Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2007-09-17
Nonlinear Sciences
Exactly Solvable and Integrable Systems
34 pages, 3 figures
Scientific paper
In this paper we investigate periodic FPU chains with an even number of particles. We show that near the equilibrium point, any such chain admits a \emph{resonant} Birkhoff normal form of order four which is \emph{completely integrable} - an important fact which helps explain the numerical experiments of Fermi, Pasta, and Ulam. We analyze the moment map of the integrable approximation of an even FPU chain. Unlike in the case of odd FPU chains these integrable systems (generically) exhibit hyperbolic dynamics. As an application we prove that any FPU chain with Dirichlet boundary conditions admits a Birkhoff normal form up to order four and show that a KAM theorem applies.
Henrici Andreas
Kappeler Thomas
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