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Three-wave case

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

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

Equation of motion $ i\dot c_l =\frac{\partial {\cal
H}}{\partial \bar c_l}$ is mostly conveniently represented in the interaction representation,
$\displaystyle i \, \dot a_l = \epsilon \sum_{m,n=1}^\infty \left( V^l_{mn} a_{m...{V}^{m}_{ln} \bar a_{n}
a_{m} e^{-i\omega^m_{ln}t } \, \delta^m_{l+n}\right),$     (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.

Dr Yuri V Lvov 2007-01-23