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This paper is concerned with the reduced-order modeling of the strongly nonlinear wave turbulence system. The motivation for such an attempt comes from the utility of the probabilistic coarse-grained model in facilitating the theoretical and numerical analysis of the true dynamical system model. One typical practice of simplifying the complex physical model is, in the spirit of Brownian motion, to replace the nonlinear interactions by white noise forcing and linear dissipation. For the case of slowly varying longwave, the resulting Mark
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