Numerical verification of the random-phase-and-amplitude formalism of weak turbulence

The random-phase-and-amplitude-formalism (RPA) has significantly extended the scope of weak turbulence studies. Because the RPA does not assume any proximity to the Gaussianity in the wave number space, it can predict, for example, how the fluctuation of the complex amplitude of each wave mode grows through nonlinear interactions with other modes and how it approaches the Gaussianity. Thus, the RPA has a great potential capability, but its validity has been assessed neither numerically nor experimentally. We compare the theoretical predictions given by the RPA with the results of direct numerical simulations (DNS) for a three-wave Hamiltonian system, thereby assessing the validity of the RPA. The predictions of the RPA agree quite well with the results of DNS in all the aspects of the statistical characteristics of the mode amplitudes studied here.