Dynamics of periodic Toda chains with a large number of particles

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 $n$ $\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.