Three-wave case

When ${\cal H} = {\cal H}_2 + {\cal H}_3$ we have Hamiltonian in a form

$${\cal H} = \sum_{n=1}^\infty \omega_n\vert c_n\vert^2 +\epsilon \sum_{l,m,n=1}^\infty \left( V^l_{mn} \bar c_{l} c_m c_n\delta^l_{m+n}+c.c.\right).$$

Equation of motion $i\dot c_l =\frac{\partial {\cal H}}{\partial \bar c_l}$ is mostly conveniently represented in the interaction representation,

$$i \, \dot a_l = \epsilon \sum_{m,n=1}^\infty \left( V^l_{mn} a_{m... ...ar{V}^{m}_{ln} \bar a_{n} a_{m} e^{-i\omega^m_{ln}t } \, \delta^m_{l+n}\right),$$ (10)

where $a_j =c_j e^{i \omega_j t}$ is the complex wave amplitude in the interaction representation, $l,m,n \in {\cal Z}^d$ are the indices numbering the wavevectors, e.g. $k_m = 2 \pi m/L$, $L$ is the box side length, $\omega^l_{mn}\equiv\omega_{k_l}-\omega_{k_m}-\omega_{k_m}$ and $\omega_l=\omega_{k_l}$ is the wave linear dispersion relation. Here, $V^l_{mn} \sim 1$ is an interaction coefficient and $\epsilon$ is introduced as a formal small nonlinearity parameter.