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

TL;DR

This paper develops a higher order asymptotic expansion for quasi unidirectional waves in the FPU chain, linking the dynamics to the KdV hierarchy and explaining the FPU paradox and eventual thermalization.

Contribution

It constructs a second and third order approximation of the wave manifold, revealing conditions under which the FPU dynamics align with the KdV hierarchy, especially for the Toda chain.

Findings

Second order dynamics governed by the first two KdV equations.

Third order dynamics aligned with the first three KdV equations under specific conditions.

Insights into the persistence of near-integrable behavior and thermalization mechanisms.

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.