System of kinetic equations for strongly nonlinear four-wave dynamics

Abstract We propose the system of kinetic equations as the new analytic tool for achieving a Gaussian approximation to the probability distribution propagated by the four-wave dynamics with strong nonlinearity. The framework forms a comprehensive generalization of the classical kinetic equation developed in the context of nonlinear waves in the weak interaction limit as well as the effective kinetic equation aimed for the wave dynamics in the regime of moderate nonlinearity. Unlike the previous formulations that exclusively determine the variance of the single wave-profile, the system of kinetic equations governs the evolution of the covariance matrix of the duet constructed via augmenting the macrovariable in significant resonance with the target variable. As it was done with the existing kinetic equations for the analysis of the model dynamics with weak and moderate nonlinearity, the system of kinetic equations will prove useful in addressing important issues concerning the statistical features exhibited by the strongly nonlinear four-wave dynamics.