universally prescribed frequency independent of any spectral asymptotics. As an application, we prove that for a class of perturbed networks with weakly coupled oscillators described by \[\frac{{{{\rm d}^2}{x_n}}}{{{\rm d}{t^2}}} + V'\left( {x_n} \right) = \varepsilon_n {W'\left( {x_{n + 1} - {x_n}} \right) - \varepsilon_{n-1}W'\left( {{x_n} - {x_{n - 1}}} \right)} ,\quad n \in \mathbb{Z},\] or even for more general perturbed networks, frequency-preserving almost periodic breathers do persist, provided that the local potential $ V $ and the coupling potential $ W $ satisfy certain assumptions. In particular, this yields the first frequency-preserving result for the Aubry--MacKay conjecture [MA94,Aub95]. |
| title | An infinite-dimensional Kolmogorov theorem and the construction of almost periodic breathers |
| topic | Dynamical Systems Mathematical Physics 37K60, 82C20, 37L60, 37K55, 37C15 |
| url | https://arxiv.org/abs/2508.13501 |