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 and the actions (I-n(N))0<n<N, both listed in the standard way, in powers of N-1 as N -> infinity. At the two edges n similar to 1 and N similar to n similar to 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 nth KdV frequencies omega(-)(n), and omega(+)(n) of two periodic potentials, q(-) respectively q(+), constructed in terms of the states considered. The frequencies cog 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. (C) 2015 Elsevier Inc. All rights reserved.