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

In this paper we construct a higher order expansion of the manifold of quasi unidirectional waves in the Fermi-Pasta-Ulam-Tsingou (FPUT) 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 FPUT potential. On the other hand, we find that only if the parameters of the FPUT 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 FPUT chain (the FPUT 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 be responsible for the eventual thermalization of the system.

Automated summary

This paper investigates the higher order expansion and dynamics approximation of quasi unidirectional waves in the FPUT chain. It reveals that the dynamics to second order is governed by a combination of the first two nontrivial equations in the KdV hierarchy, while to third order a specific parameter condition needs to be satisfied. The results suggest why the FPUT paradox persists for longer than expected and how a breakdown of integrability may lead to system thermalization.