Perturbation Theory for Fermi-pasta-ulam Chains: a Case Study in Kam Theory

Fermi-Pasta-Ulam chains have been an active research area since the famous simulations by Fermi, Pasta, and Ulam in the 1950s, which exhibited unexpected recurrence phenomena. Around the same time, new perturbation theory results began to be developed, which are now known as the KAM and Nekhoroshev theorems. These theorems provide stability results for slightly perturbed integrable systems, under appropriate conditions on the unperturbed Hamiltonian. It was soon conjectured that these theorems might provide an explanantion for the recurrence phenomena in FPU chains. This book investigates the normal form and perturbation theory of FPU chains and rigorously shows that in most cases of parameter values and boundary conditions, the KAM and Nekhoroshev theorems are in fact applicable to FPU chains. Moreover, it is shown that all types of FPU chains can be approximated up to fourth order by a completely integrable system. Finally, the foliation of the phase space into level sets of the integrals of the integrable approximation is thoroughly analyzed, thereby in particular showing that this integrable system exhibits hyperbolic dynamics.