Symmetry and resonance in periodic FPU chains

Abstract

The symmetry and resonance properties of the Fermi Pasta Ulam chain with periodic boundary conditions are exploited to construct a near-identity transformation bringing this Hamiltonian system into a particularly simple form. This `Birkhoff-Gustavson normal form' retains the symmetries of the original system and we show that in most cases this allows us to view the periodic FPU Hamiltonian as a perturbation of a nondegenerate Liouville integrable Hamiltonian. According to the KAM theorem this proves the existence of many invariant tori on which motion is quasiperiodic. Experiments confirm this qualitative behaviour. We note that one can not expect it in lower-order resonant Hamiltonian systems. So the FPU chain is an exception and its special features are caused by a combination of special resonances and symmetries.

21 pages

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• Energy localization on q-tori, long term stability and the interpretation of FPU recurrences
• Nonlinear Lattice Dynamics of Bose-Einstein Condensates
• Tail resonances of FPU q-breathers and their impact on the pathway to equipartition
• Proof of Nishida's conjecture on anharmonic lattices
• Wave Turbulence and thermalization in one-dimensional chains
• Exact discrete resonances in the Fermi-Pasta-Ulam-Tsingou system
• Birkhoff coordinates for the Toda Lattice in the limit of infinitely many particles with an application to FPU
• Modulational instability in isolated and driven Fermi--Pasta--Ulam lattices
• Adiabatic invariants for the FPUT and Toda chain in the thermodynamic limit
• Metastability and dispersive shock waves in Fermi-Pasta-Ulam system
• Direction reversing travelling waves in the Fermi-Pasta-Ulam chain
• Simply improved averaging for coupled oscillators and weakly nonlinear waves
• The $\beta$ Fermi-Pasta-Ulam-Tsingou Recurrence Problem
• An extensive resonant normal form for an arbitrary large Klein-Gordon model
• Simplified normal forms near a degenerate elliptic fixed point in two-parametric families of area-preserving maps
• Some analytic results on the FPU paradox
• Freezing of the optical--branch energy in a diatomic FPU chain
• Three- and four-wave resonances in the nonlinear quadratic Kelvin lattice
• Birkhoff normal form for the periodic Toda lattice
• Traveling and standing waves in coupled pendula and Newton's cradle
• Dispersive Fractalization in Linear and Nonlinear Fermi-Pasta-Ulam-Tsingou Lattices
• Discrete resonant Rossby/drift wave triads: an explicit parameterisation and a fast direct numerical search algorithm
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• Prethermalization in Fermi-Pasta-Ulam-Tsingou chains
• Extensive packet excitations in FPU and Toda lattices
• The inhomogeneous Fermi-Pasta-Ulam chain
• Application of Weak Turbulence Theory to FPU Model
• Near-integrability of periodic Klein-Gordon lattices
• Delocalized periodic vibrations in nonlinear electrical chains
• The 1:2:4 resonance in a particle chain
• Korteweg--de Vries limit for the Fermi--Pasta--Ulam system
• Stable motions of high energy particles interacting via a repelling potential