On the role of 5-wave resonances in the nonlinear dynamics of the Fermi–Pasta–Ulam–Tsingou lattice

We study the dynamics of the $(\alpha +\beta )$ Fermi–Pasta–Ulam–Tsingou lattice (FPUT lattice, for short) for an arbitrary number $N$ of interacting particles, in regimes of small enough nonlinearity so that a Birkhoff-Gustavson type of normal form can be found using tools from wave-turbulence theory. Specifically, we develop the theory that leads to the so-called Zakharov equation for 4-wave resonant interactions and its extension to 5-wave resonant interactions by Krasitskii, but we introduce an important new feature: by insisting that even the generic terms in these normal forms contain resonant interactions only, we obtain unique canonical transformations and normal forms that give rise to analytical approximations to the original FPUT lattice that possess a significant number of exact quadratic conservation laws, beyond the quadratic part of the Hamiltonian. We call the new equations “exact-resonance evolution equations” and examine their properties. First, because they consist of exact-resonant terms only, the evolution is very slow in the Heisenberg representation and we implement numerical methods with large time steps to obtain relevant information about the dynamics, such as Lyapunov exponents. Second, we introduce and execute successfully a number of tests, such as convergence of the normal form transformation and truncation error verification, to validate our exact-resonance evolution equations, by quantifying how well the solution to these equations serves to approximate the original FPUT lattice dynamics. Third, we use our theoretical underpinnings (based on the so-called resonant cluster matrix) in terms of the new quadratic invariants found, to propose and implement a method, based on finite-time Lyapunov exponent calculations applied to long-time dynamics, to quantify the level of nonlinearity at which the original FPUT lattice is well approximated by the exact-resonance evolution equations. Most of the numerical experiments presented are done in the case $N=9$, but the theory and the numerical methods presented are valid for arbitrary values of $N$. We conclude that, when $N$ is divisible by 3, at small enough nonlinearity the FPUT lattice’s dynamics and nontrivial hyperchaos are governed by 5-wave resonant interactions, in qualitative and quantitative agreement with the theory developed in this paper.