Dynamics of periodic Toda chains with a large number of particles

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 $(\omega^N_n){0 < n < N}$andtheactions$(I^N_n){0 < n < N},$ both listed in the standard way, in powers of $N^{-1}$ as $N\to \mathrm{\infty }$. %listed in accordance with the ordering of the frequencies at the equilibrium, %$(2\sin \frac{n\pi }{N}{)}_{0<n<N}$. 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 ${\omega }_{N-n}^N$ at order $N^{-3}$ are given by a constant multiple of the n’th 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 $(I_n^N{)}_{0<n<N},$ asymptotics of a similar nature are derived.