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

TL;DR

This paper analyzes the Birkhoff coordinates of the Toda lattice as the number of particles grows large, demonstrating their properties, and applies these results to understand energy localization and transfer in the FPU model over extended times.

Contribution

It establishes the analytic properties of Birkhoff coordinates in the large particle limit and applies these to prove energy localization in the FPU model over longer times than previously known.

Findings

Birkhoff coordinate transformation maps complex balls with radius scaling as 1/N^2.

Energy remains localized in a Fourier mode packet for exponentially decreasing wave numbers.

Energy localization persists for longer times in the FPU model than earlier results indicated.

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 $R/N^\alpha$ (in discrete Sobolev-analytic norms) into a ball of radius $R'/N^\alpha$ (with $R,R'>0$ independent of $N$) if and only if $\alpha\geq2$. 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 $R/N^2$, $0<R\ll 1$, 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.