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