Multi-wave resonances in the diatomic α-FPUT system

We examine a diatomic chain with a cubic anharmonic potential. Following the celebrated α-FPUT model, we refer to the present system as the diatomic α–FPUT model. By introducing new canonical variables, we diagonalize the harmonic part of the Hamiltonian, and, using these new variables, we analyze the nonlinear interactions between the acoustic and optical branches of the dispersion relation. In terms of the new canonical variables, dynamical equations exhibit quadratic nonlinearity, with the first resonant process being a three-wave interaction. We thoroughly investigate the dependence of these resonant interactions on the mass ratio and find that they occur when the mass ratio is less than 3. Note that in the standard α-FPUT chain, three-wave resonances do not occur. We find that these three-wave resonances in the α-diatomic chain are mostly isolated. Consequently, the resonant manifold consists of uncoupled triplets. Therefore, they do not contribute to thermalization, and we consider higher-order resonances. Over a longer time scale, four-wave resonances become significant. For this scenario, we apply the Wave Turbulence theory, deriving two coupled wave kinetic equations and the corresponding equilibrium solution.