Evolution equations for the PDF's, moments and spectrum.

In [5] the following equation for for the $N$-mode PDF was obtained for the four-wave systems,

$$\dot {\cal P} = { \pi {\epsilon}^2 } \int \vert W^{jl}_{nm}\vert^... ...a s} \right]_4 {\cal P} \right) \, d{\bf k}_j d{\bf k}_l d{\bf k}_m d{\bf k}_n,$$ (16)

where

$$\left[{\delta \over \delta s} \right]_4 = {\delta \over \delta s_... ...delta \over \delta s_l} - {\delta \over \delta s_m} -{\delta \over \delta s_n}.$$ (17)

Here $N \to \infty$ limit has already been taken and ${\delta \over \delta s_j}$ means the variational derivative. Using this equation, one can prove that RPA property holds over the nonlinear time, i.e. the $N$-mode PDF remains of the product factorized form with accuracy sufficient for the WT closures to work [5]. Using RPA, we get for the one-point marginals [5],

$${\partial P_a \over \partial t}+ {\partial F \over \partial s_j} =0,$$ (18)

with $F$ is a probability flux in the s-space,

$$F=-s_j (\gamma P_a +\eta_j {\delta P_a \over \delta s_j}),$$ (19)

where

$$\eta_j = 4 \pi \epsilon^2 \int \vert W^{jl}_{nm}\vert^2 \delta^{jl}_{nm} \... ...(\omega^{jl}_{nm}) n_l n_m n_n \, d { {\bf k}_l} d { {\bf k}_m} d { {\bf k}_n,}$$ (20)

$$\gamma_j = 4 \pi \epsilon^2 \int \vert W^{jl}_{nm}\vert^2 \delta^{jl}_{nm} \... ...n_l (n_m + n_n) - n_m n_n\Big] \, d { {\bf k}_l} d { {\bf k}_m} d { {\bf k}_n.}$$ (21)

Here we introduced the wave-action spectrum,

$$n_j = \langle A_j^2 \rangle.$$ (22)

From (14) we get the following equation for the moments $M^{(p)}_j = \langle A_j^{2p} \rangle$:

$$\dot M^{(p)}_j = -p \gamma_j M^{(p)}_j + p^2 \eta_j M^{(p-1)}_j.$$ (23)

which, for $p=1$ gives the standard wave kinetic equation (WKE),

$$\dot n_j = - \gamma_j n_j + \eta_j .$$ (24)