Validity of Wave Turbulence theory for discrete systems

Determine whether Wave Turbulence theory provides a valid and predictive description for discrete Hamiltonian systems, including lattice models such as the (alpha+beta) Fermi–Pasta–Ulam–Tsingou chain with periodic boundary conditions, by establishing conditions under which the theory’s assumptions and predictions hold or by identifying counterexamples.

Background

The paper studies weakly nonlinear dynamics in the (alpha+beta) FPUT lattice and derives exact-resonance evolution equations via canonical transformations, connecting the analysis to concepts from Wave Turbulence theory (WT). The authors emphasize that resonant interactions (particularly 5-wave resonances) govern the system’s dynamics at small nonlinearity and provide numerical validations.

Within this context, the broader question of whether WT accurately applies to discrete systems remains unresolved. This issue is significant because many predictions of WT (e.g., kinetic descriptions and energy transfer laws) rely on assumptions that may not straightforwardly extend from continuous media to discrete lattices, motivating further theoretical and numerical scrutiny.

References

As underlined in a recent report [onorato2023wave], validation of the Wave Turbulence theory to discrete systems is still an open question.

— On the role of 5-wave resonances in the nonlinear dynamics of the Fermi-Pasta-Ulam-Tsingou lattice  (2502.21293 - Comito et al., 28 Feb 2025) in Introduction (Section 1)

The conjecture states that in the limit of large torus size $L$ and vanishing strength of the nonlinearity \begin{align} \alpha:=\lambda2L{-1},\label{def:alpha} \end{align} the effective dynamics of the Fourier-space mass density $\mathbf E |\widehat u(t, k)|2$ ($k \in \mathbb{Z}_L:=L{-1}\mathbb{Z}$) is well approximated by

— The Wave Kinetic Theory for Quasilinear MMT Equation  (2608.22710 - Lü, 24 Aug 2026) in Introduction, immediately preceding Section “Statement of the results”