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 C2−functions with mesh size N−1. For such states we derive asymptotic expansions of the Toda frequencies (ωN n )0<n<N and the actions (I N n )0<n<N , both listed in the standard way, in powers of N−1 as N → ∞. At the two edges n ∼ 1 and N − n ∼ 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 ωN n and ω N N−n at order N −3 are given by a constant multiple of the n’th KdV frequencies ω− n and ω + n of two periodic potentials, q− respectively q+, constructed in terms of the states considered. The frequencies ωN n for n away from the edges are shown to be asymptotically close to the frequencies of the equilibrium. For the actions (IN n )0<n<N , asymptotics of a similar nature are derived.