Depinning of discommensurations for tilted Frenkel-Kontorova chains
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912002168848384 |
|---|---|
| author | Baesens, Claude MacKay, Robert S. Qin, Wen-Xin |
| author_facet | Baesens, Claude MacKay, Robert S. Qin, Wen-Xin |
| contents | For an untilted Frenkel-Kontorova chain and any rational $p/q$, Aubry and Mather proved there are minimising equilibrium states that are left- and right-asymptotic to neighbouring pairs of spatially periodic minimisers of type $(p,q)$. They are known as {\em discommensurations} (or kinks or fronts), {\em advancing }if the right-asymptotic equilibrium is to the right of the left-asymptotic one, {\em retreating} otherwise. Following work of Middleton, Floria \& Mazo and Baesens \& MacKay, there is a threshold tilt $F_d(p/q)\ge 0$ up to which there continue to be periodic equilibria of type $(p,q)$ and above which there is a globally attracting periodically sliding solution in the space of sequences of type $(p,q)$.
In this paper, we prove that there are values $F_d(p/q\pm)$ of tilt with $0\le F_d(p/q\pm) \le F_d(p/q)$, generically positive and less than $F_d(p/q)$, up to which there continue to be equilibrium advancing or retreating discommensurations, respectively, and such that for $F_d(p/q\pm) < F < F_d(p/q)$ there are periodically sliding discommensurations, apart perhaps from exceptional cases with both a degenerate type $(p,q)$ equilibrium and a degenerate advancing equilibrium discommensuration. We give examples, however, to show that equilibrium and periodically sliding discommensurations may co-exist, both above and below $F_d(p/q\pm)$, so the case of discommensurations is not as clean as that of periodic configurations.
On the way, we prove that $F_d(ω) \to F_d(p/q\pm)$ as $ω\searrow p/q$ or $\nearrow p/q$ respectively. Finally, we prove that $F_d(p/q\pm)=0$ is equivalent to the existence of a rotational invariant circle consisting of periodic orbits of type $(p,q)$ and right-going (respectively left-going) separatrices, for the corresponding twist map on the cylinder. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14124 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Depinning of discommensurations for tilted Frenkel-Kontorova chains Baesens, Claude MacKay, Robert S. Qin, Wen-Xin Dynamical Systems 34C12, 34C15, 34K13, 37C65, 37L60, 70F45 For an untilted Frenkel-Kontorova chain and any rational $p/q$, Aubry and Mather proved there are minimising equilibrium states that are left- and right-asymptotic to neighbouring pairs of spatially periodic minimisers of type $(p,q)$. They are known as {\em discommensurations} (or kinks or fronts), {\em advancing }if the right-asymptotic equilibrium is to the right of the left-asymptotic one, {\em retreating} otherwise. Following work of Middleton, Floria \& Mazo and Baesens \& MacKay, there is a threshold tilt $F_d(p/q)\ge 0$ up to which there continue to be periodic equilibria of type $(p,q)$ and above which there is a globally attracting periodically sliding solution in the space of sequences of type $(p,q)$. In this paper, we prove that there are values $F_d(p/q\pm)$ of tilt with $0\le F_d(p/q\pm) \le F_d(p/q)$, generically positive and less than $F_d(p/q)$, up to which there continue to be equilibrium advancing or retreating discommensurations, respectively, and such that for $F_d(p/q\pm) < F < F_d(p/q)$ there are periodically sliding discommensurations, apart perhaps from exceptional cases with both a degenerate type $(p,q)$ equilibrium and a degenerate advancing equilibrium discommensuration. We give examples, however, to show that equilibrium and periodically sliding discommensurations may co-exist, both above and below $F_d(p/q\pm)$, so the case of discommensurations is not as clean as that of periodic configurations. On the way, we prove that $F_d(ω) \to F_d(p/q\pm)$ as $ω\searrow p/q$ or $\nearrow p/q$ respectively. Finally, we prove that $F_d(p/q\pm)=0$ is equivalent to the existence of a rotational invariant circle consisting of periodic orbits of type $(p,q)$ and right-going (respectively left-going) separatrices, for the corresponding twist map on the cylinder. |
| title | Depinning of discommensurations for tilted Frenkel-Kontorova chains |
| topic | Dynamical Systems 34C12, 34C15, 34K13, 37C65, 37L60, 70F45 |
| url | https://arxiv.org/abs/2408.14124 |