# Dynamics of periodic Toda chains with a large number of particles

Bambusi, D; Kappeler, Thomas; Paul, T (2015). Dynamics of periodic Toda chains with a large number of particles. Journal of Differential Equations, 258(12):4209-4274.

## Abstract

For periodic Toda chains with a large number $N$ of particles we consider states which are $N^{-2}$-close to the equilibrium and constructed by discretizing any given $C^2$-functions with mesh size $N^{-1}$. For such states we derive asymptotic expansions of the Toda frequencies ($\omega^N_n$)$_{0<n<,N}$, and the actions ($1^N_n$)$_{0<n<N}$, both listed in the standard way, in powers of $N^{-1}$ as $N\to$$\infty. At the two edges n \sim1 and N -n\sim1, the expansions of the frequencies are computed up to order N^{-3} with an error term of higher order. Specifically, the coefficients of the expansions of \omega^N_n and \omega^N_{N-1} at order N^{-3} are given by a constant multiple of the nth KdV frequencies \omega^-_n and \omega^+_n of two periodic potentials, {_q-} respectively _{q+}, constructed in terms of the states considered. The frequencies \omega^N_n for n away from the edges are shown to be asymptotically close to the frequencies of the equilibrium. For the actions (1^N_n)_{0<n<N}, asymptotics of a similar nature are derived. For periodic Toda chains with a large number N of particles we consider states which are N^{-2}-close to the equilibrium and constructed by discretizing any given C^2-functions with mesh size N^{-1}. For such states we derive asymptotic expansions of the Toda frequencies (\omega^N_n)_{0<n<,N}, and the actions (1^N_n)_{0<n<N}, both listed in the standard way, in powers of N^{-1} as N\to$$\infty$. At the two edges $n$ $\sim$1 and $N$-$n\sim$1, the expansions of the frequencies are computed up to order $N^{-3}$ with an error term of higher order. Specifically, the coefficients of the expansions of $\omega^N_n$ and $\omega^N_{N-1}$ at order $N^{-3}$ are given by a constant multiple of the nth KdV frequencies $\omega^-_n$ and $\omega^+_n$ of two periodic potentials, ${_q-}$ respectively $_{q+}$, constructed in terms of the states considered. The frequencies $\omega^N_n$ for n away from the edges are shown to be asymptotically close to the frequencies of the equilibrium. For the actions ($1^N_n$)$_{0<n<N}$, asymptotics of a similar nature are derived.

## Citations

2 citations in Web of Science®
2 citations in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 15 June 2015 27 Jan 2016 09:25 16 Jun 2016 00:00 Elsevier 0022-0396 https://doi.org/10.1016/j.jde.2015.01.031
Permanent URL: https://doi.org/10.5167/uzh-111936

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