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→∞. 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 ωN-nN at order N-3 are given by a constant multiple of the nth KdV frequencies ωn- and ωn+ 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.
Dynamics of periodic Toda chains with a large number of particles / D. Bambusi, T. Kappeler, T. Paul. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 258:12(2015), pp. 4209-4274. [10.1016/j.jde.2015.01.031]
Dynamics of periodic Toda chains with a large number of particles
D. BambusiPrimo
;
2015
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 C2-functions with mesh size N-1. For such states we derive asymptotic expansions of the Toda frequencies (ωnN)0File | Dimensione | Formato | |
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