Korteweg-de Vries and Fermi-Pasta-Ulam-Tsingou: asymptotic integrability of quasi unidirectional waves

Abstract

In this paper we construct a higher order expansion of the manifold of quasi unidirectional waves in the Fermi-Pasta-Ulam (FPU) chain. We also approximate the dynamics on this manifold. As perturbation parameter we use $h^2=1/n^2$, where $n$ is the number of particles of the chain. It is well known that the dynamics of quasi unidirectional waves is described to first order by the Korteweg-de Vries (KdV) equation. Here we show that the dynamics to second order is governed by a combination of the first two nontrivial equations in the KdV hierarchy -- for any choice of parameters in the FPU potential. On the other hand, we find that only if the parameters of the FPU potential satisfy a condition, then a combination of the first three nontrivial equations in the KdV hierarchy determines the dynamics of quasi unidirectional waves to third order. The required condition is satisfied by the Toda chain. Our results suggest why the close-to-integrable behavior of the FPU chain (the FPU paradox) persists on a time scale longer than explained by the KdV approximation, and also how a breakdown of integrability (detachment from the KdV hierarchy) may responsible for the eventual thermalization of the system.

arXiv comment20 pages, 3 tables; accepted for publication in Phys. Rev. A: Math and Theor

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Cited by in corpus (6)

• Wave Turbulence and thermalization in one-dimensional chains
• Burgers turbulence in the Fermi-Pasta-Ulam-Tsingou chain
• Justification of the KP-II approximation in dynamics of two-dimensional FPU systems
• On the role of the Integrable Toda model in one-dimensional molecular dynamics
• Energy cascade and Burgers turbulence in the Fermi-Pasta-Ulam-Tsingou chain
• Hamiltonian field theory close to the wave equation: from Fermi-Pasta-Ulam to water waves