vectors of the three interacting modes lying over a triangle, and the three frequencies satisfying the resonant condition (see, e.g., [@nazarenko_wave_2011]) $$\begin{aligned}
    \mathbf{k} &= \mathbf{p} + \mathbf{q} \\
    \omega(\mathbf{k}) &= \omega(\mathbf{p}) + \omega(\mathbf{q}) .\end{aligned}$$ However, as there are no three wave vectors $\mathbf{k},
\mathbf{p}, \mathbf{q}$ that satisfy these two conditions when the dispersion relation is given by Eq. , three wave interactions are forbidden. Thus, only four wave interactions are present (which do satisfy their corresponding condition).

After a transformation of the fields, it is possible to write an equation for the evolution of the two-point correlator of the transformed fields. This is the so-called kinetic equation, and has the following form $$\begin{gathered}
\begin{aligned}
    \frac{\partial N_0}{\partial t} = 4 \pi &\int \lvert T_{0,1,2,3} \rvert^2
    N_0 N_1 N_2 N_3 \\
    & \left( \frac{1}{N_0} + \frac{1}{N_1}- \frac{1}{N_2}-
    \frac{1}{N_3} \right) \\
        & \delta (\mathbf{k}_0 + \mathbf{k}_0 - \mathbf{k}_2 - \mathbf{k}_3) \\
        &\delta (\omega_0 + \omega_0 - \omega_2 - \omega_3) \textrm{d}
        \mathbf{k}_{123},
        \label{kinetic_eq}
\end{aligned}\end{gathered}$$ where $N_i = N(\mathbf{k}_i)$ is the wave action spectral density (i.e., the two-point correlator of the wave action, the latter being a quantity proportional to the surface height), the deltas express the fact that interactions are between four wave vectors 