Resonant interactions in the $α$-FPUT lattice with site-dependent coefficients

Abstract

The wave turbulence framework has proven to be an effective tool for analyzing certain features of nonlinear energy transfer in one-dimensional nonlinear chains. In this work, we extend this approach to the $α$-FPUT problem when the spring stiffness $χ$ and the nonlinear coefficient $α$ are site-dependent. Although three-wave interactions are non-resonant for constant coefficients, their spatial modulation gives rise to a non-trivial resonant manifold. In this framework, we derive a new kinetic equation that suggests the possibility of substantially faster thermalization with respect to the constant coefficient case. The new kinetic equation includes also an extra term that can be associated to the Bragg-scattering mechanism, which promotes the isotropization of the wave-action spectral density function.