2013/09/21 by Bambusi, Dario, Kappeler, Thomas, Paul, Thierry
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1309.5441
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 (ωNn)0 < n < N and the actions (INn)0 < n < N, both listed in the standard way, in powers of N-1 as N → ∞. %listed in accordance with the ordering of the frequencies at the equilibrium, %(2 sin \fracnπ N)0 < n < 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 ωNn and ωNN-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 ωNn for n away from the edges are shown to be asymptotically close to the frequencies of the equilibrium. For the actions (INn)0 < n < N, asymptotics of a similar nature are derived.