Periodic orbits, localization in normal mode space, and the Fermi-Pasta-Ulam problem

Abstract

The Fermi-Pasta-Ulam problem was one of the first computational experiments. It has stirred the physics community since, and resisted a simple solution for half a century. The combination of straightforward simulations, efficient computational schemes for finding periodic orbits, and analytical estimates allows us to achieve significant progress. Recent results on $q$-breathers, which are time-periodic solutions that are localized in the space of normal modes of a lattice and maximize the energy at a certain mode number, are discussed, together with their relation to the Fermi-Pasta-Ulam problem. The localization properties of a $q$-breather are characterized by intensive parameters, that is, energy densities and wave numbers. By using scaling arguments, $q$-breather solutions are constructed in systems of arbitrarily large size. Frequency resonances in certain regions of wave number space lead to the complete delocalization of $q$-breathers. The relation of these features to the Fermi-Pasta-Ulam problem are discussed.

19 pages, 9 figures, to appear in Am. J. Phys

References in corpus (8)

• Discrete Breathers
• q-Breathers and the Fermi-Pasta-Ulam Problem
• q-Breathers in Fermi-Pasta-Ulam chains: existence, localization and stability
• Discrete breathers in Fermi-Pasta-Ulam lattices
• Low energy chaos in the Fermi-Pasta-Ulam problem
• Tail resonances of FPU q-breathers and their impact on the pathway to equipartition
• q-breathers in finite two- and three-dimensional nonlinear acoustic lattices
• Scaling properties of $q$-breathers in nonlinear acoustic lattices

Cited by in corpus (7)

• Dynamical glass in weakly non-integrable Klein-Gordon chains
• Behavior and Breakdown of Higher-Order Fermi-Pasta-Ulam-Tsingou Recurrences
• Boundary effects on quantum q-breathers in a Bose-Hubbard chain
• q-breathers in Discrete Nonlinear Schroedinger lattices
• The $\beta$ Fermi-Pasta-Ulam-Tsingou Recurrence Problem
• Periodic Orbits in Fermi-Pasta-Ulam-Tsingou Systems
• Adiabatic Gauge Potential as a Tool for Detecting Chaos in Classical Systems