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Weak nonlinearity expansion: four-wave case

Substituting (32) in (15) we get in the zeroth order $ a_l^{(0)}(T)=a_l(0)$, and the first iteration of (15) gives

$\displaystyle a_l^{(1)}(T) = - i \sum_{\alpha\mu\nu} W^{l\alpha}_{\mu\nu} \bar
... a_\nu \delta^{l\alpha}_{\mu\nu} \Delta^{l
\alpha}_{\mu\nu} + i \Omega_l a_l T.$     (31)

Iterating one more time we get
$\displaystyle a_l^{(2)} (T)$ $\displaystyle =$ $\displaystyle \sum_{\alpha\mu\nu v u}\left(
W^{\mu\nu}_{\alpha u} W^{lu}_{v\bet...
E(\tilde\omega^{l\mu\nu}_{\alpha v \beta}, \tilde\omega^{lu}_{v\beta})
    $\displaystyle \left.
2 W^{\alpha v}_{\mu\nu} W^{l u}_{v \beta}
...\mu\nu\beta}, \tilde \omega^{lu}_{v\beta})\right)
-\Omega_l^2 a_l \frac{T^2}{2}$  
    $\displaystyle +
\int \limits_0^T
\tau e^{i\Omega^{l \alpha}_{\mu\nu } \tau} d \tau
\right).$ (32)

Dr Yuri V Lvov 2007-01-23