Recurrence times in cubic and quartic Fermi-Pasta-Ulam chains: A shifted-frequency perturbation treatment

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Abstract
A shifted-frequency perturbation scheme is used to derive algebraic expressions for recurrence periods in the harmonic-mode energies of cubic and quartic Fermi-Pasta-Ulam (FPU) chains. Perturbation-theory solutions for the FPU period have been obtained for arbitrary particle numbers N. Although the convergence properties of the solutions for arbitrary N are not known, explicit evaluations of the solutions at specific values of N have been found to be in good agreement with numerical simulations and soliton theories. The perturbation-theory solutions also provide reliable results in the regime where continuum soliton theories do not apply. A beat mechanism for the superperiod phenomena in cubic FPU chains is also proposed, based on the perturbation-theory solutions. The superperiods predicted by this mechanism are generic in cubic chains but do not appear in quartic chains. This result is in agreement with our numerical simulations. An exceptional superperiod that does occur in quartic, chains with N=7, is accounted for by a resonance mechanism unique to this chain.