On the properties of energy flux in wave turbulence

We study the properties of energy flux in wave turbulence via Majda-McLaughlin-Tabak (MMT) equation with a quadratic dispersion relation. One our purposes is to resolve inter-scale $P$ stationary state elucidate its distribution and scaling spectral level. More importantly, we perform quartet-level decomposition $P=\sum_\Omega P_\Omega$, each component $P_\Omega$ representing contribution from quartet interactions frequency mismatch $\Omega$, order explain as well wave-turbulence closure model. Our results show that time series closely follows Gaussian distribution, standard deviation several times mean value $\overline{P}$. This large shown mainly result fluctuation (in time) quasi-resonances, i.e., $P_{\Omega\neq 0}$. The level $\overline{P}$ exhibits $\overline{P}^{1/3}$ $\overline{P}^{1/2}$ at high low nonlinearity, consistent kinetic dynamic scalings respectively. different laws two regimes are explained through dominance quasi-resonances ($P_{\Omega\neq 0}$) exact resonances ($P_{\Omega= former latter regimes. Finally, investigate model, which connects fourth-order correlators products pair broadening function $f(\Omega)$, sometimes argued be $sinc$ theory. numerical data behavior $f(\Omega)$ can only observed upon averaging over number quartets, but showing $f\sim 1/\Omega^\beta$ dependence $\beta$ taking values between $1.3$ $1.6$.