In this paper we study the Birkhoff coordinates (Cartesian action angle coordinates) of the Toda lattice with periodic boundary condition in the limit where the number N of the particles tends to infinity. We prove that the transformation introducing such coordinates maps analytically a complex ball of radius R/Nα (in discrete Sobolev-analytic norms) into a ball of radius R′/Nα (with R,R′>0 independent of N) if and only if α≥2. Then we consider the problem of equipartition of energy in the spirit of Fermi-Pasta-Ulam. We deduce that corresponding to initial data of size R/N2, 0<R≪1, and with only the first Fourier mode excited, the energy remains forever in a packet of Fourier modes exponentially decreasing with the wave number. Finally we consider the original FPU model and prove that energy remains localized in a similar packet of Fourier modes for times one order of magnitude longer than those covered by previous results which is the time of formation of the packet. The proof of the theorem on Birkhoff coordinates is based on a new quantitative version of a Vey type theorem by Kuksin and Perelman which could be interesting in itself.

Birkhoff coordinates for the Toda lattice in the limit of infinitely many particles with an application to FPU / Bambusi, D.; Maspero, A.. - In: JOURNAL OF FUNCTIONAL ANALYSIS. - ISSN 0022-1236. - 270:5(2016), pp. 1818-1887. [10.1016/j.jfa.2015.08.003]

Birkhoff coordinates for the Toda lattice in the limit of infinitely many particles with an application to FPU

Maspero, A.
2016

Abstract

In this paper we study the Birkhoff coordinates (Cartesian action angle coordinates) of the Toda lattice with periodic boundary condition in the limit where the number N of the particles tends to infinity. We prove that the transformation introducing such coordinates maps analytically a complex ball of radius R/Nα (in discrete Sobolev-analytic norms) into a ball of radius R′/Nα (with R,R′>0 independent of N) if and only if α≥2. Then we consider the problem of equipartition of energy in the spirit of Fermi-Pasta-Ulam. We deduce that corresponding to initial data of size R/N2, 0
270
5
1818
1887
http://www.elsevier.com/inca/publications/store/6/2/2/8/7/9/index.htt
https://arxiv.org/abs/1407.4315
Bambusi, D.; Maspero, A.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/63197
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 11
  • ???jsp.display-item.citation.isi??? 11
social impact